HandbookofLinearPartialDifferential EquationsforEngineersIntro
A survey on the stability of fractional differential equations
Eur.Phys.J.Special Topics193,27–47(2011) c EDP Sciences,Springer-Verlag2011DOI:10.1140/epjst/e2011-01379-1T HE E UROPEANP HYSICAL J OURNAL S PECIAL T OPICSReviewA survey on the stability of fractionaldifferential equationsDedicated to Prof.Y.S.Chen on the Occasion of his80th BirthdayC.P.Li1,a and F.R.Zhang1,2,b1Department of Mathematics,Shanghai University,Shanghai200444,PR China2School of Mathematics and Computational Sciences,China University of Petroleum (East China),Dongying257061,PR ChinaReceived01December2010/Received infinal form27January2011Published online4April2011Abstract.Recently,fractional calculus has attracted much attentionsince it plays an important role in manyfields of science and engineer-ing.Especially,the study on stability of fractional differential equationsappears to be very important.In this paper,a brief overview on therecent stability results of fractional differential equations and the ana-lytical methods used are provided.These equations include linear frac-tional differential equations,nonlinear fractional differential equations,fractional differential equations with time-delay.Some conclusions forstability are similar to that of classical integer-order differential equa-tions.However,not all of the stability conditions are parallel to thecorresponding classical integer-order differential equations because ofnon-locality and weak singularities of fractional calculus.Some resultsand remarks are also included.1IntroductionFractional calculus has been300years old history,the development of fractional calculus theory is mainly focused on the pure mathematicalfield.The earliest more or less systematic studies seem to have been made in the19th century by Liouville, Riemann,Leibniz,etc.[1,2].In the last two decades,fractional differential equations (FDEs)have been used to model various stable physical phenomena with anomalous decay,say that are not of exponential type[3].We can refer to[4]for the recent history of fractional calculus.As we all know,many mathematical models of real problems arising in variousfields of science and engineering are either linear systems or nonlinear systems.Nevertheless,most differential systems used to describe physical phenomena are integer-order systems.With the development of fractional calculus, it has been found that the behavior of many systems can be described by using the fractional differential systems[5–9].It is worth mentioning that many physical phenomena having memory and genetic characteristics can be described by using the fractional differential systems.In fact,real world processes generally or most likelya e-mail:lcp@b e-mail:zhangfengrongsong@28The European Physical Journal Special Topicsare fractional order systems[10,11].That is to say,a lot of physical systems showfractional dynamical behavior because of special materials and chemical properties.Recently,the theory of FDEs has been studied and some basic results areobtained including stability theory.The question of stability is of main interest inphysical and biological systems,such as the fractional Duffing oscillator[12,13],frac-tional predator-prey and rabies models[14],etc.It is known that the chaotic sequenceis non-periodic and pseudo-random.Similar to integer-order differential equations,the stability theory of FDEs is also widely applied to chaos and chaos synchroniza-tion[15–19]due to its potential applications in control processing and secure com-munication.However,a few stability results rely on a restrictive modeling of FDEs:the basic hypothesis deals with commensurability,i.e.the fractional derivative ordershave to be an integer multiple of minimal fractional order.Owing to this hypothesis,some stability results are available,based on Matignon’s theorem[20].On the otherhand,the case of incommensurate fractional order can be referred to[21,22].In this paper,we present and discuss some basic results on stability of FDEsincluding linear FDEs,nonlinear FDEs,FDEs with time-delay and the analyticalmethods used.The analysis on stability of FDEs is more complex than that of clas-sical differential equations,since fractional derivatives are nonlocal and have weaklysingular kernels.The earliest study on stability of FDEs started in[23],the authorstudied the case of linear FDEs with Caputo derivative and the same fractional orderα,where0<α≤1.The stability problem comes down to the eigenvalue prob-lem of system matrix.Corresponding to the stability result in[23],Qian et al.[22]recently studied the case of linear FDEs with Riemann-Liouville derivative and thesame fractional orderα,where0<α<1.Then,in[24–26]authors derived thesame conclusion as[23]for the case1<α<2.[21]studied the linear system withmulti-order Caputo derivative and derived a sufficient condition on Lyapunov globalasymptotical stability.In the last decades,many researchers have more interests inthe stability of linear systems and various methods have emerged in succession.Forexample,there are frequency domain methods[27–32],Linear Matrix Inequalities(LMI)methods[25,26,33,34]and conversion methods[10],[35–37].By contrast,the development of stability of nonlinear FDEs is a bit slow.Thestructural stability was studied in[38],where the system with Riemann-Liouvillederivative was considered by using Taylor polynomial.In[39]authors investigatedthe system of nonautonomous FDEs involving Caputo derivative and derived theresult on continuous dependence of solution on initial conditions.The stability inthe sense of Lyapunov has also been studied[40]by using Gronwall’s lemma andSchwartz inequality.We can also refer to[14],[41–47],where the linearization method was considered.But the rigorous theoretical derivation has not been founded.Some researchers weakened the criterion of stability,such as[48]where the L p-stability properties of nonlinear FDEs were investigated.In[49,50],the Mittag-Leffler stability and the fractional Lyapunov’s second method were proposed.At last Deng[51]derived a sufficient stability condition of nonlinear FDEs.In engineering,a kind of system is also very important,namely,the time-delaysystem.In[52–55]authors considered thefinite-time stability of FDEs with time-delay on the basis of real problems.They studied the autonomous and nonau-tonomous fractional differential systems.For the time-delay system stability,we canrefer to[21],[56–58].In[59],Petr´aˇs gave a survey on the methods for stability investigation of a certainclass of fractional differential systems with rational orders and Caputo derivative in aviewpoint of control.To complement the literature,we will review the stability resultsof FDEs with Riemann-Liouville derivative or Caputo derivative and the analyticalmethods.Furthermore,the stability of FDEs with time-delay will also be covered.Thepaper is outlined as follows.In Sec.2,wefirst recall some definitions and propositions.Perspectives on Fractional Dynamics and Control29In Sec.3,some stability conditions of linear FDEs are presented,meanwhile,we give two results.In Sec.4,some stability conditions of nonlinear FDEs are described. Sec.5deals with the stability of FDEs with time-delay.Conclusions and comments are included in Sec.6.2PreliminariesLet us denote by R the set of real numbers,R+the set of positive real numbers and by Z+the set of positive integer numbers,denote by C the set of complex numbers. We denote the real part of complex numberαby Re(α).Two kinds of fractional derivatives,i.e.,the Riemann-Liouville derivative and Caputo derivative,have been often used in fractional differential systems.We briefly introduce these two definitions of fractional derivatives which will be frequently used throughout this paper.Firstly,we introduce the definition of fractional integral[60]. Definition1.The fractional integral(or,the Riemann-Liouville integral)D−αt0,t with fractional orderα∈R+of function x(t)is defined below:D−αt0,t x(t)=1Γ(α)tt0(t−τ)α−1x(τ)dτ,where t=t0is the initial time,Γ(·)is the Euler’s gamma function.Definition2.The Riemann-Liouville derivative with orderαof function x(t)is defined below:RL Dαt0,t x(t)=1Γ(m−α)d mdt mtt0(t−τ)m−α−1x(τ)dτ=d mdt mD−(m−α)t0,t,where m−1≤α<m∈Z+.Definition3.The Caputo derivative with orderαof function x(t)is defined below:C Dαt0,t x(t)=1Γ(m−α)tt0(t−τ)m−α−1x(m)(τ)dτ=D−(m−α)t0,td mdt mx(t),where m−1<α<m∈Z+.There are also two functions that play an important role in the study on stability of FDEs.Definition4.The Mittag-Leffler function is defined byEα(z)=∞k=0z kΓ(kα+1),where Re(α)>0,z∈C.The two-parameter Mittag-Leffler function is defined byEα,β(z)=∞k=0z kΓ(kα+β),where Re(α)>0andβ∈C,z∈C.One can see Eα(z)=Eα,1(z)from the above equations.30The European Physical Journal Special TopicsDefinition 5([61]).The α-exponential function is defined as follows:e λz α=zα−1E α,α(λz α),where z ∈C \0,Re (α)>0,and λ∈C .E α,α(·)is the two-parameter Mittag-Leffler function.Proposition 1.If 0<α<2,βis an arbitrary complex number,then for an arbitrary integer p ≥1the following expansions hold:E α,β(z )=1αz (1−β)/αexp(z 1/α)−p k =11Γ(β−αk )1z k +O 1|z |p +1 ,with |z |→∞,|arg(z )|≤απ2,andE α,β(z )=−pk =11Γ(β−αk )1z k +O 1|z |p +1 ,with |z |→∞,|arg(z )|>απ2.Proof.These results were proved in [60].The following relations for the α-exponential function follow from Definition 5and Proposition 1:Proposition 2[61].If 0<α<2,z ∈C ,then the following asymptotic equivalents for e λz αas |z |reaches infinity are valid:•for |arg(λ)|≤απ2,e λz α∼λ(1−α)/ααexp(λ1/α)z ,•for |arg(λ)|>απ2,e λz α∼−λ−2Γ(−α)1z α+1.In this paper,we will consider the following general type of FDEs involving Caputo derivative or Riemann-Liouville derivativeD ¯αt 0,t x (t )=f (t,x ),(1)with suitable initial values x k =[x k 1,x k 2,...,x kn ]T ∈R n (k =0,1,...,m −1),where x (t )=[x 1(t ),x 2(t ),...,x n (t )]T ∈R n ,¯α=[α1,α2,...,αn ]T ,m −1<αi <m ∈Z +(i =1,2,...,n ),D ¯αt 0,t x (t )=[D α1t 0,t x 1(t ),...,D αn t 0,t x n (t )]T ,f :[t 0,∞)×R n →R n ,D αi t 0,t denotes either C D αi t 0,t or RL D αi t 0,t .In particular,if α1=α2=···=αn =α,then Eq.(1)can be written asD αt 0,t x (t )=f (t,x ).(2)We call Eq.(2)the same order fractional differential system,otherwise,call Eq.(1)multi-order fractional differential system.The following definitions are associated with the stability problem in the paper.Definition 6[49].The constant vector x eq is an equilibrium point of fractional dif-ferential system (1),if and only if f (t,x eq )=D ¯αt 0,t x (t )|x (t )=x eq for all t >t 0.Without loss of generality,let the equilibrium point be x eq =0,we introduce the following definition.Perspectives on Fractional Dynamics and Control31 Definition7.The zero solution of fractional differential system(1)is said to be stable if,for any initial values x k=[x k1,x k2,...,x kn]T∈R n(k=0,1,...,m−1), there exists >0such that any solution x(t)of(1)satisfies x(t) <εfor all t>t0. The zero solution is said to be asymptotically stable if,in addition to being stable, x(t) →0as t→+∞.Definition8.Let1≤p≤∞andΩ⊂[t0,∞),the same order system(2)with Riemann-Liouville derivative is L p(Ω)−stable if the solution x(t)=[x1(t),..., x n(t)]T defined by equationx(t)=x0Γ(α)(t−t0)α−1+1Γ(α)tt0(t−τ)α−1f(τ,x(τ))dτbelongs to L p(Ω).Where0<α<1,x0∈R n is the initial value,f∈C([t0,∞)×R n,R n+)is a continuous positive function.Definition9(Mittag-Leffler Stability)[49].Let B⊂R n be a domain containing the origin.The zero solution of the same order system(2)is said to be Mittag-Leffler stable ifx(t) ≤{m(x(t0))Eα(−λ(t−t0)α)}b,where t0is the initial time,α∈(0,1),λ>0,b>0,m(0)=0,m(x)≥0,and m(x)is locally Lipschitz on x∈B⊂R n with Lipschitz constant L.Definition10.A continuous functionα:[0,∞)→[0,∞)is said to belong to class-K if it is strictly increasing andα(0)=0.Definition11(Generalized Mittag-Leffler Stability).Let B⊂R n be a domain containing the origin.The zero solution of the same order system(2)is said to be generalized Mittag-Leffler stable ifx(t) ≤{m(x(t0))(t−t0)−γEα,1−γ(−λ(t−t0)α)}b,where t0is the initial time,α∈(0,1),−α<γ<1−α,λ≥0,b>0,m(0)=0,m(x)≥0,and m(x)is locally Lipschitz on x∈B⊂R n with Lipschitz constant L.We will also need the following definitions to analyze the case of FDEs with time-delay in Sec.5.First,we introduce the same order fractional differential system with multiple time delays represented by the following differential equation:⎧⎪⎨⎪⎩Dαt0,tx(t)=A0x(t)+mi=1A i x(t−τi)+B0u(t),0≤τ1<τ2<τ3<···<τi<···<τm=Δ(3)with the initial condition x(t0+t)=ψ(t)∈C[−Δ,0].Where0<α<1,Dαi t0,t denoteseither C Dαi t0,t or RL Dαi t0,t.x(t)∈R n is a state vector,u(t)∈R l is a control vector,A i(i=0,1,...,m),B0are constant system matrices of appropriate dimensions,and τi>0(i=1,2,...,m)are pure time delays.Definition12.The same order system(3)(u(t)≡0,∀t)satisfying initial condition x(t0+t)=ψ(t),−Δ≤t≤0,isfinite-time stable w.r.t.{t0,J,δ,ε,δ},δ<εif and only if:ψ c<δ,∀t∈J ,J =[− ,0]∈R32The European Physical Journal Special Topicsimplies:x (t ) <ε,∀t ∈J,where δis a positive real number and ε∈R +,δ<ε, ψ c =max −Δ≤t ≤0 ψ(t ) ,time interval J =[t 0,t 0+T ]⊂R ,quantity T may be either a positive real number or a symbol +∞.Definition 13.System given by (3)satisfying initial condition x (t 0+t )=ψ(t ),−Δ≤t ≤0,is finite-time stable w.r.t.{t 0,J,δ,ε,Δ,αu },δ<εif and only if:ψ c <δ,∀t ∈J ,J =[− ,0]∈Randu (t ) <αu ,∀t ∈Jimplies:x (t ) <ε,∀t ∈J,where δis a positive real number and ε∈R +,δ<ε,αu >0, ψ c =max −Δ≤t ≤0 ψ(t ) ,time interval J =[t 0,t 0+T ]⊂R ,quantity T may be either a positive real number or a symbol +∞.3Stability of linear fractional differential equationsIn this section,we consider the following linear system of FDEsD ¯αt 0,t x (t )=Ax (t ),(4)where x (t )=[x 1(t ),x 2(t ),...,x n (t )]T ∈R n ,matrix A ∈R n ×n ,¯α=[α1,α2,...,αn ]T ,D ¯αt 0,t x (t )=[D α1t 0,t x 1(t ),D α2t 0,t x 2(t ),...,D αn t 0,t x n (t )]T and D αi t 0,t is the Caputo derivativeor Riemann-Liouville derivative of order αi ,where 0<αi ≤2,for i =1,2,···,n .In particular,if α1=α2=···=αn =α,then fractional differential system (4)can be written as the following same order linear systemD αt 0,t x (t )=Ax (t ).(5)3.1The fundamental theoremsFor Eq.(5)and 0<α≤1,Matignon firstly gave a well-known stability result by an algebraic approach combined with the use of asymptotic results,where the necessary and sufficient conditions have been derived,the specific result is as follows [23].Theorem 1.The autonomous same order system (5)with Caputo derivative and initial value x 0=x (0),where 0<α≤1,is•asymptotically stable if and only if |arg(spec (A ))|>απ2.In this case the compo-nents of the state decay towards 0like t −α.•stable if and only if either it is asymptotically stable,or those critical eigenval-ues which satisfy |arg(spec (A ))|=απ2have geometric multiplicity one,spec (A )denotes the eigenvalues of matrix A .Perspectives on Fractional Dynamics and Control 33As we can see,in case α=1,the above stability result shows that roots of the equa-tion det(diag(λ,λ,...,λ)−A)=0lie outside the closed angular sector |arg(λ)|≤απ2,thus generalizing the well-known result for the integer case α=1.For the asymptot-ical stability in Theorem 1,the components of the state are anomalous decay,thus fractional systems have ‘memory’feature and its asymptotical stability is also called t −αstability.Exponential stability cannot be used to characterize the asymptotic stability of fractional differential systems.The case of zero eigenvalues of system matrix A is not included in Theorem 1,since the argument of zero in complex plane can be arbitrary.In view of this situation,Qian et al.recently studied the case of autonomous same order system (5)with Riemann-Liouville derivative by using the asymptotic expansions of Mittag-Leffler function,where 0<α<1.And they stated the zero eigenvalues case of Theorem 1,the corresponding conclusion is as follows [22].Theorem 2.The autonomous same order system (5)with Riemann-Liouville deriv-ative and initial value x 0=RL D α−1t 0,tx (t )|t =t 0,where 0<α<1and t 0=0,is •asymptotically stable if and only if all the non-zero eigenvalues of A satisfy |arg(spec (A ))|>απ2,or A has k -multiple zero eigenvalues corresponding to a Jordan block diag(J 1,J 2,...,J i ),where J l is a Jordan canonical form with order n l , i l =1n l =k ,and n l α<1,1≤l ≤i .•stable if and only if either it is asymptotically stable,or those critical eigenval-ues which satisfy |arg(spec (A ))|=απ2have the same algebraic and geometric multiplicities,or A has k -multiple zero eigenvalues corresponding to a Jordan block matrix diag(J 1,J 2,...,J i ),where J l is a Jordan canonical form with order n l , i l =1n l =k ,and n l α≤1,1≤l ≤i .The proof of Theorem 2is contained in [22].For the autonomous same order system(5)with Riemann-Liouville derivative and 0<α<1,if all the eigenvalues of system matrix A satisfy |arg(spec (A ))|>απ2,then the components of the state decay towards0like t −α−1which is different from the Caputo derivative case.And similar to the proof in [22],for the autonomous same order system (5)with Caputo derivative and 0<α≤1,if all the non-zero eigenvalues of A satisfy |arg(spec (A ))|≥απ2and the critical eigenvalues satisfying |arg(spec (A ))|=απ2have the same algebraic and geometric multiplicities,and the zero eigenvalue of A has the same algebraic and geometric multiplicities,then the zero solution of this system is stable from the representation of the solution.In addition,the zero solution of this system is never asymptotically stable as long as A has zero eigenvalue.The above two theorems dealt with the same order linear fractional differential system.For the multi-order linear fractional differential system (4),Deng et al.firstly studied the case that α,i s are rational numbers between 0and 1,for i =1,2,...,n ,where the following result [21,62]was introduced.Theorem 3.Suppose that α,i s are rational numbers between 0and 1,for i =1,2,...,n .Let M be the lowest common multiple (LCM)of the denominators u i of α,i s ,where αi =v i u i ,(u i ,v i )=1,u i ,v i ∈Z +,i =1,2,...,n ,and set γ=1M .Then the zero solution of system (4)with Caputo derivative and initial value x 0=x (0)is•asymptotically stable if and only if any zero solution of the polynomialdet(diag(λMα1,λMα2,...,λMαn )−A )satisfies |arg(λ)|>γπ/2,the components of the state variable (x 1(t ),x 2(t ),...,x n (t ))T ∈R n decay towards 0like t −α1,t −α2,...,t −αn ,respectively.34The European Physical Journal Special Topics•stable if and only if either it is asymptotically stable or those critical zero solutions λof the above polynomial satisfy |arg(λ)|=γπ/2have geometric multiplicity one.From Theorem 3,suppose α1=α2=···=αn =αare rational numbers between 0and 1,then the result of Theorem 3coincides with the one in Theorem 1.So,Theorem 3is an extension of Theorem 1in respect of rational orders.In [21],the authors used Laplace transform [63]and the final-value theorem to prove Theorem 3.Similarly,Odibat [64]also analyzed the stability of system (4)with 0<α=α1=α2=···=αn ≤1.In [64],the Mittag-Leffler functions and their integer-order derivatives which are analytic functions were used to obtain analytical solutions of the initial value problem (4),then the sufficient stability condition was derived by using the final-value theorem.This result [64]is consistent with the one in Theorem 1and Theorem 3.Matignon’s theorem is in fact the starting point of several results on the stability analysis.For example,Ralti et al.extended Theorem 1to the case 1<α<2[24].Remark 1.Similar to Theorem 2,if we assume that the conditions of Theorem 3hold except replacing C D ¯α0,t and the initial value x 0=x (0)by RL D ¯α0,t and the initial valuex 0=RL D ¯α0,t x (t )|t =0respectively,then the stability result which is an extension ofTheorem 2in respect of rational orders is still available.All the above conclusions are about the case of commensurate fractional order,in addition,some literatures such as [21,22]also involved the case of incommensurate fractional order.If α1,α2,...,αn are not rational numbers but real numbers between 0and 1in system (4),then we have the following result,which was introduced in [21,22]by using the final-value theorem of Laplace transform.Theorem 4.If all the roots of the characteristic equation det(diag(s α1,s α2,...,s αn )−A )=0have negative real parts,then the zero solution of system (4)is asymp-totically stable,where αi is real and lies in (0,1).From the above theorems and Proposition 2,we have the following result.Theorem 5.The autonomous same order system (5)with initial value x 0=x (0)and Riemann-Liouville derivative is asymptotically stable if and only if |arg(spec (A ))|>απ2,where n =2and 0<α≤1.Proof.From [65],this system has unique solution which can be expressed by a gen-eralization of the matrix α-exponential function as follows:x (t )=e At αC =t α−1∞k =0A kt αk Γ[(k +1)α]C.For A ,there exists an invertible matrix P from the algebra such that P −1AP =J,where J is the Jordan canonical form of the matrix A .Here are two cases to discuss:•for J = λ100λ2,x (t )=t α−1∞k =0P J k P −1t αk Γ[(k +1)α]C =P e λ1t α0e λ2t α P −1C.Perspectives on Fractional Dynamics and Control 35•for J =λ10λ,x (t )=t α−1∞k =0P J k P−1t αk Γ[(k +1)α]C =P e λt α∂∂λe λt α0e λt α P −1C.Thus,we have lim t →+∞ x (t ) =0if and only if |arg(spec (A ))|>απ2from Propo-sition 2,and the proof is complete.We can also extend Theorem 5to the case 1<α<2.Theorem 6.The autonomous same order system (5)with Riemann-Liouville deriv-ative and initial values x k =RL D α−k −10,t x (t )|t =0(k =0,1),is asymptotically stable if and only if |arg(spec (A ))|>απ2,where n =2and 1<α<2.Proof.Similar to the above proof,there exists an invertible matrix P such that P −1AP =J ,where J is the Jordan canonical form of matrix A .Let us denote y =P −1x and substitute into Eq.(5),the system can be described asRL D α0,t y (t )=Jy (t ),(6)with initial values RL D α−10,t y (t )|t =0=y 0=P −1x 0and RL D α−20,t y (t )|t =0=y 1=P −1x 1.Here are also two cases to discuss:•for J = λ100λ2,applying Laplace transform to Eq.(6),we have y i (t )=y 0i ·t α−1E α,α(λi t α)+y 1i ·t α−2E α,α−1(λi t α),where y 0i is the i -th component of y 0,y 1i is the i -th component of y 1,i =1,2.According to Proposition 1,y i (t )∼1αy 0i λ(1−α)/αi +y 1i λ(2−α)/αi exp λ1αi t ,with t →∞,|arg(λi )|≤απ2,andy i (t )∼−y 0i λ−2i Γ(−α)t −α−1−y 1i λ−2i Γ(−1−α)t −α−2,with t →∞and |arg(λi )|>απ2.•for J = λ10λ ,similarly,y 1(t )=y 01·t α−1E α,α(λt α)+y 11·t α−2E α,α−1(λt α)+y 02·t 2α−1E (1)α,α(λt α)+y 12·t 2α−2E (1)α,α−1(λt α),y 2(t )=y 02·t α−1E α,α(λt α)+y 12·t α−2E α,α−1(λt α).From Proposition 1,one getsy 02·t 2α−1E (1)α,α(λt α)+y 12·t 2α−2E (1)α,α−1(λt α)∼y 02+y 12α d dσ σ(1−α)/α+σ(2−α)/α exp σ1αt σ=λ,36The European Physical Journal Special Topicswith t→∞,|arg(λ)|≤απ2,andy02·t2α−1E(1)α,α(λtα)+y12·t2α−2E(1)α,α−1(λtα)∼2y02Γ(−α)λ−3t−α−1+2y12Γ(−1−α)λ−3t−α−2,with t→∞and|arg(λ)|>απ2.The proof is complete. Recently,the stability analysis of linear FDEs is much involved.Some new methods are presented,such as the popular Laplace transform methods(i.e.,the frequency domain methods)[27–32],the Linear Matrix Inequalities(LMI)methods[25,26,33, 34]and the conversion methods[10],[35–37].These summaries are as follows.3.2The frequency domain methods for stability analysis of linear FDEsThe stability analysis of linear FDEs is very important in the control area,where the frequency domain method is often used.It is known that the key point of the stability analysis for linear FDEs is to determine the location of the roots of charac-teristic equations in the complex plane.It seems difficult to calculate all the roots of the characteristic equations because of the equations with fractional power options. Therefore,the authors in[27]made the following variable substitution.In general, the characteristic equation of a linear fractional differential equation has the formni=1a i sαi=0,(7)where0<αi≤1,for i=1,2,...,n.Whenαi=u i vi is rational,where(u i,v i)=1,the above equation may be rewritten asni=1a i s i m=0,(8)where m is an integer and m=LCM{v1,v2,...,v n}.Translating this equation into the W-plane yields:ni=1a i W i=0,(9) where W=s1m.The steps for stability analysis are as follows[27,31]:1.For given a i,calculate the roots of Eq.(9)andfind the minimum absolute phaseof all roots|θW min|.2.The condition for stability is|θW min|>π2m,while the condition for oscillation is|θW min|=π2m,otherwise the system is unstable.3.Roots in the primary sheet of the W-plane which have corresponding roots in thes-plane can be obtained byfinding all roots which lie in the region|θW|<πm, then applying the inverse transformation s=W m.Evidently the time response of the system can be easily related to these roots.Recently,Trigeassou et al.[32]has presented a new frequency domain method based on Nyquist’s criterion to test the stability of FDEs.The stability of FDEs with one, two derivatives was investigated,some analytic results for systems with N fractional derivatives were given.The detailed analysis can be referred to[32].Thefirst fre-quency domain method is simple,but it is hard to calculate the roots of Eq.(9). The second frequency domain method is based on Nyquist’s criterion(a graphical approach),so the latter is more intuitive.3.3LMI conditions for stability analysis of linear FDEsIn what follows,we will survey the Linear Matrix Inequalities(LMI)conditions on stability of linear FDEs.LMI has played an important role in control theory since the1960s due to its particular form.The main issue when dealing with LMI is the convexity of the optimization set,however,the stability domain of the same order system described by Eq.(5)with order1≤α<2is a convex set,various LMI methods for defining such a region have already been developed[25,26,33,34].Hence a LMI-based theorem for the stability of fractional differential system(5)with order 1≤α<2can be introduced as follows[25,26,34]:Theorem7.A fractional differential system described by Eq.(5)with order1≤α< 2is asymptotically stable if and only if there exists a matrix P=P T>0,P∈R n×n,such that(A T P+P A)sin(απ2)(A T P−P A)cos(απ2)(P A−A T P)cos(απ2)(A T P+P A)sin(απ2)<0.Unfortunately,the stability domain is not convex when0<α<1.The LMI condi-tions can thus not be directly derived.Here there are three different ideas how one can apply these methods to obtain LMI conditions for the stability of linear FDEs. The following three theorems have also been proven in[25,26,34].Theorem8.A fractional differential system described by Eq.(5)with order0<α< 1is asymptotically stable if there exists a matrix P>0,P∈R n×n,such thatA1α TP+PA1α<0.The derivation of above Theorem8(see also[33])is based on an algebraic transfor-mation of the system(5)combined with Lyapunov’s second method.This method has conservatism(explanation can be seen in[26]),so it is a sufficient condition.In order to avoid the conservatism,a new stability theorem based on a geometric analysis of the stability domain was proposed as follows.Theorem9.A fractional differential system described by Eq.(5)with order0<α< 1is asymptotically stable if and only if there exists a positive definite matrix P∈S, where S denotes the set of symmetric matrices,such that−(−A)12−α TP+P−(−A)12−α<0.However,LMI of Theorem9is not linear in relation to matrix A,thus limiting its use in the more specific control problems.In order to overcome this problem,a third condition is proposed.It based on the fact that instability domain is a convex subset of the complex plane when0<α<1.Theorem10.A fractional differential system described by Eq.(5)with order0<α<1is asymptotically stable if and only if there does not exist any nonnegative rank one matrix Q∈C n×n such that(AQ+QA T)sin(απ2)(AQ−QA T)cos(απ2)(AQ−QA T)cos(απ2)(AQ+QA T)sin(απ2)≥0.The following statement is based on stability domain decomposition[34].。
Differential equation
Differential equationNot to be confused with Difference equation.Stokes differential equations used to simulate airflow around an obstruction.ClassificationSolutionVisualization of heat transfer in a pump casing, created by solving the heat equation. Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.A differential equation is amathematical equation that relatessome function of one or more variableswith its derivatives. Differentialequations arise whenever adeterministic relation involving somecontinuously varying quantities(modeled by functions) and their ratesof change in space and/or time(expressed as derivatives) is known orpostulated. Because such relations areextremely common, differentialequations play a prominent role inmany disciplines includingengineering, physics, economics, andbiology.Differential equations aremathematically studied from severaldifferent perspectives, mostlyconcerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have beendeveloped to determine solutions with a given degree of accuracy.ExampleFor example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time.In some cases, this differential equation (called an equation of motion) may be solved explicitly.An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity.Directions of studyThe study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions. Differential equations play an important role in modelling virtually every physical, technical, or biological process, from celestial motion, to bridge design, to interactions between neurons. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.Mathematicians also study weak solutions (relying on weak derivatives), which are types of solutions that do not have to be differentiable everywhere. This extension is often necessary for solutions to exist.The study of the stability of solutions of differential equations is known as stability theory.NomenclatureThe theory of differential equations is well developed and the methods used to study them vary significantly with the type of the equation.Ordinary and partial•An ordinary differential equation (ODE) is a differential equation in which the unknown function (also known as the dependent variable) is a function of a single independent variable. In the simplest form, the unknown function is a real or complex valued function, but more generally, it may be vector-valued or matrix-valued: this corresponds to considering a system of ordinary differential equations for a single function.Ordinary differential equations are further classified according to the order of the highest derivative of the dependent variable with respect to the independent variable appearing in the equation. The most important cases for applications are first-order and second-order differential equations. For example, Bessel's differential equation(in which y is the dependent variable) is a second-order differential equation. In the classical literature a distinction is also made between differential equations explicitly solved with respect to the highest derivative and differential equations in an implicit form. Also important is the degree, or (highest) power, of the highest derivative(s) in the equation (cf. : degree of a polynomial). A differential equation is called a nonlinear differential equation if its degree is not one (a sufficient but unnecessary condition).• A partial differential equation (PDE) is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives. The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second-order linear equations, is of utmost importance. Some partial differentialequations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.Linear and non-linearBoth ordinary and partial differential equations are broadly classified as linear and nonlinear.• A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products of the unknown function and its derivatives are not allowed) and nonlinear otherwise. The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogeneous linear differential equations are a further subclass for which the space of solutions is a linear subspace i.e. the sum of any set of solutions or multiples of solutions is also a solution. The coefficients of the unknown function and its derivatives in a linear differential equation are allowed to be (known) functions of the independent variable or variables; if these coefficients are constants then one speaks of a constant coefficient linear differential equation.•There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Nonlinear differential equations can exhibit verycomplicated behavior over extended time intervals, characteristic of chaos. Even the fundamental questions of existence, uniqueness, and extendability of solutions for nonlinear differential equations, and well-posedness of initial and boundary value problems for nonlinear PDEs are hard problems and their resolution in special cases is considered to be a significant advance in the mathematical theory (cf. Navier–Stokes existence and smoothness).However, if the differential equation is a correctly formulated representation of a meaningful physical process, then one expects it to have a solution.Linear differential equations frequently appear as approximations to nonlinear equations. These approximations are only valid under restricted conditions. For example, the harmonic oscillator equation is an approximation to the nonlinear pendulum equation that is valid for small amplitude oscillations (see below).ExamplesIn the first group of examples, let u be an unknown function of x, and c and ω are known constants.•Inhomogeneous first-order linear constant coefficient ordinary differential equation:•Homogeneous second-order linear ordinary differential equation:•Homogeneous second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:•Inhomogeneous first-order nonlinear ordinary differential equation:•Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:In the next group of examples, the unknown function u depends on two variables x and t or x and y.•Homogeneous first-order linear partial differential equation:•Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:•Third-order nonlinear partial differential equation, the Korteweg–de Vries equation:Related concepts• A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times.• A stochastic differential equation (SDE) is an equation in which the unknown quantity is a stochastic process and the equation involves some known stochastic processes, for example, the Wiener process in the case of diffusion equations.• A differential algebraic equation (DAE) is a differential equation comprising differential and algebraic terms, given in implicit form.Connection to difference equationsSee also: Time scale calculusThe theory of differential equations is closely related to the theory of difference equations, in which the coordinates assume only discrete values, and the relationship involves values of the unknown function or functions and values at nearby coordinates. Many methods to compute numerical solutions of differential equations or study the properties of differential equations involve approximation of the solution of a differential equation by the solution of a corresponding difference equation.Universality of mathematical descriptionMany fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics, differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turns out that many diffusion processes, while seemingly different, are described by the same equation; the Black–Scholes equation in finance is, for instance, related to the heat equation.Notable differential equationsPhysics and engineering•Newton's Second Law in dynamics (mechanics)•Euler–Lagrange equation in classical mechanics•Hamilton's equations in classical mechanics•Radioactive decay in nuclear physics•Newton's law of cooling in thermodynamics•The wave equation•Maxwell's equations in electromagnetism•The heat equation in thermodynamics•Laplace's equation, which defines harmonic functions•Poisson's equation•Einstein's field equation in general relativity•The Schrödinger equation in quantum mechanics•The geodesic equation•The Navier–Stokes equations in fluid dynamics•The Diffusion equation in stochastic processes•The Convection–diffusion equation in fluid dynamics•The Cauchy–Riemann equations in complex analysis•The Poisson–Boltzmann equation in molecular dynamics•The shallow water equations•Universal differential equation•The Lorenz equations whose solutions exhibit chaotic flow.Biology•Verhulst equation – biological population growth•von Bertalanffy model – biological individual growth•Lotka–Volterra equations – biological population dynamics•Replicator dynamics – found in theoretical biology•Hodgkin–Huxley model – neural action potentialsEconomics•The Black–Scholes PDE•Exogenous growth model•Malthusian growth model•The Vidale–Wolfe advertising modelReferences•P. Abbott and H. Neill, Teach Yourself Calculus, 2003 pages 266-277•P. Blanchard, R. L. Devaney, G. R. Hall, Differential Equations, Thompson, 2006• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955• E. L. Ince, Ordinary Differential Equations, Dover Publications, 1956•W. Johnson, A Treatise on Ordinary and Partial Differential Equations[2], John Wiley and Sons, 1913, in University of Michigan Historical Math Collection [3]• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations (2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003. ISBN 1-58488-297-2.•R. I. Porter, Further Elementary Analysis, 1978, chapter XIX Differential Equations•Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems[4]. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.[1]/w/index.php?title=Template:Differential_equations&action=edit[2]/cgi/b/bib/bibperm?q1=abv5010.0001.001[3]/u/umhistmath/[4]http://www.mat.univie.ac.at/~gerald/ftp/book-ode/External links•Lectures on Differential Equations (/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/) MIT Open CourseWare Videos•Online Notes / Differential Equations (/classes/de/de.aspx) Paul Dawkins, Lamar University•Differential Equations (/diffeq/diffeq.html), S.O.S. Mathematics•Differential Equation Solver (/tools/differential_equation_solver/) Java applet tool used to solve differential equations.•Introduction to modeling via differential equations (/mat/u-u/en/ differential_equations_intro.htm) Introduction to modeling by means of differential equations, with critical remarks.•Mathematical Assistant on Web (http://user.mendelu.cz/marik/maw/index.php?lang=en&form=ode) Symbolic ODE tool, using Maxima•Exact Solutions of Ordinary Differential Equations (http://eqworld.ipmnet.ru/en/solutions/ode.htm)•Collection of ODE and DAE models of physical systems (/research/models.htm) MATLAB models•Notes on Diffy Qs: Differential Equations for Engineers (/diffyqs/) An introductory textbook on differential equations by Jiri Lebl of UIUC•Khan Academy Video playlist on differential equations (/math/ differential-equations) Topics covered in a first year course in differential equations.•MathDiscuss Video playlist on differential equations (/category/courses/ solutions-differential-equations/homogeneous-linear-systems/)Article Sources and Contributors8Article Sources and ContributorsDifferential equation Source: /w/index.php?oldid=610771276 Contributors: 17Drew, After Midnight, Ahoerstemeier, Alarius, Alfred Centauri, Amahoney, AndreiPolyanin, Andres, AndrewHowse, Andycjp, Andytalk, AngryPhillip, Anonymous Dissident, Antoni Barau, Antonius Block, Anupam, Apmonitor, Arcfrk, Asdf39, Asyndeton, Attilios,Babayagagypsies, Baccala@, Baccyak4H, Bejohns6, Bento00, Berland, Bidabadi, Bigusbry, BillyPreset, Bob.v.R, Bolatbek, Brandon, Bryanmcdonald, Btyner, Bygeorge2512,Callumds, Charles Matthews, Christian75, Chtito, Cispyre, Cmprince, Coginsys, ConMan, Cxz111, Cybercobra, DAJF, Danski14, Dbroadwell, Ddxc, Delaszk, DerHexer, Dewritech, Difu Wu, Djordjes, DominiqueNC, Donludwig, Dpv, Dr sarah madden, Drmies, DroEsperanto, Duoduoduo, Dysprosia, EconoPhysicist, Elwikipedista, Epicgenius, EricBright, Erin.Annette.Brown,Estudiarme, F=q(E+v^B), Fintor, Fioravante Patrone, Fioravante Patrone en, Flameturtle, Friend of the Facts, FutureTrillionaire, Gabrielleitao, Gandalf61, Gauss, Genedronek, Geni, Giftlite,GoingBatty, Gombang, Grenavitar, Haham hanuka, Hamiltondaniel, Harry, Haruth, Haseeb Jamal, Heikki m, Holmes1900, Ilya Voyager, Iquseruniv, Iulianu, Izodman2012, J arino, J.delanoy, Ja 62, Jak86, JamesBWatson, Jao, Jarble, Jauhienij, Jayden54, Jeancey, Jersey Devil, Jim Sukwutput, Jim.belk, Jim.henderson, JinJian, Jitse Niesen, JohnOwens, Johndoeisnotmyname, JorisvS,Julesd, K-UNIT, Kayvan45622, KeithJonsn, Kensaii, Khalid Mahmood, Klaas van Aarsen, Kr5t, Krushia, LOL, Lambiam, Lavateraguy, Lethe, LibLord, Linas, Lumos3, Madmath789, Mandarax, Mankarse, MarSch, Martastic, Martynas Patasius, Maschen, Math.geek3.1415926, Matqkks, Mattmnelson, Maurice Carbonaro, Maxis ftw, Mazi, McVities, Mduench, Mets501, Mh, MichaelHardy, Mindspillage, MisterSheik, Mohan1986, Mossaiby, Mpatel, MrOllie, Mtness, Mysidia, Nik-renshaw, Nkayesmith, Norm mit, Okopecz, Oleg Alexandrov, Opelio, Pahio, Parusaro, Paul August, Paul Matthews, Paul Richter, PavelSolin, Pgk, Phoebe, Pine, Pinethicket, Pratyya Ghosh, PseudoSudo, Qwerty Binary, Qzd800, R'n'B, Rama's Arrow, Randomguess, Reallybored999, RexNL, Reyk, RichMorin, Robin S, Romansanders, Rosasco, Ruakh, SDC, SFC9394, SakeUPenn, Salix alba, Sam Staton, Sampathsris, Sardanaphalus, Senoreuchrestud, Silly rabbit, Siroxo,Skakkle, Skypher, SmartPatrol, Snowjeep, Spirits in the Material, Starwiz, Suffusion of Yellow, Sverdrup, Symane, TVBZ28, TYelliot, Tannkrem, Tbhotch, Tbsmith, TexasAndroid, Tgeairn, The Hybrid, The Thing That Should Not Be, Timelesseyes, Tranum1234567890, Tsirel, Tuseroni, User A1, Vanished User 0001, Vishwanathnm, Vthiru, Waffleguy4, Waldir, Waltpohl, Wavelength, Wclxlus, Wihenao, Willtron, Winterheart, Wsears, XJaM, Yafujifide, Zepterfd, ﺪﺟﺎﺳ ﺪﺠﻣﺍ ﺪﺟﺎﺳ, 363 anonymous editsImage Sources, Licenses and ContributorsFile:Airflow-Obstructed-Duct.png Source: /w/index.php?title=File:Airflow-Obstructed-Duct.png License: Public Domain Contributors: Original uploader was User A1 at en.wikipediaFile:Elmer-pump-heatequation.png Source: /w/index.php?title=File:Elmer-pump-heatequation.png License: Creative Commons Attribution-Sharealike 3.0Contributors: Christian1985, Crimerob, Kri, User A1, 2 anonymous editsLicenseCreative Commons Attribution-Share Alike 3.0///licenses/by-sa/3.0/。
拉普拉斯变换相关书籍英文
Laplace Transform: A Comprehensive StudyIntroductionLaplace Transform is a powerful mathematical tool used in various fields such as engineering, physics, and mathematics. It provides a way to transform differential equations into algebraic equations, making them easier to solve. In this article, we will explore the concept of Laplace Transform in detail and discuss its applications.Basic Definition and Properties1.The Laplace Transform is defined for a function f(t) defined for t≥ 0 by the equation: > F(s) = L{f(t)} = ∫[0, ∞] e^(-st) f(t) dtHere, s is a complex variable and F(s) is the Laplace Transform of f(t).2.Linearity Property: For any constants a and b, and functions f(t)and g(t), we have: > L{af(t) + bg(t)} = aF(s) + bG(s)This property allows us to apply Laplace Transform to linearcombinations of functions.3.Shifting Property: If F(s) is the Laplace Transform of f(t),then: > L{e^(-at)f(t)} = F(s + a)This property enables us to manipulate the time domain bymultiplying the function with an exponential term.4.Differentiation Property: If F(s) is the Laplace Transform of f(t),then: > L{df(t)/dt} = sF(s) - f(0)This property allows us to transform derivatives in the timedomain into algebraic expressions in the Laplace domain.Inverse Laplace Transform1.The Inverse Laplace Transform of F(s) is denoted by L^{-1}{F(s)}and is defined as: > f(t) = L^{-1}{F(s)} = (1/2πi) ∫[-i∞, i∞]e^(st) F(s) dsThis transformation allows us to retrieve the original functionf(t) from its Laplace Transform F(s).mon Pairs: There are several common Laplace Transform pairsthat are frequently used for solving differential equations. Some examples include:–L{1} = 1/s, L^{-1}{1/s} = 1–L{t^n} = n!/(s^(n+1)), L{-1}{n!/(s(n+1))} = t^nThese pairs provide a shortcut for transforming common functionsin the time domain.Applications of Laplace Transform1.Solving Ordinary Differential Equations (ODEs): Laplace Transformis extensively used to solve linear ordinary differentialequations with constant coefficients. By applying LaplaceTransform to both sides of the equation, we can convert theequation into an algebraic equation that is easier to solve. Once solved, the inverse Laplace Transform is applied to obtain thesolution in the time domain.2.Circuit Analysis: Laplace Transform is widely used in electricalcircuit analysis. By transforming the circuit equations into theLaplace domain, we can analyze the behavior of the circuits interms of complex impedance and transfer functions. This simplifies the analysis of complex circuits and facilitates the design ofcircuits for specific applications.3.Control Systems: The Laplace Transform plays a crucial role in theanalysis and design of control systems. By transforming theequations governing the dynamics of the system, we can analyzestability, transient response, and steady-state behavior. Laplace Transform also allows us to design controllers and compensators to achieve desired system performance.4.Signal Processing: Laplace Transform is utilized in signalprocessing to analyze and manipulate signals. By transformingsignals into the frequency domain, we can analyze their spectralcharacteristics and apply various filters or modifications.Laplace Transform is particularly useful for analyzing continuous-time signals and systems.ConclusionIn conclusion, Laplace Transform is a fundamental mathematical tool that has numerous applications in various fields. Its ability to transform differential equations into algebraic equations simplifies problem-solving and analysis. By understanding the basic properties and applications of Laplace Transform, we can effectively solve complex problems in engineering, physics, and mathematics.。
一阶常微分方程
Chapter 1 First-order ordinary differential equations (ODE)一階常微分方程1.1 基本概念()x f y =或()t f y =，y 是x 或t 的函數，y 是因變數(dependent variable )，x 或t 是自變數(independent variable )◎ 微分方程(differential equations)：一方程式包含有因變數y 關於自變數x 或t的導數(derivatives)y y ′ ,&或微分(differentials)dy 。
◎ 常微分方程(ordinary differential equations, ODE)：一微分方程包含有一個或數個因變數（通常為()x y ）關於僅有一個自變數x 的導數。
Ex. 222)2(2 ,09 ,cos y x y e y y x y y x y x +=′′+′′′′=+′′=′◎ 偏微分方程(partial differential equations, PDE)：一微分方程包含至少有一個因變數關於兩個以上自變數的部分導數。
Ex. 02222=∂∂+∂∂yux u◎ 微分方程的階數：在微分方程式中所出現最高階導數的階數。
◎ 線性微分方程：在微分方程式中所出現的因變數因變數因變數或其導數僅有一次式(first degree)而無二次以上的乘積（自變數可以有二次以上的乘積）。
Ex. x y y x y cos 24=+′+′′ 因變數：y ，自變數：x ，二階線性常微分方程 x y y y y cos 24=+′+′′ 因變數：y ，自變數：x ，二階非線性常微分方程 222)2(2 y x y e y y x x +=′′+′′′′ 因變數：y ，自變數：x ，三階非線性常微分方程□ 一階常微分方程(first-order ordinary differential equations)隱式形式(implicit form) 表示 0),,(=′y y x F (4)顯式形式(explicit form) 表示 ),(y x f y =′Ex. 隱式形式ODE 0423=−′−y y x ，當0≠x 時，可表示為顯式形式234y x y =′□ 解的概念(concept of solution)在某些開放間隔區間b x a <<，一函數)(x h y =是常微分方程常微分方程0),,(=′y y x F 的解，其函數)(x h 在此區間b x a <<是明確(defined)且可微分的(differentiable)，其)(x h 的曲線（或圖形）是被稱為解答曲線(solution curve)。
数学专业词汇(英序+中序)学位考试用
阿贝尔的，交换的Abelian augmented matrix增广矩阵阿贝尔的，交换的Abelian augmented matrix增广矩阵鞍点saddle point asymptotic渐进的鞍点saddle point asymptotic渐进的凹函数concavefunctionasymptote渐进线凹函数concavefunctionasymptote渐进线八边形octagon asymmetrical非对称的八边形octagon asymmetrical非对称的半单纯的semisimple associative law结合律半单纯的semisimple associative law结合律半定的semi-definite ascending上升的半定的semi-definite ascending上升的半径radius arrangement排列半径radius arrangement排列半平面half plane arithmetic算术半平面half plane arithmetic算术半群semigroup argument 幅角，幅度，自变量，论证半群semigroup argument幅角，幅度，自变量，论证半圆semicircle area面积半圆semicircle area面积伴随的adjoint arc length弧长伴随的adjoint arc length弧长伴随算子adjointoperatorapothem边心距伴随算子adjointoperatorapothem边心距伴随转置的adjugate apex顶点伴随转置的adjugate apex顶点包含inclusion aperiodic非周期的包含inclusion aperiodic非周期的包络envelope antisymmetric反对称的包络envelope antisymmetric反对称的胞腔复形cell complex antiderivative原函数胞腔复形cell complex antiderivative原函数保序的order-preservinganticlockwise逆时针的保序的order-preservinganticlockwise逆时针的被积函数integrand annihilator零化子被积函数integrand annihilator零化子被加数summand angular velocity角速度被加数summand angular velocity角速度本原的primitive angle of rotation旋转角本原的primitive angle of rotation旋转角比例proportion angle of incidence入射角比例proportion angle of incidence入射角闭包closure angle of elevation仰角闭包closure angle of elevation仰角闭球closed ball angle ofdepression 俯角闭球closed ball angle ofdepression俯角边界periphery angle ofcircumference圆周角边界peripheryangle ofcircumference圆周角边界的，边界boundary analytic space复空间边界的，边界boundary analytic space复空间边心距apothem analytic geometry解析几何边心距apothem analytic geometry解析几何边缘marginal analytic function解析函数边缘marginal analytic function解析函数变分，变差variation analytic extension解析开拓变分，变差variation analytic extension解析开拓变分法calculus ofvariationsamplitude幅角，振幅变分法calculus ofvariationsamplitude幅角，振幅变换式transformation alternative互斥的变换式transformation alternative互斥的标尺，尺度scale alternate series交错级数标尺，尺度scale alternate series交错级数标量场scalar field almost everywhere几乎处处标量场scalar field almost everywhere几乎处处标志flag algebraic topology代数拓扑标志flag algebraic topology代数拓扑标准型canonical form algebraicexpression代数式标准型canonical formalgebraicexpression代数式表格式的tabular algebraic代数的表格式的tabular algebraic代数的并，并集union affine仿射(几何学)的并，并集union affine仿射(几何学)的并，求并运算cup admissible error容许误差并，求并运算cup admissible error容许误差病态ill-condition admissible容许的病态ill-condition admissible容许的波wave adjugate伴随转置的波wave adjugate伴随转置的波动方程wave equation adjoint operator伴随算子波动方程wave equation adjoint operator伴随算子波函数wave function adjoint伴随的波函数wave function adjoint伴随的波形wave form adjacency邻接波形wave form adjacency邻接不变的invariant additive加法，加性不变的invariant additive加法，加性不成立的inconsistent acute angle锐角不成立的inconsistent acute angle锐角不等的unequal accumulation point聚点不等的unequal accumulation point聚点不等式inequality accidential error偶然误差不等式inequality accidential error偶然误差不定积分indefiniteintegralaccessible point可达点不定积分indefiniteintegralaccessible point可达点不定性uncertainty abstract space抽象空间不定性uncertainty abstract space抽象空间不动点fixed point abstract algebra抽象代数不动点fixed point abstract algebra抽象代数不可比的incomparable absolute value绝对值不可比的incomparable absolute value绝对值不可数的noncountable absoluteintegrable 绝对可积不可数的noncountable absoluteintegrable绝对可积不可约的irreducible absoluteconvergent 绝对收敛不可约的irreducible absoluteconvergent绝对收敛不连通的disconnected Abelian阿贝尔的，交换的不连通的disconnected Abelian阿贝尔的，交换的不连续的discontinuous balance equation平衡方程不连续的discontinuous balance equation平衡方程不舍入的unrounded bandwidth带宽不舍入的unrounded bandwidth带宽不稳定instability barycenter重心不稳定instability barycenter重心不稳定的unstable base基不稳定的unstable base基不相交的disjoint base vectors基向量不相交的disjoint base vectors基向量不相容的，互斥的incompatible biased error有偏误差不相容的，互斥的incompatible biased error有偏误差部分和partial sum biased statistic有偏统计量部分和partial sum biased statistic有偏统计量参数parameter bilinear双线性的参数parameter bilinear双线性的侧面side face bijective双射的侧面side face bijective双射的侧面积lateral area bilateral shift双侧位移的侧面积lateral area bilateral shift双侧位移的测地线geodesic binomial二项式测地线geodesic binomial二项式测度measure bisector二等分线，平分线测度measure bisector二等分线，平分线层sheaf boundary边界的，边界层sheaf boundary边界的，边界差，差分difference bounded有界的差，差分difference bounded有界的长方形，矩形rectangle broken line折线长方形，矩形rectangle broken line折线超定的overdetermined bundle丛，把，卷超定的overdetermined bundle丛，把，卷超平面hyperplane calculus微积分超平面hyperplane calculus微积分超限的transfinite calculus ofvariations变分法超限的transfinitecalculus ofvariations变分法超越的transcendental cancellation消去超越的transcendental cancellation消去乘法multiplication canonical典型的，标准的乘法multiplication canonical典型的，标准的抽象代数abstractalgebracanonical form标准型抽象代数abstractalgebracanonical form标准型抽象空间abstract space cap交，求交运算抽象空间abstract space cap交，求交运算稠密性，密度density capacity容量稠密性，密度density capacity容量初始状态original state cardinal number基数初始状态original state cardinal number基数除不尽的indivisible Cartesiancoordinates笛卡尔坐标除不尽的indivisibleCartesiancoordinates笛卡尔坐标除法division category范畴，类型除法division category范畴，类型垂直perpendicular cell单元，方格，胞腔垂直perpendicular cell单元，方格，胞腔纯的pure cell complex胞腔复形纯的pure cell complex胞腔复形次可加的subadditive character特征标次可加的subadditive character特征标次序的ordinal characterization特征次序的ordinal characterization特征丛，把，卷bundle circuit环路，线路，回路丛，把，卷bundle circuit环路，线路，回路簇variety circular ring圆环簇variety circular ring圆环大样本large sample circulatingdecimal 循环小数大样本large sample circulatingdecimal循环小数代数的algebraic clockwise顺时针方向的代数的algebraic clockwise顺时针方向的代数式algebraicexpressionclosed ball闭球代数式algebraicexpressionclosed ball闭球代数拓扑algebraictopologyclosure闭包代数拓扑algebraictopologyclosure闭包带宽bandwidth cluster point聚点带宽bandwidth cluster point聚点单侧的one-side coefficient系数单侧的one-side coefficient系数单侧的unilateral cofinal共尾的单侧的unilateral cofinal共尾的单纯形simplex cohomology上同调单纯形simplex cohomology上同调单位unit coincidence重合，叠和单位unit coincidence重合，叠和单位圆unit circle collinear共线的单位圆unit circle collinear共线的单一同态monomorphism collective集体的单一同态monomorphism collective集体的单元，方格，胞腔cell columnar rank列秩单元，方格，胞腔cell columnar rank列秩导数derivative combinatorialtheory组合理论导数derivativecombinatorialtheory组合理论倒数的，互反的reciprocal common tangent公切线倒数的，互反的reciprocal common tangent公切线道路空间path space commutative交换的道路空间path space commutative交换的等边的equilateral compact紧的等边的equilateral compact紧的等度连续的equicontinuous compact operator紧算子等度连续的equicontinuous compact operator紧算子等价equivalence compatibility相容性等价equivalence compatibility相容性等角equiangular compatible events相容事件等角equiangular compatible events相容事件等距的isometric complementary余的，补的等距的isometric complementary余的，补的笛卡尔坐标Cartesiancoordinatescomplete完全的，完备的笛卡尔坐标Cartesiancoordinatescomplete完全的，完备的递减的decreasing complex analysis复变函数论递减的decreasing complex analysis复变函数论典型的，标准的canonical complex potential复位势典型的，标准的canonical complex potential复位势点积dot product composite复合的点积dot product composite复合的调和的harmonic concave function凹函数调和的harmonic concave function凹函数迭代iteration concentric circles同心圆迭代iteration concentric circles同心圆顶点apex concurrent共点顶点apex concurrent共点顶点vertex conditional number条件数顶点vertex conditional number条件数定方向orient confidenceinterval 置信区间定方向orient confidenceinterval置信区间定积分definiteintegralconformal共形的定积分definiteintegralconformal共形的定义definition conic圆锥的定义definition conic圆锥的动力系统dynamic system conjugate共轭的动力系统dynamic system conjugate共轭的动态规划dynamicprogrammingconnected连通的动态规划dynamicprogrammingconnected连通的动态模型dynamic model connected domain连通域动态模型dynamic model connected domain连通域端点end point consistence相容，一致端点end point consistence相容，一致对策game constrained约束的对策game constrained约束的对称的symmetric continuable可延拓的对称的symmetric continuable可延拓的对顶角vertical angle continuity连续性对顶角vertical angle continuity连续性对合involution contour周线，回路，轮廓线对合involution contour周线，回路，轮廓线对和，卷积convolution convergence收敛性对和，卷积convolution convergence收敛性对偶dual convexity凸形对偶dual convexity凸形对偶基reciprocalbasisconvolution对和，卷积对偶基reciprocalbasisconvolution对和，卷积对数logarithm coordinate坐标对数logarithm coordinate坐标对应correspondence coprime互质的，互素的对应correspondence coprime互质的，互素的钝角obtuse angle correspondence对应钝角obtuse angle correspondence对应多边形polygon coset陪集多边形polygon coset陪集多极multipole countable可数的多极multipole countable可数的多项式polynomial counterexample反例多项式polynomial counterexample反例多元分析multi-analysis covariance协方差多元分析multi-analysis covariance协方差多值函数many-valuedfunctioncovariant共变的多值函数many-valuedfunctioncovariant共变的二次的quadratic covering覆盖二次的quadratic covering覆盖二次曲面quadricsurface critical临界的二次曲面quadricsurfacecritical临界的二等分线，平分线bisector cubic root立方根二等分线，平分线bisector cubic root立方根二分法dichotomy cup并，求并运算二分法dichotomy cup并，求并运算二项式binomial curl旋度二项式binomial curl旋度二重积分doubleintegral curvature曲率二重积分doubleintegralcurvature曲率发散的divergent curve曲线发散的divergent curve曲线法方向normaldirectioncyclic循环的法方向normaldirectioncyclic循环的法向导数normalderivativedecade十进制的法向导数normalderivativedecade十进制的反对称的antisymmetric decagon十边形反对称的antisymmetric decagon十边形反例counterexample decimal小数的，十进制的反例counterexample decimal小数的，十进制的反三角函数inversecircularfunctiondecision theory决策论反三角函数inversecircularfunctiondecision theory决策论反射reflection decomposable可分解的反射reflection decomposable可分解的反演inversion decreasing递减的反演inversion decreasing递减的泛的，通用的universal decrement减量泛的，通用的universal decrement减量泛函functional deduction推论，归纳法泛函functional deduction推论，归纳法范畴，类型category defect亏量，缺陷范畴，类型category defect亏量，缺陷范数norm deficiency亏格范数norm deficiency亏格方向导数directionalderivativedefinition定义方向导数directionalderivativedefinition定义方向角directionangle definite integral定积分方向角directionangledefinite integral定积分仿射(几何学)的affine deflation压缩仿射(几何学)的affine deflation压缩非对称的asymmetrical deflection挠度，挠率，变位非对称的asymmetrical deflection挠度，挠率，变位非交换的noncommutative degenerate退化的非交换的noncommutative degenerate退化的非空的nonempty deletedneighborhood去心邻域非空的nonemptydeletedneighborhood去心邻域非齐次的inhomogeneous denominator分母非齐次的inhomogeneous denominator分母非奇异的nonsingular density稠密性，密度非奇异的nonsingular density稠密性，密度非线性的nonlinear density function密度函数非线性的nonlinear density function密度函数非周期的aperiodic denumerable可数的非周期的aperiodic denumerable可数的分布，广义函数distribution departure偏差，偏离分布，广义函数distribution departure偏差，偏离分段的piecewise dependent相关的分段的piecewise dependent相关的分解resolution dependent variable因变量分解resolution dependent variable因变量分裂的splitting derangement重排分裂的splitting derangement重排分母denominator derivation求导分母denominator derivation求导分歧的ramified derivative导数分歧的ramified derivative导数分数，分式fraction descent下降分数，分式fraction descent下降幅角，幅度，自变量，论证argument determinant行列式幅角，幅度，自变量，论证argument determinant行列式幅角，振幅amplitude diagram图，图表幅角，振幅amplitude diagram图，图表俯角angle ofdepressiondiameter直径俯角angle ofdepressiondiameter直径负的，否定的negative diamond菱形负的，否定的negative diamond菱形复变函数论complexanalysisdichotomy二分法复变函数论complexanalysisdichotomy二分法复合的composite diffeomorphism微分同胚复合的composite diffeomorphism微分同胚复空间analytic space differentiable可微的复空间analytic space differentiable可微的复位势complexpotentialdifferential微分复位势complexpotentialdifferential微分赋范的normed differentialgeometry微分几何赋范的normeddifferentialgeometry微分几何赋值valuation difference差，差分赋值valuation difference差，差分赋值，值的计算evaluation digit数字赋值，值的计算evaluation digit数字覆盖covering dimension维数覆盖covering dimension维数改变，传transfer directed graph有向图改变，传transfer directed graph有向图概率probability directed set有向集概率probability directed set有向集格点lattice point direct prodect直积格点lattice point direct prodect直积格子lattice direct sum直和格子lattice direct sum直和根号radical sign direction angle方向角根号radical sign direction angle方向角公切线common tangent directionalderivative方向导数公切线common tangentdirectionalderivative方向导数公式formula disc圆盘公式formula disc圆盘共变的covariant disconnected不连通的共变的covariant disconnected不连通的共点concurrent discontinuous不连续的共点concurrent discontinuous不连续的共轭的conjugate discrete离散的共轭的conjugate discrete离散的共尾的cofinal discriminant判别式共尾的cofinal discriminant判别式共线的collinear disjoint不相交的共线的collinear disjoint不相交的共形的conformal disorder混乱，无序共形的conformal disorder混乱，无序估计量estimator dissection剖分估计量estimator dissection剖分孤立点isolated point dissipation损耗孤立点isolated point dissipation损耗骨架skeleton distribution分布，广义函数骨架skeleton distribution分布，广义函数固有周期natural period divergent发散的固有周期natural period divergent发散的卦限octant divisor因子，除数卦限octant divisor因子，除数拐点inflectionpointdivision除法拐点inflectionpointdivision除法关键词keyword domain区域，定义域关键词keyword domain区域，定义域管状的tubular dot product点积管状的tubular dot product点积惯性律inertia law double integral二重积分惯性律inertia law double integral二重积分光滑流形smoothmanifold dual对偶光滑流形smoothmanifolddual对偶广义逆generalizedinversedynamic model动态模型广义逆generalizedinversedynamic model动态模型归纳，普遍化generalization dynamicprogramming 动态规划归纳，普遍化generalization dynamicprogramming动态规划归纳的inductive dynamic system动力系统归纳的inductive dynamic system动力系统归纳定义inductivedefinitioneccentricity离心率归纳定义inductivedefinitioneccentricity离心率规范化正交的orthonormal econometrics计量经济学规范化正交的orthonormal econometrics计量经济学轨道orbit edge棱，边轨道orbit edge棱，边轨道trajectory eigenvalue特征值轨道trajectory eigenvalue特征值轨迹locus(pl.loci)eigenvector特征向量轨迹locus(pl.loci)eigenvector特征向量函数function eigenspace特征空间函数function eigenspace特征空间函子，算符functor element元素函子，算符functor element元素和sum ellipse椭圆和sum ellipse椭圆核kernel embed嵌入核kernel embed嵌入恒等，恒同identical empirical equation经验公式恒等，恒同identical empirical equation经验公式恒等式，单位元identity empiricalassumption经验假设恒等式，单位元identityempiricalassumption经验假设横截、transverse endomorphism自同态横截、transverse endomorphism自同态弧长arc length end point端点弧长arc length end point端点弧度radian entropy熵弧度radian entropy熵互不相交mutuallydisjointentire function整函数互不相交mutuallydisjointentire function整函数互斥的alternative envelope包络互斥的alternative envelope包络互斥事件exclusiveeventsepimorphism满同态互斥事件exclusiveeventsepimorphism满同态互反性reciprocity equiangular等角互反性reciprocity equiangular等角互质的，互素的coprime equilateral等边的互质的，互素的coprime equilateral等边的划分，分类partition equicontinuous等度连续的划分，分类partition equicontinuous等度连续的环路，线路，回路circuit equilibrium平衡环路，线路，回路circuit equilibrium平衡黄金分割golden section equivalence等价黄金分割golden section equivalence等价混乱，无序disorder error estimate误差估计混乱，无序disorder error estimate误差估计迹trace estimator估计量迹trace estimator估计量积分integral evaluation赋值，值的计算积分integral evaluation赋值，值的计算基base even number偶数基base even number偶数基本群fundamentalgroupexact sequence正合序列基本群fundamentalgroupexact sequence正合序列基数cardinalnumber exact solution精确解基数cardinalnumberexact solution精确解基向量base vectors excenter外心基向量base vectors excenter外心极，极点pole excision切割，分割极，极点pole excision切割，分割极大的，最大的maximal exclusive events互斥事件极大的，最大的maximal exclusive events互斥事件极限limit exhaustive穷举的极限limit exhaustive穷举的极小的，最小的minimal expansion展开，展开式极小的，最小的minimal expansion展开，展开式极坐标polarcoordinatesexpectation期望极坐标polarcoordinatesexpectation期望集体的collective experimental error实验误差集体的collective experimental error实验误差几何的geometrical explicit function显函数几何的geometrical explicit function显函数几何级数geometricseriesexponent指数几何级数geometricseriesexponent指数几乎处处almosteverywhereextension扩张，外延几乎处处almosteverywhereextension扩张，外延计量经济学econometrics face面计量经济学econometrics face面加plus factor因子加plus factor因子加法，加性additive factorial阶乘加法，加性additive factorial阶乘架，标架frame fallacy谬误架，标架frame fallacy谬误假设hypothesis fiducial置信假设hypothesis fiducial置信间断，间隙gap field域，场间断，间隙gap field域，场减minus field theory域论减minus field theory域论减法subtraction figure图形，数字减法subtraction figure图形，数字减量decrement finite有限的减量decrement finite有限的简单曲线simple curve finite group有限群简单曲线simple curve finite group有限群简化，化简reduce finite iteration有限迭代简化，化简reduce finite iteration有限迭代渐进的asymptotic finite rank有限秩渐进的asymptotic finite rank有限秩渐进线asymptote finitely covered有限覆盖渐进线asymptote finitely covered有限覆盖交，求交运算cap fitting拟合交，求交运算cap fitting拟合交，相交intersection fixed point不动点交，相交intersection fixed point不动点交错级数alternateseriesflag标志交错级数alternateseriesflag标志交点point ofintersectionflat space平旦空间交点point ofintersectionflat space平旦空间交换的commutative formula公式交换的commutative formula公式角速度angularvelocityfraction分数，分式角速度angularvelocityfraction分数，分式阶，级，次序order frame架，标架阶，级，次序order frame架，标架阶乘factorial free boundary自由边界阶乘factorial free boundary自由边界节点的nodal frequency频数，频率节点的nodal frequency频数，频率结合律associativelaw front side正面结合律associativelawfront side正面解solution function函数解solution function函数解析函数analyticfunctionfunctional泛函解析函数analyticfunctionfunctional泛函解析几何analyticgeometryfunctor函子，算符解析几何analyticgeometryfunctor函子，算符解析开拓analyticextensionfundamental group基本群解析开拓analyticextensionfundamental group基本群介值intermediatevaluefuzzy模糊的介值intermediatevaluefuzzy模糊的紧的compact gain增益，放大率紧的compact gain增益，放大率紧算子compactoperatorgame对策紧算子compactoperatorgame对策浸入immersion gap间断，间隙浸入immersion gap间断，间隙经验公式empiricalequationgeneral topology一般拓扑学经验公式empiricalequationgeneral topology一般拓扑学经验假设empiricalassumptiongeneral term通项经验假设empiricalassumptiongeneral term通项精确解exact solution generalized普遍的，推广的精确解exact solution generalized普遍的，推广的径向radial generalizedinverse 广义逆径向radial generalizedinverse广义逆局部坐标localcoordinatesgeneralization归纳，普遍化局部坐标localcoordinatesgeneralization归纳，普遍化矩moment generating line母线矩moment generating line母线矩阵matrix(pl.matrices)genus亏格矩阵matrix(pl.matrices)genus亏格距离空间metric space geodesic测地线距离空间metric space geodesic测地线聚点accumulationpointgeometrical几何的聚点accumulationpointgeometrical几何的聚点cluster point geometric series几何级数聚点cluster point geometric series几何级数决策论decisiontheory golden section黄金分割决策论decisiontheorygolden section黄金分割绝对可积absoluteintegrablegraph图形，网格绝对可积absoluteintegrablegraph图形，网格绝对收敛absoluteconvergenthalf plane半平面绝对收敛absoluteconvergenthalf plane半平面绝对值absolute value harmonic调和的绝对值absolute value harmonic调和的开方radication hexagon六边形开方radication hexagon六边形开球open ball hereditary可传的开球open ball hereditary可传的可测的measurable holomorphic全纯的可测的measurable holomorphic全纯的可传的hereditary homeomorphism同胚可传的hereditary homeomorphism同胚可传递的transitive homogeneous齐次的可传递的transitive homogeneous齐次的可达点accessiblepointhomology同调可达点accessiblepointhomology同调可定向的orientable homotopy同伦可定向的orientable homotopy同伦可分的separable hyperbola双曲线可分的separable hyperbola双曲线可分解的decomposable hyperplane超平面可分解的decomposable hyperplane超平面可积的integrable hypothesis假设可积的integrable hypothesis假设可加的summable ideal理想可加的summable ideal理想可交换的permutable idempotent幂等的可交换的permutable idempotent幂等的可解的solvable identical恒等，恒同可解的solvable identical恒等，恒同可逆的invertible identity恒等式，单位元可逆的invertible identity恒等式，单位元可求长的rectifiable ill-condition病态可求长的rectifiable ill-condition病态可数的countable image像点，像可数的countable image像点，像可数的denumerable imaginary axis虚轴可数的denumerable imaginary axis虚轴可微的differentiable imbedding嵌入可微的differentiable imbedding嵌入可延拓的continuable imitation模仿，模拟可延拓的continuable imitation模仿，模拟空，零null immersion浸入空，零null immersion浸入亏格deficiency impulse function脉冲函数亏格deficiency impulse function脉冲函数亏格genus inclination斜角，倾角亏格genus inclination斜角，倾角亏量，缺陷defect inclined plane斜面亏量，缺陷defect inclined plane斜面扩张，外延extension inclusion包含扩张，外延extension inclusion包含类型，型，序型type incomparable不可比的类型，型，序型type incomparable不可比的棱，边edge incompatible不相容的，互斥的棱，边edge incompatible不相容的，互斥的棱柱prism inconsistent不成立的棱柱prism inconsistent不成立的棱锥，棱锥体pyramid indefiniteintegral 不定积分棱锥，棱锥体pyramid indefiniteintegral不定积分离散的discrete independence 无关（性），独立（性）离散的discrete independence无关（性），独立（性）离心率eccentricity index指数，指标离心率eccentricity index指数，指标李代数Lie algebra indivisible除不尽的李代数Lie algebra indivisible除不尽的理想ideal inductive归纳的理想ideal inductive归纳的立方根cubic root inductivedefinition归纳定义立方根cubic rootinductivedefinition归纳定义立体几何solid geometry induced诱导的立体几何solid geometry induced诱导的立体形solid figure inequality不等式立体形solid figure inequality不等式连通的connected inertia law惯性律连通的connected inertia law惯性律连通域connecteddomaininference推理，推论连通域connecteddomaininference推理，推论连续性continuity infimum下确界连续性continuity infimum下确界联合分布jointdistributioninfinite无穷大的联合分布jointdistributioninfinite无穷大的良序的well-ordered infinite decimal无穷小数良序的well-ordered infinite decimal无穷小数良序集normal set infinite series无穷级数良序集normal set infinite series无穷级数两两的pairwise infinitesimal无穷小的两两的pairwise infinitesimal无穷小的量，数量quantity inflection point拐点量，数量quantity inflection point拐点列表，造表tabulation information theory信息论列表，造表tabulation information theory信息论列秩columnar rank inhomogeneous非齐次的列秩columnar rank inhomogeneous非齐次的邻接adjacency injection内射邻接adjacency injection内射邻域neighborhood inner point内点邻域neighborhood inner point内点临界的critical instability不稳定临界的critical instability不稳定菱形diamond integer整数菱形diamond integer整数菱形lozenge integrable可积的菱形lozenge integrable可积的零zero integrand被积函数零zero integrand被积函数零点zeros integral积分零点zeros integral积分零化子annihilator intermediate value介值零化子annihilator intermediate value介值零因子nil-factor intersection交，相交零因子nil-factor intersection交，相交零因子zero divisor interval区间零因子zero divisor interval区间流线stream-line intrinsic内在的，内蕴的流线stream-line intrinsic内在的，内蕴的流形manifold invariant不变的流形manifold invariant不变的留数，残数residue inverse circularfunction反三角函数留数，残数residueinverse circularfunction反三角函数六边形hexagon inverse image逆像，原像六边形hexagon inverse image逆像，原像卵形线oval inversion反演卵形线oval inversion反演逻辑logic invertible可逆的逻辑logic invertible可逆的螺线spiral involution对合螺线spiral involution对合脉冲函数impulsefunctionirrational无理的，无理数脉冲函数impulsefunctionirrational无理的，无理数满射的surjective irreducible不可约的满射的surjective irreducible不可约的满同态epimorphism isolated point孤立点满同态epimorphism isolated point孤立点蔓叶线leaf isometric等距的蔓叶线leaf isometric等距的密度函数densityfunctionisomorphic同构的密度函数densityfunctionisomorphic同构的幂等的idempotent iteration迭代幂等的idempotent iteration迭代幂零的nilpotent joint distribution联合分布幂零的nilpotent joint distribution联合分布面face kernel核面face kernel核面积area keyword关键词面积area keyword关键词谬误fallacy knot纽结谬误fallacy knot纽结模，模数modulus known已知的模，模数modulus known已知的模仿，模拟imitation large sample大样本模仿，模拟imitation large sample大样本模糊的fuzzy last term末项模糊的fuzzy last term末项模型model lateral area侧面积模型model lateral area侧面积末项last term lattice格子末项last term lattice格子末行terminal row lattice point格点末行terminal row lattice point格点母体parentpopulationlaw of identity同一律母体parentpopulationlaw of identity同一律母线generatingline leadingcoefficient首项系数母线generatinglineleadingcoefficient首项系数内点inner point leaf蔓叶线内点inner point leaf蔓叶线内射injection least squaressolution最小二乘解内射injectionleast squaressolution最小二乘解内在的，内蕴的intrinsic lemma引理内在的，内蕴的intrinsic lemma引理挠度，挠率，变位deflection Lie algebra李代数挠度，挠率，变位deflection Lie algebra李代数挠率torsion lifting提升挠率torsion lifting提升挠曲，扭转twist likelihood似然的挠曲，扭转twist likelihood似然的拟范数quasi-norm limit极限拟范数quasi-norm limit极限拟合fitting linear combination线性组合拟合fitting linear combination线性组合拟群quasi-group linear filter线性滤波拟群quasi-group linear filter线性滤波拟正规quasi-normal linear fractiontransformation线性分拟正规quasi-normallinear fractiontransformation线性分逆时针的anticlockwise linear filter线性滤波逆时针的anticlockwise linear filter线性滤波逆像，原像inverse image式变换式变换逆像，原像inverse image式变换式变换纽结knot linear functional线性泛函纽结knot linear functional线性泛函偶然误差accidentialerrorlinear operator线性算子偶然误差accidentialerrorlinear operator线性算子偶数even number linearly dependent线性相关偶数even number linearly dependent线性相关排队论queuing theory linearlyindependent线性无关排队论queuing theorylinearlyindependent线性无关排列arrangement local coordinates局部坐标排列arrangement local coordinates局部坐标判别式discriminant locus(pl.loci)轨迹判别式discriminant locus(pl.loci)轨迹抛物线parabola logarithm对数抛物线parabola logarithm对数陪集coset lower bound下界陪集coset lower bound下界偏差，偏离departure logic逻辑偏差，偏离departure logic逻辑偏的，部分的partial lozenge菱形偏的，部分的partial lozenge菱形偏斜度skewness lunar新月型偏斜度skewness lunar新月型偏心的off-centre main diagonal主对角线偏心的off-centre main diagonal主对角线偏序partialorderingmanifold流形偏序partialorderingmanifold流形频数，频率frequency mantissa尾数频数，频率frequency mantissa尾数平旦空间flat space many-valuedfunction多值函数平旦空间flat spacemany-valuedfunction多值函数平衡equilibrium map into映入平衡equilibrium map into映入平衡方程balanceequationmap onto映到平衡方程balanceequationmap onto映到平角straight angle mapping映射平角straight angle mapping映射平均，中数mean marginal边缘平均，中数mean marginal边缘平面的planar master equation主方程平面的planar master equation主方程平面区域plane domain mathermaticalanalysis数学分析平面区域plane domainmathermaticalanalysis数学分析平面曲线plane curve mathematicalexpectation数学期望平面曲线plane curvemathematicalexpectation数学期望平面束plane pencil matrix(pl.matrices)矩阵平面束plane pencilmatrix(pl.matrices)矩阵平行parallel maximal极大的，最大的平行parallel maximal极大的，最大的平行四边形parallelogram maximum norm最大模平行四边形parallelogram maximum norm最大模平行线parallel lines mean平均，中数平行线parallel lines mean平均，中数平移translation measurable可测的平移translation measurable可测的剖分dissection measure测度剖分dissection measure测度普遍的，推广的generalized mesh网络普遍的，推广的generalized mesh网络谱spectrum metric space距离空间谱spectrum metric space距离空间谱论spectraltheory midpoint中点谱论spectraltheorymidpoint中点期望expectation minus减期望expectation minus减齐次的homogeneous minimal极小的，最小的齐次的homogeneous minimal极小的，最小的奇偶性odevity model模型奇偶性odevity model模型奇偶性pairity modulus模，模数奇偶性pairity modulus模，模数奇数odd number moment矩奇数odd number moment矩奇异值singularvalues monomorphism单一同态奇异值singularvaluesmonomorphism单一同态嵌入embed multi-analysis多元分析嵌入embed multi-analysis多元分析嵌入imbedding multiplication乘法嵌入imbedding multiplication乘法切割，分割excision multipole多极切割，分割excision multipole多极切空间tangent space mutual相互的切空间tangent space mutual相互的切向量tangent vector mutually disjoint互不相交切向量tangent vector mutually disjoint互不相交穷举的exhaustive natural boundary自然边界穷举的exhaustive natural boundary自然边界求导derivation naturalequivalence 自然等价求导derivation naturalequivalence自然等价球面，球形sphere natural number自然数球面，球形sphere natural number自然数区间interval natural period固有周期区间interval natural period固有周期区域region negative负的，否定的区域region negative负的，否定的区域，定义域domain neighborhood邻域区域，定义域domain neighborhood邻域曲率curvature nil-factor零因子曲率curvature nil-factor零因子曲线curve nilpotent幂零的曲线curve nilpotent幂零的趋势trend nodal节点的趋势trend nodal节点的取样sampling noncommutative非交换的取样sampling noncommutative非交换的去心邻域deletedneighborhoodnondense疏的，无处稠密的去心邻域deletedneighborhoodnondense疏的，无处稠密的全纯的holomorphic nonempty非空的全纯的holomorphic nonempty非空的全微分perfectdifferentialnoncountable不可数的全微分perfectdifferentialnoncountable不可数的全序集totallyordered setnonlinear非线性的全序集totallyordered setnonlinear非线性的权重，重量weight nonsingular非奇异的权重，重量weight nonsingular非奇异的扰动，摄动perturbation norm范数扰动，摄动perturbation norm范数容量capacity normal正规的，法线容量capacity normal正规的，法线容许的admissible normal derivative法向导数容许的admissible normal derivative法向导数容许的permissible normal direction法方向容许的permissible normal direction法方向容许误差admissibleerrornormaldistribution正态分布容许误差admissibleerrornormaldistribution正态分布入射角angle ofincidencenormal family正规族入射角angle ofincidencenormal family正规族。
The Canadian Board of Examiners for Professional Surveyors
January 2007 Note: this booklet is current as of January 2007 but some sections are currently under review and may be revised. Users should periodicallycheck the CBEPS website for updates.The Canadian Board of Examiners for Professional SurveyorsSCHEDULE IItem 1: MathematicsLearning OutcomesStudy Guide & Sample QuestionsItem 2: Least-Squares Estimation & Data AnalysisLearning OutcomesStudy Guide & Sample QuestionsItem 3: Advanced Surveying (including Survey Astronomy) Learning OutcomesStudy Guide & Sample QuestionsAnswers to Questions from Previous I-3 Examinations Item 4: Remote Sensing and Applied PhotogrammetryLearning OutcomesItem 5: Spatial Database Management Systems (Informatics) Item 6: Map Projections and CartographyItem 7: Cadastral StudiesLearning OutcomesStudy GuideSCHEDULE IIItem 1:Geodetic PositioningLearning OutcomesItem 2: Hydrographic Surveying and OceanographyItem 3:- Survey LawLearning OutcomesItem 4: Land Use Planning and Environmental Management Learning OutcomesItem 5: Land Information Systems and ManagementLearning OutcomesStudy Guide & Sample QuestionsNotes on Your Examination-TakingItem 6:- Business: Law, Administration and EconomicsLearning OutcomesStudy Guide-- SCHEDULE I --Schedule I - Item 1Functions, continuity and limits, Differentiation and applications, Integration and applications, Numerical integration, Plane curves, tangency and curvature, Sequences and series, Series expansions, Partial differentiation, Multiple integrals and numerical approximations, Vector operations and analytical geometry, First and second order linear differential equations and solutions, Introduction to matrix algebra, linear equations and transformations, vector spaces and geometry, quadratic forms, rotation matrices.Textbooks:a. Advanced Engineering Mathematics, 8th Edition, Kreyszig, E. (1992) JohnWiley & Sons, Toronto ISBN 0-471-55380-8 b. Calculus with Analytic Geometry, 5th Edition, Edwards, C.H. and Penney,D.E. (1998) Prentice Hall Inc. ISBN 0-13-736331-1References:a. Single-Variable Calculus, Stewart J., 4th Edition (1999), Brooks/ColePublishing Co., Pacific Grove, CA ISBN 0-534-35562-5 b. Calculus of a Single Variable,Early Transcendental Functions, Larson R.,Hostetler R. P., Edwards B.H., and Heyd, D. E., 2nd Edition, 1999, HoughtonMifflin Co. BostonISBN 0-395-93321-8 c. Introduction to Applied Mathematics, Strang, G. (1986) Wellesley-CambridgePressISBN 0-961-40880-4 Note: programmable calculators may be used in this examination.LEARNING OUTCOMES FOR SCHEDULE I – ITEM 1 Recommended Pre-examination Studies:First-year University MathematicsCalculus and Analytic GeometryLinear and Matrix AlgebraNumerical ComputationsLearning Outcomes:1.Functions, continuity and limits.• Nature of mathematical functions• Nature of continuity of a function at one point• Nature of mathematical limits2. Differentiation and applications.• Nature of differentiability of a function at one point• Differentiation of simple functions• Interpretations of derivatives of a function3. Integration and applications. Numerical integration.• Meaning of integration of a function• Integration of simple functions• Nature of indefinite and definite integrals• Numerical evaluation of definite integrals4. Plane curves, tangency and curvature.• Representations of plane curves• Direction of tangent to a curve at one point• Curvature of a curve at one point5. Sequences and series. Series expansions.• Nature of sequences and series• Convergence of sequences and series• Tests of convergence for sequences and series• Taylor series expansions for simple functions6. Partial differentiation.• Nature of partial differentiation• Partial differentiation of simple functions• Gradient and Laplacian operations7. Multiple integrals and numerical approximations.• Nature of multiple indefinite and definite integration• Numerical approximation techniques for multiple integrals8. Vector operations and analytical geometry.• Nature of real and complex vectors• Scalar and vector products of vectors• Analytical geometry in terms of vectors9. First and second order linear differential equations and solutions.• Nature of linear ordinary differential equations• Nature of linear partial differential equations• Solution methods for simple ordinary differential equations• Solution methods for simple partial differential equations10. Introduction to matrix algebra, linear equations and transformations.• Nature of matrices and simple matrix algebra• Matrix representation of linear algebraic equations and solutions• Matrix representation of linear transformations11. Vector spaces and geometry.• Nature of vector real and complex spaces• Geometry in real and complex spaces12. Quadratic forms, rotation matrices.• Nature of quadratic forms and applications• Rotation matrices in two and three dimensionsA STUDY GUIDE FOR SCHEDULE I – ITEM 11. Functions, continuity and limits.• Nature of mathematical functions• Nature of continuity of a function at one point• Nature of mathematical limitsSample Questions:a) What is a mathematical function? Give examples.b) Define the continuity of a function f(x) at some point x = x o.c) What is a discontinuity of a function f(x) at some point x = x o?d) Define the limit for a function f(x) as x approaches x o (i.e. as x x o)e) Distinguish between the limit from the left and the limit from the rightfor a function f(x) as x approaches x o on the real line.2. Differentiation and applications.• Nature of differentiability of a function at one point• Differentiation of simple functions• Interpretations of derivatives of a functionSample Questions:a) What is differentiability of function f(x) at some point x = x o?b) Distinguish between continuity and differentiability of f(x) at a point x = x o.c) Differentiate simple functions e.g. sin x, cosh x, e x, log e x, log10 xd) What are the 2nd, 3rd, … derivatives of the previous functions?e) For a function f(x), what is the interpretation of its derivatives f′(x), f″(x),…?3. Integration and applications. Numerical integration.• Meaning of integration of a function• Integration of simple functions• Nature of indefinite and definite integrals• Numerical evaluation of definite integralsSample Questions:a) What is the indefinite integral of a function f(x)? Give examples.b) What are definite integrals of the same function f(x)? Give examples.c) Evaluate the integral of f(x) = sin x between x = 0 and x = /2.d) What is the interpretation of the previous integral?e) Approximate the previous integral with a discretization over 0 x /2.4. Plane curves, tangency and curvature.• Representations of plane curves• Direction of tangent to a curve at one point• Curvature of a curve at one pointSample Questions:a) What is a plane curve? Give examples.b) What are implicit and parametric forms of a simple sinusoid?c) What are arclength, slope and curvature for a simple sinusoid?d) Given the function f(x) = sin x, what is the direction of tangent at some x = x o?e) Given the function f(x) = sin x, what is its curvature at some x = x o ?5. Sequences and series. Series expansions.• Nature of sequences and series• Convergence of sequences and series• Tests of convergence for sequences and series• Taylor series expansions for simple functionsSample Questions:a) What is the difference between a sequence and a series? Give examples.b) What is the limit of 1, 1/2, 1/3, … , 1/n, … as n ?c) How do verify that 1 + 1/2 + 1/3 + … + 1/n + … diverges?d) How do verify that 1 + 1/4 + 1/9 + … + 1/n2 + … converges?e) What is a Taylor expansion of a function f(x) = sin x at some point x = /2 ?6. Partial differentiation.• Nature of partial differentiation• Partial differentiation of simple functions• Gradient and Laplacian operationsSample Questions:a) What is a partial differentiation of some function f(x, y, z) ? Give examples.b) Given f(x, y, z) = e x sin(y+z), what are the corresponding partial derivatives?c) What is the total derivative of the previous f(x, y, z)?d) What is ∇f(x, y, z) for the previous f(x, y, z)?e) What is ∆f(x, y, z) ∇2f(x, y, z) for the previous f(x, y, z)?7. Multiple integrals and numerical approximations.• Nature of multiple indefinite and definite integration• Numerical approximation techniques for multiple integralsSample Questions:a) What are multiple indefinite and definite integrals of a function? Giveexamples.b) What is the area of the surface f(x, y) = sin xy inside 0 x /2 and 0 y/2 ?c) What is the volume under f(x, y) = sin xy inside 0 x /2 and 0 y /2 ?d) What are some methods for approximating those integrals by summations?e) What are quadrature methods? Give some simple examples.8. Vector operations and analytical geometry.• Nature of real and complex vectors• Scalar and vector products of vectors• Analytical geometry in terms of vectorsSample Questions:a) What are real and complex vectors? Give examples.b) Express real and complex vectors in cartesian and polar coordinates.c) What is the inner or scalar product of two arbitrary vectors? Give examples.d) What is the cross or vector product of two arbitrary vectors? Give examples.e) What is the length of a vector? What is the angle between two vectors?9. First and second order linear differential equations and solutions.• Nature of linear ordinary differential equations• Nature of linear partial differential equations• Solution methods for simple ordinary differential equations• Solution methods for simple partial differential equationsSample Questions:a) What is the difference between ordinary and partial differential equations?b) What are the orders in ordinary and partial differential equations? Giveexamples.c) What are the differential equations for f(x) = sin x ? g(x) = sinh x? h(x) = e x ?d) What are the solution methods for the previous differential equations?e) Solve the equations ∆U(x, y) = 0 and ∆V(x, y, z) = 0 for Cartesian x, yand z.10. Introduction to matrix algebra, linear equations and transformations.• Nature of matrices and simple matrix algebra• Matrix representation of linear algebraic equations and solutions• Matrix representation of linear transformationsSample Questions:a) What is a matrix? What is a column or row vector? Give examples.b) Given two matrices A and B, are A + B, AB, A2 + B2, (A + B)2 meaningful?c) What is the determinant of a square matrix? What is the matrix inverse?d) What are Cramer’s rule and Gaussian elimination for linear algebraicsystems?e) What is the matrix algebra of linear systems of equations? Give examples.11. Vector spaces and geometry.• Nature of vector real and complex spaces• Geometry in real and complex spacesSample Questions:a) What is a real vector space? What is a complex vector space? Giveexamples.b) What are linear dependence and independence of a set of real or complexvectors?c) What is a basis for a real or complex vector space? Give simple examples.d) What is the dimension of a real or complex vector space? Give simpleexamples.e) What is a subspace of a vector space? What is a projection onto asubspace?12. Quadratic forms, rotation matrices.• Nature of quadratic forms and applications• Rotation matrices in two and three dimensionsSample Questions:a) What is a quadratic form? What is it used for? Give examples.b) What are eigenvalues and eigenvectors for a square matrix?c) What are singular values for an arbitrary matrix?d) What is a singular value decomposition (SVD) for an arbitrary matrix?e) What are rotation matrices in two and three dimensions? Examples?Schedule I - Item 2Mathematical modelling; Error propagation and linearization, Concept of adjustment, Least-squares adjustment, Variance-covariance propagation, Pre-analysis of survey measurements, Normal, Chi-square, t (Student) and F distributions, Confidence intervals, Statistical testing of estimates, residuals and variances, Error ellipses and ellipsoids, General least-squares adjustment, Conditions and combined cases, Constraints, Sequential adjustment techniques, Least-squares applications to plane and curvilinear coordinates, Network design, Statistical testing and analysis of estimates, residuals and variances, Error detection, Introduction to least-squares prediction and filtering.Textbooks:a. Analysis and Adjustment of Survey Measurements, Mikhail, E.M. and Gracie,G. (1981) Reprints available from the Dept. of Geomatics Engineering, Universityof Calgary, Calgary, Alberta, ($15.00)b. Observations and Least-Squares, Mikhail, E.M. (1976). Reprints available fromthe Dept. of Geomatics Engineering, University of Calgary, Calgary, Alberta,($30.00)c. Adjustment Computations: Statistics and Least Squares in Surveying andGISWolf, P. R. and Ghilani, C. D. (1997); John Wiley and Sons, TorontoISBN 0-471-16833-5 References:a. Geodesy: The Concepts, Part III (Methodology), 2nd Edition, 1986, Vanicek,P. & Krakiwsky, E.J. North-Holland, New York. ISBN 0-444-87775-4ISBN 0-444-87777-0 b. Statistical and Data Analysis in Geology, Davis, J.C., (1986), 2nd Edition,John Wiley & Sons, Toronto ISBN 0-471-08079-9 Note: programmable calculators may be used in this examination.LEARNING OUTCOMES FOR SCHEDULE I – ITEM 2Recommended Pre-examination Studies:Introductory probability & statisticsSchedule 1 - Item 1 - MathematicsCourses or extensive experience in plane surveying (horizontal and vertical relative positioning, topographic surveying, route design and setting out and quantity calculations, familiarity with commonly used surveying instruments, procedures, and recording.)Learning Outcomes:1. Apply knowledge of matrix theory, statistics and estimation.• Manipulation of matrix algebra involved in adjustment of observations• Concept of linear system and linearization• Fundamental of probability and statistics• Principle of least square estimation and properties2. Analyze measurement errors and modeling, perform random errorpropagation and pre-analysis of survey measurements.• Types of errors and characteristics• Types of models and characteristics• Law of random error propagation, variance and covariance matrix• Pre-analysis of survey measurements3. Formulate least squares adjustment problems (condition, parametric andcombined cases).• Formulate parametric adjustment models (functional and stochastic)• Formulate condition adjustment models (functional and stochastic)• Formulate combined adjustment models (functional and stochastic)4. Derive adjustment equations of difference cases and conduct least squareadjustment to geomatics problems such as levelling, traverse, triangulation and trilateration networks.• Derive parametric adjustment equations• Derive condition adjustment equations• Derive combined adjustment equations• Application to geomatics problems such as levelling, traverse, triangulation and trilateration networks.5. Assess the quality of the adjustment solutions (variance factor, variance-covariance matrix, error ellipse).• Estimation of variance factor• Determination of variance-covariance matrix of parameters obtained from least square adjustment• Concept of error ellipse and its determination6. Perform statistical tests on mean and variance to detect and identify outliersin observations (normal, Chi-square, t Student and F distributions, statistical hypotheses, type I and II errors).• Perform statistical tests on mean and variance to detect and identify outliers in observations• Determine the confidence interval of adjusted parameters• Select appropriate testing methods (normal, Chi-square, t Student and F distributions)• Confidence level and error probability of statistical decisions (significance level, test power, type I and II errors)STUDY GUIDE FOR SCHEDULE I – ITEM 21. Apply Knowledge of matrix theory, statistics and estimation.• Manipulation of matrix algebra involved in adjustment of observations • Concept of linear system and linearization • Fundamental of probability and statistics• Principle of least square estimation and propertiesSample questions (e.g. see related chapters in “Analysis and Adjustment of Survey Measurements” by Mikhail and Gracie (1981))Define and explain briefly the following terms:• Expectation • Variance• Unbiasedness of an estimator • Redundancy of a linear system • Root mean square error• Null hypothesis and alternative hypothesis • Type I error and Type II error • Accuracy and precision • Confidence interval2. Analyze measurement errors and modeling, perform random error propagation and pre-analysis of survey measurements. • Types of errors and characteristics • Types of models and characteristics• Law of random error propagation, variance and covariance matrix • Pre-analysis of survey measurementsSample questions (e.g. see Chapter 2 and 6 of “Analysis and Adjustment ofSurvey Measurements” by Mikhail and Gracie (1981)) 1) Given a leveling network below where A and B are two control points with known height, 1h and 2h are two height difference measurements with standard deviation of 1σ and 2σ, respectively and 1σ = 2 2σ. Determine the value of 1σand 2σ so that the standard deviation of the height solution for point P using the least squares adjustment is equal to 2mm.B2) A surveyor’s error (one sigma) due only to reading is determined to be "5.1 when making observations with a particular instrument. After repeatedly pointing on a distant target with the same instrument, the surveyor determines the combined error due to both reading and pointing to be "6.2. What is the surveyor’s pointing error?3) Given the variance-covariance matrix of the measurement vector=21 :=32313132C and two functions of :21x += and 13y =, determine y x xy C ,C ,C7. Formulate least squares adjustment problems (condition, parametric and combinedcases).• Formulate parametric adjustment models (functional and stochastic) • Formulate condition adjustment models (functional and stochastic) • Formulate combined adjustment models (functional and stochastic)Sample questions (e.g. see Chapter 3 and 4 of “Analysis and Adjustment of SurveyMeasurements” by Mikhail and Gracie (1981))1) The angles shown in the following figure are measured with a theodolite, and their observed values and standard deviations are listed in the table. Formulate the functional and stochastic models to determine the adjusted values for these angles using the condition approach.1β2β3β4β5β6β1P 2P 3P 4PAngleObserved valueStandard Deviation1β 44o 50’44” "0..3 2β 46o 10’25” "5.1 3β 45o 55’12” "5.1 4β 43o 04’03” "5.1 5β 48o 32’45” "5.1 6β42o 27’42”"0..32) Assume that the points P1 and P2 in the problem 1) are two control points whose known coordinates are provided in the following table. Formulate the functional and stochastic models to determine the coordinates of the points P3 and P4 using parametric approach.Control Points X(m) Y(m)P1 0.00 0.00 P2 1000.00 0.008. Derive adjustment equations of different cases and conduct least square adjustment to geomatics problems such as levelling, traverse, triangulation and trilateration networks.• Derive parametric adjustment equations • Derive condition adjustment equations • Derive combined adjustment equations• Application to geomatics problems such as levelling, traverse, triangulation and trilateration networks.Sample questions (e.g. see Chapter 3 and 4 of “Analysis and Adjustment of SurveyMeasurements” by Mikhail and Gracie (1981))1) Given the following mathematical modelsC C 0)x , x ,(fC 0) x ,(f 221x 2122111 ==where i f , i x , i and i C represent mathematical model vectors, unknown parameter vectors, observation vectors and covariance matrices. a) Linearize the mathematical models b) Formulate the variation functionc) Derive the most expanded form of the least squares normal equation system.2) Given the angle measurements at a station along with their standard deviations:Angle Measurement Standard Deviation 134o 38’56” 6.7” 83o 17’35” 9.9”142o 03’14”4.3”Perform a least squares adjustment to the problem usinga) Conditional equations (condition adjustment) b) Observation equations (parametric adjustment)9. Assess the quality of the adjustment solutions (variance factor, variance-covariance matrix, error ellipse).• Estimation of variance factor• Determination of variance-covariance matrix of parameters obtained from least square adjustment• Concept of error ellipse and its determinationSample questions (e.g. see Chapter 6 and 8 of “Analysis and Adjustment of SurveyMeasurements” by Mikhail and Gracie (1981)) 1) Given the variance-covariance matrix of the horizontal coordinates (x, y) of a survey station, determine the semi-major, semi-minor axis and the orientation of the standard error ellipse associated with this station.σ=0.5100.0250.0250.380C 20x where 2cm 0=.2) Calculate the variance-covariance matrix and the standard deviations of the adjusted angles in the following network. Assume all angles are measured with the same precision ("0.1=σβ).1β2β3β4β5β6β1P 2P 3P 4P10. Perform statistical tests on mean and variance to detect and identify outliers inobservations (normal, Chi-square, t Student and F distributions, statistical hypotheses, type I and II errors).• Perform statistical tests on mean and variance to detect and identify outliers in observations• Determine the confidence interval of adjusted parameters• Select appropriate testing methods (normal, Chi-square, t Student and F distributions)• Confidence level and error probability of statistical decisions (significance level, test power, type I and II errors)Sample questions (e.g. see Chapter 8 of “Analysis and Adjustment of Survey Measurements” by Mikhail and Gracie (1981))1) An angle is measured 10 times. Each measurement is independent and madewith the same precision. The sample standard deviation is s = "7.3. Test at a significance level of 5% the hypothesis that the population standard deviation of the measurements is 2.0” against the alternative that is not 2.0”.2) A baseline of calibrated length (µ) 1153.00m is measured 5 times. Eachmeasurement is independent and made with the same precision. The sample mean (x) and sample standard deviation(s) are calculated from the measurements:x=1153.39m s = 0.06ma) Describe the major steps to test the mean value.b) Test at the 10% level of confidence if the measured distance is significantlydifferent from the calibrated distance.Schedule I - Item 3! " Precision horizontal and vertical control; simulation [pre-analysis] and design of surveys; error analysis of resultant data; application of optical and electromagnetic measuring principles and techniques; calibration of surveying instruments; utilization of surveying instruments; engineering surveys [including high precision, deformation, and mining surveys].The celestial sphere and its coordinate system; altitude and azimuth; declination, hour angle, right ascension; celestial latitude and longitude; time (sidereal, apparent, universal); equation of time; determination of azimuth; use of ephemerides, almanacs and star catalogues.Textbooks:a. Surveying: Theory and Practice, Anderson, J. M. and Mikhail, E. M. 7th Edition,1998 McGraw-Hill, New York ISBN 0-07-015914-9 b. Chapter 33 (“Modern Surveying Techniques for Mining and Civil Engineering.” byA.Chrzanowski) in J.A. Hudson [edit] Comprehensive Rock Engineering, v.3 RockTesting and Site Characterization. 1993, Pergammon Press, ISBN 0-08-042066-4 c. Chapter 16 (“Engineering and Mining Surveys.” by A. Chrzanowski) in G. McGrathand L.M. Sebert [edit] Mapping a Northern Land, 1947 -1994. 1999, McGill-Queen’s University Press, ISBN 0-7735-1689-1 d. Chapter 20 (“Mining Surveys” by A. Chrzanowski and A.J. Robinson) in Davis, R.E.,Foote, F.S., Anderson, J.M., and Mikhail, E.M. Surveying: Theory and Practice, 6th, 1981, McGraw-Hill, ISBN 0-07-015790-1 [out of print, replaced by 7th edition by Anderson and Mikhail, which no longer includes this chapter].e. Introduction to Geodetic Astronomy, Thomson, D. B., Lecture Notes 49, 1981,University of New Brunswick, Department of Geodesy and Geomatics Engineering, (ordering particulars through <http://gge.unb.ca/Pubs/Pubs.html>)f. The Star Almanac for Land Surveyors, (current year), Her Majesty's NauticalAlmanac Office, London, England, available through <>Copies of textbook items b., c. and d. can be obtained as a package entitled “Joint Boards Item I-3, Selected Textbook Material”, prepaid by cheque or postal money order for $ 10.00, [includes shipping by regular post and GST], payable to “Geodesy and Geomatics Engineering”. Contact: Publications, Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, NB, E3B 5A3 or via <delorey@unb.ca>.References:a. Elementary Surveying, Wolf, P.R. and Ghilani, C.D., 10th Edition, 2002, PrenticeHall Canada Inc. ISBN 0-321-01461-8b. The Astronomical Almanac, (current year), U.S. Naval Observatory, NauticalAlmanac Office, Washington, D.C. available through <> or <>Note: programmable calculators may be used in this examination.LEARNING OUTCOMES FOR SCHEDULE I – ITEM 3Recommended Pre-examination Studies:Schedule I - Item 1 - MathematicsSchedule I - Item 2 – Least-Squares Estimation and Data AnalysisCourses or extensive experience in plane surveying (horizontal and vertical relative positioning, topographic surveying, route design and setting out and quantity calculations, familiarity with commonly used surveying instruments, procedures, and recording.)Learning Outcomes:Design the appropriate combination of equipment and procedures for a data gathering task that will ensure that the gathered data meets the quality requirements of relative positioning (horizontally or vertically or in three dimensions simultaneously)Assess data gathered for the calibration of surveying instruments, for measurements of high precision, or for mining surveying.Gather, process, and assess astronomic observations for azimuth.Student should have an ability to:Discuss the concept of precision as it relates to surveying processes including the sources and types of errors.Translate specifications such as maximum misclosures into a choice of equipment and procedures.Compose specifications and requirements for gathering survey related data.Differentiate between the process that results in position information and the process that requires repeated positioning for local deformation monitoring.。
伍德里奇计量经济学导论第四版
15CHAPTER 3TEACHING NOTESFor undergraduates, I do not work through most of the derivations in this chapter, at least not in detail. Rather, I focus on interpreting the assumptions, which mostly concern the population. Other than random sampling, the only assumption that involves more than populationconsiderations is the assumption about no perfect collinearity, where the possibility of perfect collinearity in the sample (even if it does not occur in the population should be touched on. The more important issue is perfect collinearity in the population, but this is fairly easy to dispense with via examples. These come from my experiences with the kinds of model specification issues that beginners have trouble with.The comparison of simple and multiple regression estimates – based on the particular sample at hand, as opposed to their statistical properties – usually makes a strong impression. Sometimes I do not bother with the “partialling out” interpretation of multiple regression.As far as statistical properties, notice how I treat the problem of including an irrelevant variable: no separate derivation is needed, as the result follows form Theorem 3.1.I do like to derive the omitted variable bias in the simple case. This is not much more difficult than showing unbiasedness of OLS in the simple regression case under the first four Gauss-Markov assumptions. It is important to get the students thinking aboutthis problem early on, and before too many additional (unnecessary assumptions have been introduced.I have intentionally kept the discussion of multicollinearity to a minimum. This partly indicates my bias, but it also reflects reality. It is, of course, very important for students to understand the potential consequences of having highly correlated independent variables. But this is often beyond our control, except that we can ask less of our multiple regression analysis. If two or more explanatory variables are highly correlated in the sample, we should not expect to precisely estimate their ceteris paribus effects in the population.I find extensive treatments of multicollinearity, where one “tests” or somehow “solves” the multicollinearity problem, to be misleading, at best. Even the organization of some texts gives the impression that imperfect multicollinearity is somehow a violation of the Gauss-Markovassumptions: they include multicollinearity in a chapter or part of the book devoted to “violation of the basic assumptions,” or something like that. I have noticed that master’s students who have had some undergraduate econometrics are often confused on the multicollinearity issue. It is very important that students not confuse multicollinearity among the included explanatory variables in a regression model with the bias caused by omitting an important variable.I do not prove the Gauss-Markov theorem. Instead, I emphasize its implications. Sometimes, and certainly for advanced beginners, I put a special case of Problem 3.12 on a midterm exam, where I make a particular choice for the function g (x . Rather than have the students directly 课后答案网ww w.kh d aw .c om16compare the variances, they should appeal to the Gauss-Markov theorem for the superiority of OLS over any other linear, unbiased estimator.SOLUTIONS TO PROBLEMS3.1 (i Yes. Because of budget constraints, it makes sense that, the more siblings there are in a family, the less education any one child in the family has. To find the increase in the number of siblings that reduces predicted education by one year, we solve 1 = .094(Δsibs , so Δsibs = 1/.094 ≈ 10.6.(ii Holding sibs and feduc fixed, one more year of mother’s education implies .131 years more of predicted education. So if a mother has four more years of education, her son is predicted to have about a half a year (.524 more years of education. (iii Since the number of siblings is the same, but meduc and feduc are both different, the coefficientson meduc and feduc both need to be accounted for. The predicted difference in education between B and A is .131(4 + .210(4 = 1.364.3.2 (i hsperc is defined so that the smaller it is, the lower the student’s standing in high school. Everything else equal, the worse the student’s standing in high school, the lower is his/her expected college GPA. (ii Just plug these values into the equation:n colgpa= 1.392 − .0135(20 + .00148(1050 = 2.676.(iii The difference between A and B is simply 140 times the coefficient on sat , because hsperc is the same for both students. So A is predicted to have ascore .00148(140 ≈ .207 higher.(iv With hsperc fixed, n colgpaΔ = .00148Δsat . Now, we want to find Δsat such that n colgpaΔ = .5, so .5 = .00148(Δsat or Δsat = .5/(.00148 ≈ 338. Perhaps not surprisingly, a large ceteris paribus difference in SAT score – almost two and one-half standard deviations – is needed to obtain a predicted difference in college GPA or a half a point.3.3 (i A larger rank for a law school means that the school has less prestige; this lowers starting salaries. For example, a rank of 100 means there are 99 schools thought to be better.课后答案网ww w.kh d aw .c om17(ii 1β > 0, 2β > 0. Both LSAT and GPA are measures of the quality of the entering class. No matter where better students attend law school, we expect them to earn more, on average. 3β, 4β > 0. The numbe r of volumes in the law library and the tuition cost are both measures of the school quality. (Cost is less obvious than library volumes, but should reflect quality of the faculty, physical plant, and so on. (iii This is just the coefficient on GPA , multiplied by 100: 24.8%. (iv This is an elasticity: a one percent increase in library volumes implies a .095% increase in predicted median starting salary, other things equal. (v It is definitely better to attend a law school with a lower rank. If law school A has a ranking 20 less than law school B, the predicted difference in starting salary is 100(.0033(20 = 6.6% higher for law school A.3.4 (i If adults trade off sleep for work, more work implies less sleep (other things equal, so 1β < 0. (ii The signs of 2β and 3β are not obvious, at least to me. One could argue that more educated people like to get more out of life, and so, other things equal,they sleep less (2β < 0. The relationship between sleeping and age is more complicated than this model suggests, and economists are not in the best position to judge such things.(iii Since totwrk is in minutes, we must convert five hours into minutes: Δtotwrk = 5(60 = 300. Then sleep is predicted to fall by .148(300 = 44.4 minutes. For a week, 45 minutes less sleep is not an overwhelming change. (iv More education implies less predicted time sleeping, but the effect is quite small. If we assume the difference between college and high school is four years, the college graduate sleeps about 45 minutes less per week, other things equal. (v Not surprisingly, the three explanatory variables explain only about 11.3% of the variation in sleep . One important factor in the error term is general health. Another is marital status, and whether the person has children. Health (however we measure that, marital status, and number and ages of children would generally be correlated with totwrk . (For example, less healthy people would tend to work less.3.5 Conditioning on the outcomes of the explanatory variables, we have 1E(θ =E(1ˆβ + 2ˆβ = E(1ˆβ+ E(2ˆβ = β1 + β2 = 1θ.3.6 (i No. By definition, study + sleep + work + leisure = 168. Therefore, if we change study , we must change at least one of the other categories so that the sum is still 168. 课后答案网ww w.kh d aw .c om18(ii From part (i, we can write, say, study as a perfect linear function of the otherindependent variables: study = 168 − sleep − work − leisure . This holds for every observation, so MLR.3 violated. (iii Simply drop one of the independent variables, say leisure :GPA = 0β + 1βstudy + 2βsleep + 3βwork + u .Now, for example, 1β is interpreted as the change in GPA when study increases by one hour, where sleep , work , and u are all held fixed. If we are holding sleep and work fixed but increasing study by one hour, then we must be reducing leisure by one hour. The other slope parameters have a similar interpretation.3.7 We can use Table 3.2. By definition, 2β > 0, and by assumption, Corr(x 1,x 2 < 0.Therefore, there is a negative bias in 1β: E(1β < 1β. This means that, on average across different random samples, the simple regression estimator underestimates the effect of thetraining program. It is even possible that E(1β is negative even though 1β > 0.3.8 Only (ii, omitting an important variable, can cause bias, and this is true only when the omitted variable is correlated with the included explanatory variables. The homoskedasticity assumption, MLR.5, played no role in showing that the OLS estimators are unbiased.(Homoskedasticity was used to o btain the usual variance formulas for the ˆjβ. Further, the degree of collinearity between the explanatory variables in the sample, even if it is reflected in a correlation as high as .95, does not affect the Gauss-Markov assumptions. Only if there is a perfect linear relationship among two or more explanatory variables is MLR.3 violated.3.9 (i Because 1x is highly correlated with 2x and 3x , and these latter variables have largepartial effects on y , the simple and multiple regression coefficients on 1x can differ by largeamounts. We have not done this case explicitly, but given equation (3.46 and the discussion with a single omitted variable, the intuition is pretty straightforward.(ii Here we would expect 1β and 1ˆβ to be similar (subject, of course, to what we mean by “almost uncorrelated”. The amount of correlation between 2x and 3x does not directly effect the multiple regression estimate on 1x if 1x is essentially uncorrelated with 2x and 3x .(iii In this case we are (unnecessarily introducing multicollinearity into the regression: 2x and 3x have small partial effects on y and yet 2x and 3x are highly correlated with 1x . Adding2x and 3x like increases the standard error of the coefficient on 1x substantially, so se(1ˆβis likely to be much larger than se(1β . 课后答案网ww w.kh d aw .c om19(iv In this case, adding 2x and 3x will decrease the residual variance without causingmuch collinearity (because 1x is almost uncorrelated with 2x and 3x , so we should see se(1ˆβ smaller than se(1β. The amount of correlation between 2x and 3x does not directly affect se(1ˆβ.3.10 From equation (3.22 we have111211ˆ,ˆni ii ni i r yr β===∑∑where the 1ˆi rare defined in the problem. As usual, we must plug in the true model for y i : 1011223311211ˆ(.ˆni i i i ii ni i r x x x u r βββββ==++++=∑∑The numerator of this expression simplifies because 11ˆni i r=∑ = 0, 121ˆni i i r x =∑ = 0, and 111ˆni i i r x =∑ = 211ˆni i r =∑. These all follow from the fact that the 1ˆi rare the residuals from the regression of 1i x on 2i x : the 1ˆi rhave zero sample average and are uncorrelated in sample with 2i x . So the numerator of 1βcan be expressed as2113131111ˆˆˆ.n n ni i i i i i i i rr x r u ββ===++∑∑∑Putting these back over the denominator gives 13111113221111ˆˆ.ˆˆnni i ii i nni i i i r x rur r βββ=====++∑∑∑∑课后答案网ww w.kh d aw .c om20Conditional on all sample values on x 1, x 2, and x 3, only the last term is random due to its dependence on u i . But E(u i = 0, and so131113211ˆE(=+,ˆni i i ni i r xr βββ==∑∑which is what we wanted to show. Notice that the term multiplying 3β is the regressioncoefficient from the simple regression of x i 3 on 1ˆi r.3.11 (i 1β < 0 because more pollution can be expected to lower housing values; note that 1β isthe elasticity of price with respect to nox . 2β is probably positive because rooms roughlymeasures the size of a house. (However, it does not allow us to distinguish homes where each room is large from homes where each room is small. (ii If we assume that rooms increases with quality of the home, then log(nox and rooms are negatively correlated when poorer neighborhoods have more pollution, something that is often true. We can use Ta ble 3.2 to determine the direction of the bias. If 2β > 0 andCorr(x 1,x 2 < 0, the simple regression estimator 1βhas a downward bias. But because 1β < 0, this means that the simple regression, on average, overstates the importance of pollution. [E(1β is more negative than 1β.] (iii This is what we expect from the typical sample based on our analysis in part (ii. The simple regression estimate, −1.043, is more negative (larger in magnitude than the multiple regression estimate, −.718. As those estimates are only for one sample, we can never know which is closer to 1β. But if this is a “typical” sample, 1β is closer to −.718.3.12 (i For notational simplicity, define s zx = 1(;ni i i z z x =−∑ this is not quite the samplecovariance between z and x because we do not divide by n – 1, but we are only using it tosimplify notation. Then we can write 1β as11(.niii zxz z ys β=−=∑This is clearly a linear function of the y i : take the weights to be w i = (z i −z /s zx . To show unbiasedness, as usual we plug y i = 0β + 1βx i + u i into this equation, and simplify: 课后答案网w w w .k h d aw .c o m21 11 1 011 111(( (((n ii i i zxnni zx i ii i zxniii zxz z x u s z z s z z u s zz u s ββββββ====−++=−++−=−=+∑∑∑∑where we use the fact that 1(ni i z z =−∑ = 0 always. Now s zx is a function of the z i and x i and theexpected value of each u i is zero conditional on all z i and x i in the sample. Therefore, conditional on these values,1111(E(E(niii zxz z u s βββ=−=+=∑because E(u i = 0 for all i . (ii From the fourth equation in part (i we have (again conditional on the z i and x i in the sample,2111222212Var ((Var(Var((n ni i i i i i zx zxnii zxz z u z z u s s z z s βσ===⎡⎤−−⎢⎥⎣⎦==−=∑∑∑because of the homoskedasticit y assumption [Var(u i = σ2 for all i ]. Given the definition of s zx , this is what we wanted to show.课后答案网ww w.kh d aw .c om22(iii We know that Var(1ˆβ = σ2/21[(].ni i x x =−∑ Now we can rearrange the inequality in the hint, drop x from the sample covariance, and cancel n -1everywhere, to get 221[(]/ni zx i z z s =−∑ ≥211/[(].ni i x x =−∑ When we multiply through by σ2 we get Var(1β ≥ Var(1ˆβ, which is what we wanted to show.3.13 (i The shares, by definition, add to one. If we do not omit one of the shares then the equation would suffer from perfect multicollinearity. The parameters would not have a ceteris paribus interpretation, as it is impossible to change one share while holding all of the other shares fixed. (ii Because each share is a proportion (and can be at most one, when all other shares are zero, it makes little sense to increase share p by one unit. If share p increases by .01 – which is equivalent to a one percentage point increase in the share of property taxes in total revenue – holding share I , share S , and the other factorsfixed, then growth increases by 1β(.01. With the other shares fixed, the excluded share, share F , must fall by .01 when share p increases by .01.SOLUTIONS TO COMPUTER EXERCISESC3.1 (i Prob ably 2β > 0, as more income typically means better nutrition for the mother and better prenatal care. (ii On the one hand, an increase in income generally increases the consumption of a good, and cigs and faminc could be positively correlated. On the other, family incomes are also higher for families with more education, and more education and cigarette smoking tend to benegatively correlated. The sample correlation between cigs and faminc is about −.173, indicating a negative correlation.(iii The regressions without and with faminc aren 119.77.514bwghtcigs =−21,388,.023n R ==and n 116.97.463.093bwghtcigs faminc =−+21,388,.030.n R ==课后答案网ww w.kh d aw .c om23The effect of cigarette smoking is slightly smaller when faminc is added to the regression, but the difference is not great. This is due to the fact that cigs and faminc are not very correlated, and the coefficient on faminc is practically small. (The variable faminc is measured in thousands, so $10,000 more in 1988 income increases predicted birth weight by only .93 ounces.C3.2 (i The estimated equation isn 19.32.12815.20price sqrft bdrms =−++288,.632n R ==(ii Holding square footage constant, n price Δ = 15.20 ,bdrms Δ and so n price increases by 15.20, which means $15,200.(iii Now n price Δ = .128sqrft Δ + 15.20bdrms Δ = .128(140 + 15.20 = 33.12, or $33,120. Because the size of the house is increasing, this is a much larger effect than in (ii. (iv About 63.2%. (v The predicted price is –19.32 + .128(2,438 + 15.20(4 = 353.544, or $353,544. (vi From part (v, the estimated value of the home based only on square footage and number of bedrooms is $353,544. The actual selling price was $300,000, which suggests the buyer underpaid by some margin. But, of course, there are many other features of a house (some that we cannot even measure that affect price, and we have not controlled for these.C3.3 (i The constant elasticity equation isn log( 4.62.162log(.107log(salary sales mktval =++ 2177,.299.n R ==(ii We cannot include profits in logarithmic form because profits are negative for nine of the companies in the sample. When we add it in levels form we getn log( 4.69.161log(.098log(.000036salary sales mktval profits =+++2177,.299.n R ==The coefficient on profits is very small. Here, profits are measured in millions, so if profits increase by $1 billion, which means profits Δ = 1,000 – a huge change – predicted salaryincreases by about only 3.6%. However, remember that we are holding sales and market value fixed.课后答案网ww w.kh d aw .c om24Together, these variables (and we could drop profits without losing anything explain almost 30% of the sample variation in log(salary . This is certainly not “most” of the variation.(iii Adding ceoten to the equation givesn log( 4.56.162log(.102log(.000029.012salary sales mktval profits ceoten =++++2177,.318.n R ==This means that one more year as CEO increases predicted salary by about 1.2%. (iv The sample correlation between log(mktval and profits is about .78, which is fairly high. As we know, this causes no bias in the OLS estimators, although it can cause their variances to be large. Given the fairly substantial correlation between market value andfirm profits, it is not too surprising that the latter adds nothing to explaining CEO salaries. Also, profits is a short term measure of how the firm is doing while mktval is based on past, current, and expected future profitability.C3.4 (i The minimum, maximum, and average values for these three variables are given in the table below:Variable Average Minimum Maximum atndrte priGPA ACT 81.71 2.59 22.516.25 .86131003.93 32(ii The estimated equation isn 75.7017.26 1.72atndrtepriGPA ACT =+− n = 680, R 2 = .291.The intercept means that, for a student whose prior GPA is zero and ACT score is zero, the predicted attendance rate is 75.7%. But this is clearly not an interesting segment of thepopulation. (In fact, there are no students in the college population with priGPA = 0 and ACT = 0, or with values even close to zero. (iii The coefficient on priGPA means that, if a student’s prior GPA is one point higher (say, from 2.0 to 3.0, the attendance rate is about 17.3 percentage points higher. This holds ACT fixed. The negative coefficient on ACT is, perhaps initially a bit surprising. Five more points on the ACT is predicted to lower attendance by 8.6 percentage points at a given level of priGPA . As priGPAmeasures performance in college (and, at least partially, could reflect, past attendance rates, while ACT is a measure of potential in college, it appears that students that had more promise (which could mean more innate ability think they can get by with missing lectures. 课后答案网ww w.kh d aw .c om(iv We have atndrte = 75.70 + 17.267(3.65 –1.72(20 ≈ 104.3. Of course, a student cannot have higher than a 100% attendance rate. Getting predictions like this is always possible when using regression methods for dependent variables with natural upper or lower bounds. In practice, we would predict a 100% attendance rate for this student. (In fact, this student had an actual attendance rate of 87.5%. (v The difference in predicted attendance rates for A and B is 17.26(3.1 − 2.1 − (21 − 26 = 25.86. C3.5 The regression of educ on exper and tenure yields n = 526, R2 = .101. ˆ Now, when we regres s log(wage on r1 we obtain ˆ log( wage = 1.62 + .092 r1 n = 526, R2 = .207. (ii The slope coefficientfrom log(wage on educ is β1 = .05984. ˆ ˆ (iv We have β1 + δ 1 β 2 = .03912 +3.53383(.00586 ≈ .05983, which is very close to .05984; the small difference is due to rounding error. C3.7 (i The results of the regression are math10 = −20.36 + 6.23log(expend − .305 lnchprg 课 (iii The slope coefficients from log(wage on educ and IQ are ˆ = .03912 and β = .00586, respectively. ˆ β1 2 后答案 C3.6 (i The slope coefficient from the regression IQ on educ is (rounded to five decimal places δ1 = 3.53383. n = 408, R2 = .180. 25 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 网ˆ As expected, the coefficient on r1 in the second regression is identical to the coefficient on educ in equation (3.19. Notice that the R-squared from the above regression is below that in (3.19. ˆ In effect, the regression of log(wage on r1 explains log(wage using only the part of educ that is uncorrelated with exper and tenure; separate effects of exper and tenure are not included. ww w. kh da w. co m ˆ educ = 13.57 − .074 exper + .048 ten ure + r1 .The signs of the estimated slopes imply that more spending increases the pass rate (holding lnchprg fixed and a higher poverty rate (proxied well by lnchprg decreases the pass rate (holding spending fixed. These are what we expect. (ii As usual, the estimated intercept is the predicted value of the dependent variable when all regressors are set to zero. Setting lnchprg = 0 makes sense, as there are schools with low poverty rates. Setting log(expend = 0 does not make sense, because it is the same as setting expend = 1, and spending is measured in dollars per student. Presumably this is well outside any sensible range. Not surprisingly, the prediction of a −20 pass rate is nonsensical. (iii The simple regression results are failing to account for the poverty rate leads to an overestimate of the effect of spending. C3.8 (i The average of prpblck is .113 with standarddeviation .182; the average of income is 47,053.78 with standard deviation 13,179.29. It is evident that prpblck is a proportion and that income is measured in dollars. (ii The results from the OLS regression are psoda = .956 + .115 prpblck + .0000016 income 后 If, say, prpblck increases by .10 (ten percentage points, the price of soda is estimated toincrease by .0115 dollars, or about 1.2 cents. While this does not seem large, there are communities with no black population and others that are almost all black, in which case the difference in psoda is estimated to be almost 11.5 cents. (iii The simple regression estimate on prpblck is .065, so the simple regression estimate is actually lower. This is because prpblck and income are negatively correlated (-.43 and income has a positive coefficient in the multiple regression. (iv To get a constant elasticity, income should be in logarithmic form. I estimate the constant elasticity model: 26 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 课答案 n = 401, R2 = .064. 网ww ˆ (v We can use equation (3.23. Because Corr(x1,x2 < 0, which means δ1 < 0 , and β 2 < 0 , ˆ the simple regression estimate, β , is larger than the multiple regression estimate, β . Intuitively, 1 w. kh (iv The sample correl ation between lexpend and lnchprg is about −.19 , which means that, on average, high schools with poorer students spent less per student. This makes sense, especially in 1993 in Michigan, where school funding was essentially determined by local property tax collections. da w. n = 408, R2 = .030 and the estimated spending effect is larger than it was in part (i –almost double. co 1 m math10 = −69.34 + 11.16 log(expendlog( psoda = −.794 + .122 prpblck + .077 log(income n = 401, R2 = .068. If prpblck increases by .20, log(psoda is estimated to increase by .20(.122 = .0244, or about 2.44 percent. ˆ (v β prpblck falls to about .073 when prppov is added to the regression. (vi The correlation is about −.84 , which makes sense because poverty rates are determined by income (but not directly in terms of median income. (vii There is no argument that they are highly correlated, but we are using them simply as controls to determine if the is price discrimination against blacks. In order to isolate the pure discrimination effect, we need to control for as many measures of income as we can; including both variables makes sense. C3.9 (i The estimated equation is (iv The estimated equation is gift = −7.33 + 1.20 mailsyear − .261 giftlast + 16.20 propresp + .527 avggift Aft er controlling for the average past gift level, the effect of mailings becomes even smaller: 1.20 guilders, or less thanhalf the effect estimated by simple regression. (v After controlling for the average of past gifts – which we can view as measuring the “typical” generosity of the person and is positively related to the current gift level – we find that the current gift amount is negatively related to the most recent gift. A negative relationship makes some sense, as people might follow a large donation with a smaller one. 27 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 课 n = 4,268, R 2 = .2005 后 (iii Because propresp is a proportion, it makes little sense to increase it by one. Such an increase can happen only if propresp goes from zero to one. Instead, consider a .10 increase in propresp, which means a 10 percentage point increase. Then, gift i s estimated to be 15.36(.1 ≈ 1.54 guilders higher. 答案 (ii Holding giftlast and propresp fixed, one more mailing per year is estimated to increase gifts by 2.17 guilders. The simple regression estimate is 2.65, so the multiple regression estimate is somewhat smaller. Remember, the simple regression estimate holds no other factors fixed. 网 ww The R-squared is now about .083, compared with about .014 for the simple regression case. Therefore, the variables giftlast and propresp help to explain significantly more variation in gifts in the sample (although still just over eight percent. w. n = 4,268, R 2= .0834 kh gift = −4.55 + 2.17 mailsyear + .0059 giftlast + 15.36 propresp da w. co m。
一类含p-Laplace算子的奇异拟线性问题解的多重性
u
∈ W01,
p
(Ω)
时，
u p ∇u p ∈ L1 (Ω) 不一定成立。为解决这一困难，参考文献[10]中的方法，我们进行变量替换 v = g −1 (u ) ，
g 定义为
g (s) = −g (−s) s ∈ (−∞,0],
( )
= g ′ ( s )
1
( ) 1+ 2 p−1 g s p 1 p
引理 2.3：函数 s → ( g (s))−γ g′(s), s > 0 满足下列性质： 1) ( g (s))−γ g′(s) 在 (0,∞) 递减； 2) lim ( g (s))−γ g′(s) = +∞ ；
s→0+
3) 存在常数 C1 > 0 使得对所有 s ≥ 1 ，成立 0 < ( g (s))−γ g′(s) ≤ C1 。
Received: Mar. 7th, 2021; accepted: Mar. 29th, 2021; published: Apr. 9th, 2021
Abstract
In this paper we combine sub-supersolution technique and minimax methods to study the existence and multiplicity of solutions for a class of singular quasilinear elliptic equations. Firstly, by suitable hypotheses on the nonlinearity term and singular term, we obtain the existence of non-
概率方法解拟线性偏微分方程
_v gIBOg Z> e
w
eh (2.4) .{* 0. a1=
s t
u(s, ytx (s)) = u(t, x) +
ux (r, ytx (r))σdW (r).
(2.7)
ZF .{ u(s, ytx (s)), s ∈ [t, 0] P1 3 Ll{ t < 0, a > 0, x ∈ R +f
2
|u(t, x)| ≤ u0 . (2.3) (2.1), (2.2)
w5f WL]vU) >_` ]Kj p t < 0, x ∈ R 3p p (−∞, 0) × R |Pl{D ut , ux
ut (t, x) + f (u(t, x))ux (t, x) +
|6U) (1.2) > uxx +3p9LdF
oi hg"}z~ u0 tv_ f x
; f%s
Abstract It is proved by stochastic analysis method that if u0 is bounded and measurable and f > 0 then the solution of Cauchy problem of the quasilinear parabolic 2 partial differential equation u0 + f (u)ux + σ 2 uxx = 0, u(0, x) = u0 (x) converges to the entropy solution of the quasilinear partial differential equation u0 + f (u)ux = 0, u(0, x) = u0 (x), as σ → 0, that is the validity of “vanishing viscosity” method for the equation. This is the well known result due to Oleinik by using difference method in 1975. The result of this paper suggests the importance of probabilistic method for solving nonlinear partial differential equations. Keywords Quasilinear partial differential equation, Entropy condition, Stochastic partial differential equation, Itˆ o formula, Bounded variation function 1991 MR Subject Classification 60H15, 35F25, 35K10 Chinese Library Classification O211.63, O175.29
代数英语
(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。
0+||zero-dagger; 读作零正。
1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。
AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。
BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。
B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。
C0 类函数||function of class C0; 又称“连续函数类”。
CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。
Cp统计量||Cp-statisticC。
Linear_differential_equations_pdf_
Pre-requisites:throughout this chapter,the following basic properties and notions are assumed to be known:∙derivatives and antiderivatives;∙usual functions and in particular,their derivatives;∙complex numbers;roots of a second degree polynomial.Contents1General definitions and standard vocabulary2 2First order linear differential equations32.1Exponential functions and their characterization (4)2.2Structure of the set of solutions and sum principle (6)2.3Solutions to the associated homogeneous equation (7)2.4Variation of parameters (8)2.5Solution satisfying a given initial condition:existence and uniqueness (14)3Second order linear differential equations with constant coefficients153.1Definition and structure of the set of solutions (15)3.2Solutions to the associated homogeneous equation (16)3.3Finding a particular solution when the second member is of type exponential-polynomial (20)3.4Variation of parameters (22)3.5Solution satisfying a given initial condition:existence and uniqueness (24)1General definitions and standard vocabularyThroughout this chapter,unless specified otherwise, will be used to state results that are valid with either =ℝor =ℂ.Definition 1.0.1Let be a positive integer.A differential equation of order is an equation in which the unknown is a function (with domain (to be determined)and (at least) times differentiable on )and of the form:( ): ( , , ′,⋅⋅⋅, ( −1), ( ))=0where for each ∈[∣0, ]∣, ( )is the -th derivative of and is a function of ( +2)variables,and is not constant with respect to the last variable.Example 1.0.1You have probably already studied a number of differential equations,in particular during your physics class.Here a few examples of differential equations:∙ ′+2 =0is a (linear)differential equation of order 1(setting ( , , ′)= ′+2 );∙ ′( )− ( )− 2=0is a (linear)differential equation or order 1;∙ ′=1+ 2is a also a differential equation of order 1(but non linear);∙ ′′+ 2 =0is a (linear)differential equation of order 2;∙ (6)− (3)+2 2=cos( )is a (non linear)differential equation of order 6.Remark:the condition “which is not constant with respect to the last variable”is a technical condition.It is there to guarantee that the equation is really of order .For instance,if we set: ( , , , )= + then the differential equation ( , , ′, ′′)=0is + ′=0,which is really of order 1and not 2.Definition 1.0.2We say a function with domain (a non trivial interval)is a solution to ( )if is at least times differentiable on and:∀ ∈ , ( , ( ), ′( ),⋅⋅⋅, ( )( ))=0Therefore,a solution to the differential equation ( )is really a couple ( , ),where is a real interval and a function at least times differentiable on .Example 1.0.2The function with domain [0;1]given by ( )=exp( )for all ∈[0;1]is a solution of ′= .However,it is clearly not the only one as we could define for any interval , ( )=exp( )for ∈ which would also be a solution to the same differential equation,but on .This is why,in our search for solutions,we focus on solutions with maximal domain.Definition 1.0.3To solve the differential equation ( )is to find all maximal solutions,i.e.find all solutions whose domain’s are maximal (that is,there is no solution to ( )with domain such that ⊊ and ∣ = ).Example 1.0.3In our previous example,the maximal solutions of ′= are the functions with domain ℝgiven by ( )= exp( )for all real number .Obviously,there are a number of differential equations we do not know how to solve explicitly.This is where numerical analysis can be very convenient:there are many methods to give approximations of solutions,even though we do not know an exact form.We will study this later in the year.However,there are one kind of differential equations for which,under“reasonable conditions”,we always know how to solve explicitly:namely,linear equations.The object of the next sections is to study these equations,but only when their order is either1or2.Before we do so,we will give a formal definition of what a linear differential equation actually is.Definition1.0.4A differential equation is said to be linear if it is of the form: ( )− =0,where is the unknown, is a function and is a linear map,i.e. satisfies:( 1+ 2)= ( 1)+ ( 2)and ( 1)= ( 1)for all“unknowns” 1and 2and all scalar constants ∈ .In that case,we say that:∙ is the second member of( );∙the differential equation( ): ( )=0is the homogeneous equation associated to( ).Remark:it is often convenient to group both properties in the definition of a linear map.More specifically, is a linear map if and only if,for all unknowns 1, 2and all , ∈ , ( 1+ 2)= 1+ 2.Example1.0.4Setting ( )= ′− ,we have for all unknowns 1and 2,and all , ∈ :( 1+ 2)=( 1+ 2)′−( 1+ 2)= ′1+ ′2− 1− 2= ( ′1− 1)+ ( ′2− 2)= ( 1)+ ( 2)Therefore, ′− =0is indeed a linear differential equation.Example1.0.5We stated earlier that ′=1+ 2is not a linear differential equation.Indeed,we have here: ′− 2=1 so that ( )= ′− 2and =1(constant function).However, (2 )=2 ′−4 2=2( ′− 2)if =0.Therefore, is not a linear map(the condition must be verified for all unknowns).2First order linear differential equationsDefinition2.0.5A linear differential equation of order1is a differential equation of the form:( ) ′( )+ ( ) ( )= ( )where , and are three functions defined on a real interval .We will only work withfirst order linear differential equations which are“resolved”in ′,i.e.of the form:′( )+ ( ) ( )= ( )Remark:often,we write ′+ ( ) = ( )which theoretically is abusive.The proper forms would either be ′( )+ ( ) ( )= ( )for all in a certain interval or ′+ = (equality amongst functions).However,this abuse is currently accepted and we will always write the equations in that form.Proposition2.0.1Afirst order linear differential equation is indeed a linear differential equation.Hence,the name.Proof:To begin,we observe that the equation can be written in the form: ( )= ,where =and ( )= ( ) ′+ ( ) (again,we should write ( ): −→ ( ) ′( )+ ( ) ( )).Now,let 1and 2be two unknowns and let , be two scalars in .We then have:( 1+ 2)= ( )( 1+ 2)′+ ( )( 1+ 2)= ( )( ′1+ ′2)+ ( )( 1+ 2)= ( ( ) ′1+ ( ) 1)+ ( ( ) ′2+ ( ) 2)= ( 1)+ ( 2)Thus, is a linear map and the equation is indeed linear.⊠2.1Exponential functions and their characterizationTheorem2.1.1Let ∈ .Then the function :ℝ−→−→exp( )is the only(maximal)solution of ′=that satisfies the initial condition (0)=1.Proof:Suppose that is a solution of ′= that verifies (0)=1and let ( )= ( )exp(− ).Then is differentiable onℝ(product of two functions that are)and for all real number ,′( )= ′( )exp(− )− ( )exp(− )=exp(− )( ′( )− ( ))=0Hence, is constant onℝ.Furthermore, (0)= (0)exp(0)=1×1=1so that for all ∈ℝ,( )exp(− )=1.Multiplying both sides by exp( )=0,wefind that: ( )=exp( )forall real number .This proves that if is a solution of ′= that satisfies (0)=1then= .Conversely,one can easily check that satisfies ′ = and (0)=1.⊠Remark:this is one way of characterizing the exponential functions(real or complex)as the only solutions of ′= with initial condition (0)=1.Theorem2.1.2(Second characterization)The only functions differentiable onℝthat verify:∀ , ∈ℝ, ( + )= ( )× ( )(∗)are either exponential functions with ∈ℂor identically zero.Proof:⇐=The identically zero function and exponential functions obviously are differentiable onℝand verify(∗).=⇒Let be a function which is differentiable onℝand verifies the functional equation(∗).Next,let ∈ℝand consider::ℝ−→−→ ( + )− ( ) ( )As is differentiable on ℝ, is also.Furthermore,by hypothesis, and hence, ′are identically zero.Therefore,by differentiating:∀ ∈ℝ, ′( + )= ′( ) ( )Setting =0,we have: ′( )= ′(0) ( ),i.e. ′= with = ′(0).We now consider two possibilities:∙either is identically zero (and the result follows);∙else there is at least one real number 0such that ( 0)=0.Setting = 0and =0in (∗),we have: ( 0)= ( 0+0)= (0)× ( 0).However,by hypothesis, ( 0)=0;hence, (0)=1.We have therefore shown that satisfies ′= and (0)=1.According to our last theorem, = for some ∈ℂ.⊠Remark:one can actually show that the conditions we imposed are a bit strong.Indeed,we need not suppose differentiable.The hypothesis continuous on ℝis sufficient,and actually,even continuous at one point is enough.Exercise 2.1.1Let be a function that verifies (∗).Prove that:(1) is identically zero or does not vanish at all and (0)=1;(2)if is continuous at (with ∈ℝ),then is continuous on ℝ;(3)if is not identically zero,then there exists >0such that∫( )d =0and that therefore, is differentiableon ℝ.Remark:this second characterization of exponential functions is very useful.For instance,this is how one can prove that continuous random variable without memory: ( > + ∣ > )= ( > )for all , ⩾0has necessarily an exponential distribution.Remark:we can also mention a few applications of these theorems to Physics:∙radioactive decay:take for instance plutonium isotope Pu-239.Physics laws state that if ( )is the number ofradioactive atoms (or otherwise the mass (multiplying by the molar mass)),then the activity -given by =−dd-is proportional to .This leads to the relation −d d = .Here,ln 2is what we call the half-life,i.e.thetime required for half the atoms in a sample of radioactive material to decay.∙Newton’s law of cooling states that the rate of change in the temperature of an object is proportional to the difference between the object’s temperature and the temperature of the surrounding medium,i.e.dd=ℎ( 0− )where is the temperature of the object, 0the temperature of the surrounding medium,andℎis what we call the Newton coefficient.2.2Structure of the set of solutions and sum principleTheorem2.2.1Let( )be a linear differential equation,( ): ( )= with a linear map.Assume that is a particular solution of( ).Then for any differentiable function , is a solution of( )if and only if − is a solution of( ).In other words,setting (respectively( ))the set of solutions of( )(respectively( )),one has:= +Proof:Let be any differentiable function.Then,is a solution of( )⇐⇒ ( )=⇐⇒ ( )= ( )(because is a solution of(E))⇐⇒ ( )− ( )=0⇐⇒ ( − )=0(because is linear)⇐⇒ − is a solution of( )⊠this fundamental theorem has a very important practical signification.In order to solve alinear differential equation,one mustfind one solution(what we call a particular solution)and then add all the solutions to the homogeneous equation.Remark:this is why in the next paragraphs,we will focus on how to solve the homogeneous equation and then how to find a particular solution.Remark:for your information,we will say that is an affine space-that is the sum of“a point”( )and a“vector space”( ).Affine spaces and vector spaces are fundamental algebraic structures which we will study next year. Proposition2.2.1(Sum principle)Let be a linear map, 1be a solution of ( )= 1and 2a solution of ( )= 2,where 1and 2are two functions.Then:(i) 1+ 2is a solution of ( )= 1+ 2;(ii)for all ∈ , 1is a solution of ( )= 1.Proof:All these properties are direct consequences of the fact that is linear.Indeed,( 1+ 2)= ( 1)+ ( 2)= 1+ 2and( 1)= ( 1)= 1⊠Example2.2.1Say we want to solve the differential equation( ): ′− =cos( )+3 2 .We then proceed in three steps:∙Step1:wefind a particular solution of ′− =cos( ).∙Step2:wefind a particular solution of ′− = 2 .∙Step3:we solve the homogeneous equation(here the solutions are −→ )We can therefore conclude that the solutions of( )are the functions:= 1+3 2particular solutionby the sum principle +solutions of( )There are now two questions that arise naturally:1.How do we solve the homogeneous equation?2.How do wefind a particular solution if there is no evident one?That’s what we will now focus on.2.3Solutions to the associated homogeneous equationReminder(which we have already reviewed in the chapter on usual functions)Definition2.3.1Let be a function.We say that a function is an antiderivative of on the real interval if is differentiable on and ′( )= ( )for all in .Theorem2.3.1Any continuous function on an interval has antiderivatives.Furthermore,if ∈ ,then −→∫( )d is the only antiderivative of on which vanishes at .Proposition2.3.1Let be a continuous function on the real interval (with values in )and consider the differential equation( ): ′+ ( ) =0.Let be any antiderivative of on .Then the solutions of( )are the functions with domain given by:∀ ∈ , ( )= exp(− ( )),with ∈Proof:First,we notice that since we assumed to be continuous on , does indeed have an-tiderivatives.Therefore, exists.Next,let be any function and set = exp( ).As theexponential function(real or complex)does not vanish,we have:= exp( )⇐⇒ = exp(− )Hence, is differentiable on if and only if is.We then have:∈ ⇐⇒∀ ∈ , ′( )+ ( ) ( )=0⇐⇒∀ ∈ , ′( )exp(− ( ))− ( ) ( )exp−( ( ))+ ( ) ( )exp(− ( ))=0⇐⇒∀ ∈ , ′( )exp(− ( ))=0⇐⇒∀ ∈ , ′( )=0⇐⇒∃ ∈ ,∀ ∈ , ( )=⇐⇒∃ ∈ ,∀ ∈ , ( )= exp(− ( ))⊠Remark:please note that this result holds for =ℂas well;and also,that this proposition states that the maximal solutions are all defined on !Example2.3.1Consider the following differential equations:∙ ′= where ∈ℂ.We already know from the characterization of exponential functions that the solutions are ( )= (0)exp( ).Setting this aside,( )can be written as ′− =0and if we apply the previous proposition,we know that −→− is continuous onℝand −→ is an antiderivative onℝ.Therefore,the solutions are of the form:( )= exp( ),with ∈We do indeedfind the same solutions,which is reassuring!∙( ): ′+11+ 2=0. −→11+ 2is continuous onℝand has −→arctan( )for antiderivative.Therefore,the solutions of( )are the functions defined on by:∀ ∈ , ( )= exp(−arctan( )), ∈Remark:solving a homogeneous differential equation of order one basically boils down tofinding an antiderivative of a given function.Proposition2.3.2With the previous notations,(i)solutions of( ),other than zero,do not vanish on ;(ii) ={ − , ∈ }is a“one-dimensional vector space”or a“line”:all elements of are“proportional”(“collinear”)to − .Proof:(ii)is simply a formal way of writing the set of solutions.As for( ),let be a solution of( ).Then by the previous proposition,there is a constant ∈ such that = exp(− ).Moreover,we know that the exponential never vanishes;therefore,if ( 0)=0for some0∈ ,then =0and =0(identically zero).⊠2.4Variation of parametersNow we consider the full equation( ): ′+ ( ) = ( ).Let 0be a non-zero solution of( )(( 0)is a basis of ).From the previous paragraph,we know that the solutions of the homogeneous equation( )are of the form 0,with ∈ .We are therefore going to try tofind a particular solution of the same form,but by“variation of the parameter ”,i.e.of the form:( )= ( ) 0( )By( )in the previous proposition,we know that for all ∈ , 0( )=0;hence,= 0⇐⇒ = 0and therefore, is differentiable on if and only if is.We then deduce that for any differentiable function : = 0; ′= ′ 0+ ′0et ′+ = ′ 0+( ′+ )0( 0solution of( ))= ′ 0Hence,is a solution of( )⇐⇒ ′ 0= ⇐⇒ ′= 0We can now state our result formally:Theorem 2.4.1Consider the differential equation ( ): ′+ ( ) = ( ),where and are two continuous functions on .Let be an antiderivative of and an antiderivative of → ( ) ( ).Then a particular solution of ( )is given by:∀ ∈ , ( )= ( ) − ( )In particular,if 0∈ ,then the solutions of ( )aregiven by:∀ ∈ , ( )=(∫( ) ( )d ) − ( )+ − ( )Proof:We choose 0=exp(− ).The previous calculations show that:is a solution of ( )⇐⇒ ′= exp( )⇐⇒= +Substituting in = exp(− )yields the results.⊠△!Caution:beware!All the theorems we stated apply for equations which are “resolved”in ′.If they are not,i.e.of the form ( ) ′+ ( ) = ( ),then one most solve the equation on intervals on which does not vanish,wherewe can write ′+ ( ) ( ) = ( )( ).Example 2.4.1Consider a resistor and a capacity mounted in series with a generator delivering a constant tension .We know that the tension (or voltage)then satisfies the following differential equation:( 1):dd+ = As = =0,( 1)is equivalent tod d + =.We now apply our method.∙Solutions to the associated homogeneous equation:−→1 is continuous on ℝand has −→for antiderivative.Therefore,the solutions to ( 1, )are thefunctions given by:∀ ∈ℝ, ( )= exp(−), ∈ℝ∙Search for a particular solution:we could simply apply the variation of parameters method.However,it is much simpler here to search for a trivial solution.The second member is constant:we therefore try to find a constant solution,i.e.such that dd =0.This yields: = .∙ConclusionThis proves that the tension is of the form:∀ ∈[0;+∞[, ( )= + −.We can then determine the value of by using the initial condition.Example2.4.2Consider the equation( 2): ′− =cos( ).∙Solutions to the associated homogeneous equation:It is clear that the solutions to the homogeneous equation are given by: ( )= for all ∈ℝ,with ∈ℝ.∙Search for a particular solution:–First method:variation of parametersApplying the variation of parameters,we set for all inℝ, ( )= ( ) where is a differentiable function onℝ.Then:is a solution of( 2)⇐⇒∀ ∈ℝ, ′ ( ) =cos( )⇐⇒∀ ∈ℝ, ( )= − cos( )We must therefore determine an antiderivative onℝof −→ − cos( ).To do this,we can,for example, use integration by parts twice:∫ 0 − cos( )d =[− − cos( )]−∫(− − )(−sin( ))d=− − cos( )+1−[− − sin( )]−∫− cos( )dThus,2∫− cos( )d =− − cos( )+ − sin( )+1So that −→ −2(sin( )−cos( ))is an antiderivative of −→ − cos( ).Hence,one particular solution of( 2)is given by:∀ ∈ℝ, ( )=12(sin( )−cos( ))–Second method:using complex numbersConsider the new equation:( ): ′− = .If wefind a particular solution ,thenℜ ( )will be a particular solution of( 2)1.Next,we try tofind a solution of the form = with ∈ℂ(for more details,see section3.3).We then have the following equivalences:solution of( )⇐⇒∀ ∈ℝ, − =⇐⇒( −1) =1⇐⇒ =1 −1⇐⇒ =− −12Thus,a particular solution of( 2)is given by:for all ∈ℝ,( )=ℜ (−1+2)=−12cos( )+12sin( )–Third method:trying tofind a trivial solutionGiven the second member,it might seem reasonable to look for a solution of the form: −→ cos( )+ sin( ).∙ConclusionWe can now conclude that the solutions to( 2)are given by:∀ ∈ℝ, ( )= −2sin( )− −2cos( )+ − , ∈ℝ1Careful!This method works here because the coefficients are all real!Otherwise,we could not say thatℜ ( ′− )=ℜ ( )′−ℜ ( ).Example2.4.3Consider( 3): ′+1+ 2=11+ 2.∙Step1:solutions to the associated homogeneous equation−→1+ 2is continuous onℝand has −→12ln(1+ 2)for antiderivative.Thus, ={ℝ−→ℝ−→ exp(−12ln(1+ 2)), ∈ℝ}={−→√1+ 2, ∈ℝ}∙Step2:search for a particular solutionApplying the previous theorem,we know that a particular solution is of the form: −→( )√1+ 2where is anantiderivative of −→ ( ) − ( )=1√1+ 2.Here,we can choose =Argsh which leads to:∀ ∈ℝ, ( )=Argsh( )√1+ 2∙ConclusionThis shows that the solutions to the equation( 3)are the functions given by:∀ ∈ℝ, ( )=Argsh( )+√1+ 2,with ∈ℝExercise2.4.1Solve the differential equation: ′+ = 3.Finally,to conclude this paragraph,we will study one example of equation which is not“resolved”in ′and see how one goes about determining maximal solutions.Example2.4.4We wish to solve the differential equation:( ):(1− 2) ′−2 =sin( ).First,notice that1− 2=0if and only if =±1.Therefore,even though( )is defined onℝ,we cannot simply apply our theorems onℝ.We must solve the equation on each interval where1− 2does not vanish,i.e.on 1=]−∞;−1[, 2=]−1;1[and 3=]1;+∞[,and then determine whether or nor,there are any solutions onℝ.∙Step1:Solutions to( )on each interval where1− 2does not vanishLet ∈{1,2,3}.We then have:solution of( )on ⇐⇒ ′+22−1=sin( )1− 2–Solutions to the associated homogeneous equation−→22−1is continuous on and −→ln∣ 2−1∣is an antiderivative of that function on .Therefore,the solutions to( )(the homogeneous equation)are given by:∀ ∈ , ( )= exp (−ln∣ 2−1∣)=∣ 2−1∣, ∈ℝFurthermore,on the interval , 2−1has constant sign.We can thus remove the absolute value and incorporate the sign of 2−1in the sign of the constant .In other words,the solutions to( )are ofthe form:∀ ∈ , ( )=1− 2, ∈ℝ△!Caution:it is extremely important to realize that the parameter that wefind solving the equation depends of the interval on which we are solving the equation.Thus,the notation to indicate this fact.–Search for a particular solutionApplying the variation of parameter method,we set ( )= ( )1−for in ,with a differentiablefunction on .Then,is a solution of( )on ⇐⇒∀ ∈ , ′( )1−=sin( )1−⇐⇒∀ ∈ , ′ ( )=sin( ) Hence,a particular solution of( )on is given by:∀ ∈ , ( )=−cos( ) 1− 2–ConclusionThis proves that the solution of( )on are the functions of the form:∀ ∈ , ( )= −cos( )1− 2,with ∈ℝRemark:we actually could have solved( )much faster by noticing that(1− 2) ′−2 =((1− 2) )′so that( )is equivalent to(1− 2) =−cos( )+ .∙Solving( )completely–Necessary conditionsSuppose is a solution of( )(i.e.a solution onℝ).Then,its restriction to each interval (for ∈{1,2,3}) is a solution of( )on .According to what we have just shown,there are three real constants 1, 2and3such that:( )=⎧⎨⎩1−cos( )1− 2if ∈]−∞;−1[ 2−cos( )1−if ∈]−1;1[ 3−cos( )1− 2if ∈]1;+∞[Also,setting =±1in the equation,we see that (±1)=−sin(1)2so that in fact,if is solution of( )onℝ,then necessarily:( )=⎧⎨⎩1−cos( )1−if ∈]−∞;−1[−sin(1)2if =−12−cos( )1− 2if ∈]−1;1[−sin(1)2if =13−cos( )1− 2if ∈]1;+∞[Furthermore,if is a solution of( ),we know that necessarily, is differentiable at±1,hence continuous.Moreover,lim→−1<−1(1− 2)=0−and lim→−1<−1( 1−cos( ))= 1−cos(1).Therefore,for to be continuousfrom the left at−1,it is necessary that 1−cos(1)=0,i.e. 1=cos(1).Proceeding in the same way,one shows that for to be continuous(period)at−1and1,one must have 1= 2= 3=cos(1).Therefore, we have shown that if a solution to( )exists,it is necessarily given by:( )=⎧⎨⎩cos(1)−cos( )1− 2if =±1−sin(1)2if =±1–Search for sufficient conditionsAssume that is defined onℝby the previous piecewise expression.We wish to determine whether or not is differentiable onℝand solution of( ).First,it is clear that is differentiable onℝ∖{−1,1}and solution of( )on each interval]−∞;−1[,]−1;1[ and]1;+∞[.We must therefore study the differentiability at±1:by construction,if is differentiable at ±1,then will automatically satisfy the equation for =±1.Next,let be any real number such that∣ ∣=1.Then,( )=cos(1)−cos( )(1− )(1+ )=−12×sin(−12)−12×sin(+12)+12Setting ( )=⎧⎨⎩sin( )if =01if =0,we see that in fact,for all real number (including =±1):( )=−12(−12)× (+12)Thus,to prove the differentiability of at ±1,it is sufficient to prove that is differentiable on ℝ.I leave that fact as an exercise for you to do 2.∙ConclusionWe have proved that the differential equation ( )has one and only one solution on ℝwhich is given by:∀ ∈ℝ, ( )=⎧ ⎨ ⎩cos(1)−cos( )1− 2if =±1−sin(1)2if =±1Important facts to remember from this example:(1)when solving a first order differentiable equation which isn’t “resolved”in ′,we cannot directly apply ourtheorems (notice that we found only one solution as opposed to an infinite number of solutions);(2)in this case,it is important to remember to solve on each interval where the function (coefficient in frontof ′)does not vanish,and that the constants (or parameters)we find depend on the interval;(3)then comes the big piece of work:determining necessary conditions on the constants so that a function canbe a solution on the full interval,and then,conversely,checking that such a function is indeed differentiable and a solution to the differentiable equation.(4)note that these equations require a whole lot more work then the other form....2.5Solution satisfying a given initial condition:existence and uniquenessTheorem 2.5.1Consider the differential equation ( ): ′+ ( ) = ( ),where and are two continuous functions on .Let 0∈ and let 0be any scalar in .Then the equation ( )has one and only one solution satisfying the initial condition ( 0)= 0.Furthermore,we can give an exact expression for this solution:∀ ∈ , ( )=( 0+∫ 0( ) ∫ 0 ( )d d )−∫ 0 ( )d Proof:Let be a solution of ( ).Consider : −→ given by ( )=∫( )d so that isthe only antiderivative of on that vanishes at 0.By theorem 2.4.1,we know that thereexists a scalar ∈ such that:∀ ∈ , ( )=(∫ 0 ( ) ( )d ) − ( )+ − ( )Thus,( 0)= 0⇐⇒(∫ 0 0 ( ) ( )d ) − ( 0)+ − ( 0)= 0⇐⇒= 0⊠2hint:start by proving that for any non-negative , − 36⩽sin( )⩽ .3Second order linear differential equations with constant coefficients3.1Definition and structure of the set of solutionsDefinition3.1.1A second order linear differential equation with constant coefficients is by definition a differential equation of the form:( ): ′′+ ′+ = ( )where , , are three scalars in such that =0and a function.Proposition3.1.1A second order linear differential equation with constant coefficients is indeed a linear differ-ential equation.Proof:Set ( )= ′′+ ′+ ;then( )is equivalent to ( )= .Now we show that is alinear map.Let 1, 2be two functions twice differentiable on a real interval and let ,[]be two scalars in .Then:( 1+ 2)= ( 1+ 2)′′+ ( 1+ 2)′+ ( 1+ 2)= ( ′′1+ ′′2)+ ( ′1+ ′2)+ ( 1+ 2)= ( ′′1+ ′1+ 1)+ ( ′′2+ ′2+ 2)= ( 1)+ ( 2)This proves that is a linear map and therefore,that( )is a linear differential equation.⊠As for the structure of the set of solutions,if you read the proof of theorem2.2.1and proposition2.2.1,you can easily see that they do not depend on the order of the differential equation,but solely on the fact that it is linear.Hence, they also apply for second order linear differential equations:Theorem3.1.1Let( )be a linear differential equation,( ): ( )= with a linear map.Assume that is a particular solution of( ).Then for any differentiable function , is a solution of( )if and only if − is a solution of( ).In other words,setting (respectively( ))the set of solutions of( )(respectively( )),one has:= +Proposition3.1.2(Sum principle)Let be a linear map, 1be a solution of ( )= 1and 2a solution of ( )= 2,where 1and 2are two functions.Then:(i) 1+ 2is a solution of ( )= 1+ 2;(ii)for all ∈ , 1is a solution of ( )= 1.Once again,we are now left with two problems:finding the solutions to the homogeneous equation andfinding a particular solution.3.2Solutions to the associated homogeneous equationWe now focus on the associated homogeneous equation:( ): ′′+ ′+ =0We know that for linear differential equations of order 1with constant coefficients,the solutions of the homogeneous equation are exponentials.Also,we know that when differentiating an exponential −→exp( )(for ∈ ),we get a function which is proportional to the same exponential.Therefore,it is perfectly logical to try to find solutions to the homogeneous equation of the form: ( )= with ∈ .Let ∈ .The function is at least twice differentiable on ℝ, ′ = and ′′ = 2 .Thus,the followingpropositions are equivalent:is a solution of ( )⇐⇒′′ + ′ + =0⇐⇒2 + + =0⇐⇒ ×( 2+ + )=0Considering the fact that the exponential function is never zero (or does not vanish),we find that:is a solution of ( )⇐⇒ 2+ + =0Definition 3.2.1The characteristic polynomial associated to ( )(or to ( ))is the second degree polynomial:( )= 2+ +One also says that 2+ + =0is the characteristic equation of ( )or of ( ).Our previous results show that is a solution of ( )if and only if ( )=0.Therefore,the number of different exponentials which will be solutions depends of the number of roots that has in .Hence,one must distinguish two possibilities.∙First case: has at least one root inLet 1be a root of in .Relations between roots and coefficients show that has another root 2(eventually 2= 1is a root with multiplicity 2)that verifies 1+ 2=−.Next,let be any function twice differentiable on ℝ.Copying our variation of parameter,we set: ( )= ( ) − 1 ,or equivalently, ( )= ( ) 1 .Then is twice differentiable on ℝas well and for all real number ,′( )= ′( ) 1( )+ ( ) 1 1( )′′( )= ′′( ) 1( )+2 1 ′( ) 1( )+ 21 ( ) 1( )Consequently,collecting terms and replacing in the equation,we find that:is a solution of ( )⇐⇒ 1×( ′′+2 1 ′+ 21+ ′+ 1 + )=0⇐⇒ ′′+(2 1+ ) ′+( 21+ 1+ ) =0Furthermore,by definition, 1is a root of so that ( 1)=0,i.e 21+ 1+ =0.Also,substituting =− ( 1+ 2)we find that 2 1+ = ( 1− 2)and thus:is solution of ( )⇐⇒∀ ∈ℝ, ′′+ ( 1− 2) ′=0⇐⇒∀ ∈ℝ, ′′+( 1− 2) ′=0⇐⇒∀ ∈ℝ, ′( )= ( 2− 1) ,for some ∈We must now find ,i.e.find antiderivatives of −→ ( 2− 1) .which again introduces two separate cases:First sub-case: 2= 1(i.e. has two distinct roots in )。
数学英语词汇(微积分)
微积分——Differential and integral calculus第一章函数——Chapter 1. Function邻域 Neighborhoodneighborhood开邻域 Open函数 Functionfunction；One-valued function；Monotropic function；单值函数 Monodromefunction；Many-value function；Multivalued function多值函数 Multiform分段 Piecewisefunction分段函数 Piecewisefunction显函数 Explicit定义域Domain of definition复合 Composition；function复合函数 Composite反函数 Inversefunction单调 Monotone单调性 Monotonicity单调序列 Monotonesequence单调数列Monotone sequence of numbersfunction单调函数 Monotonicincreasing；Monotonic increasing单调增加 Monotone单调增加函数Monotonically increasing functiondecreasing；Monotonic decreasing单调减少 Monotone单调减少函数Monotonically decreasing functionmonotone严格单调 Strictly严格单调递减Strictly monotone decreasing严格单调递增Strictly monotone increasing有界的 Boundedbound上界 Upperbound下界 Lower上有界的 Boundedabovebelow下有界的 Bounded上确界 Supremumlowerbound下确界 Greatestvariable有界变量 Bounded奇偶性 Odevityfunction偶函数 Even奇函数 Oddfunctionfunction狄利克雷函数 Dirichlet’sfunction初等函数 Elementary基本函数 Fundamentalfunctionfunction基本初等函数 Fundamentalelementary指数 Exponentfunction指数函数 Exponential对数 Logarithmfunction对数函数 Logarithmfunctioncircular反三角函数 Inverse第二章极限与连续——Chapter 2. Limit and continuity序列 Sequencenumbersof数列 Sequence极限 Limit单侧 One-sided单侧极限 One-sidedlimit；Unilateral limitlimit左极限 Left收敛 Converge收敛性 Convergence发散 Diverge趋于零 Vanishingquantity无穷小量 Infinitelysmall无穷小的阶Order of infinitesimaldivisor零因子 Null无穷大量Infinitely large quantityinfinity正无穷大 Plusinfinity负无穷大 Minuslimit无穷极限 Infinite无穷大的阶Order of infinity单调收敛定理Monotone convergence theorem复利 Compoundinterest连续性 Continuity；Property of continuitycontinuous；Continuity from the left左连续 Leftcontinuous；Continuity from the right右连续 Rightpoint连续点 Continuitypoint不连续点 Discontinuitydiscontinuityof间断点 Point第一类不连续点Discontinuity point of the first kind第二类不连续点Discontinuity point of the second kinddiscontinuity可去间断点 Removablepoint；Moving singularitysingular可去奇点 Movablediscontinuity无穷间断点 Infinitediscontinuity跳跃间断点 Jumpvalue最大值 Greatesttheoremvalue介值定理 Intermediatepoint零点 Null第三章导数与微分——Chapter 3. Derivative and differential瞬时 Instantaneousspeed瞬时速度 Instantaneous导数 Derivativederivative单侧导数 One-sidedderivative；Derivative on the left左导数 Leftderivative；Derivative on the right；Progressive derivative右导数 Rightrule链式法则 Chain尖点 Cusp曲线的尖点Cusp of a curveorder高阶 Higherderivative；Higher derivative高阶导数 Higher-order微分 Differential可微的 Differentiablefunction不可微函数 Non-differentiableoperation微分运算 Differential求微分 Differentiatecoefficient微商 Differentialdifferential；Higher differential；Differentials of higher order 高阶微分 Higher-orderderivative相对导数 Relative弹性 Elasticity；Resilience第四章微分中值定理与导数应用——Chapter 4. differential mid-value theorem and application of derivative 中值 Mid-valuetheorem罗尔定理 Rolle’s未定式 Indeterminateformvalue；Extremal极值 Extreme极值点 Extremepointmaximum局部极大值 Localminimum局部极小值 Local凹 Concave凸 Convexpoint；knee拐点 Inflection渐近线 Asymptote；Asymptotic lineasymptote水平渐近线 Horizontalasymptote铅垂渐近线 Vertical第五章不定积分——Chapter 5. Indefinite integralfunction原函数 Primary积分 Integralintegral不定积分 Indefinite求积分 Quadrature可积的 Integrable曲线簇Family of curve积分法 Integration代换 Substitution换元积分法Integration by substitution分部积分法Integration by parts部分分式积分法Integration by partial fraction部分分式展开Partial fraction expansion第六章定积分——Chapter 6. Definite integralintegral定积分 Definite积分中值Integral mean value牛顿-莱布尼兹公式 Newton-Leibniz’sformulaofrotation旋转体 Bodyintegral广义积分 ImproperfunctionΓ函数 Gamma第七章级数——Chapter 7. Series级数 Seriesseries无穷级数 Infinite级数的项Term of a series部分和 Partialsumseries发散级数 Divergentseries振荡级数 Oscillatory几何级数 Geometricseries；Geometrical seriesseries正项级数 Positivetermseries调和级数 Harmoniccriterion达朗贝尔准则 D’Alembert’s绝对收敛 Absolutelyconvergent绝对收敛级数Absolutely convergent seriesconvergent条件收敛 Conditional条件收敛级数Conditional convergent series函数级数Series of functionsseries幂级数 Powerradius收敛半径 Convergencedomain；Region of convergence收敛域 Convergenceconvergence收敛区间 Intervalof逐项微分Term by term differentiationexpansion级数展开 Series幂级数展开Power series expansion；Expansion into power-seriesseries泰勒级数 Taylorformula泰勒公式 Taylor’sseries马克劳林级数 Maclaurin第八章多元函数微分学——Chapter 8. Differential for function of many variables 有界区域 Boundedregionregion开区域 Openregion连通区域 Connect不连通的 Disconnectedpoint；Interior point内点 Innerpoint外点 Outerpoint边界点 Frontiersurface柱面 Cylindrical母线 Generator柱的母线Element of cylinder；Generator of cylinder椭球 Ellipsoidcylinder椭圆柱面 Ellipticcylinder抛物柱面 Parabolic双曲面 Hyperboloid单叶双曲面Hyperboloid of one sheet双叶双曲面Hyperboloid of two sheets鞍点 Saddle齐次 Homogeneousfunction齐次函数 Homogeneouslimit；Two-limit二重极限 Double多重极限 Multiplelimitslimitn重极限 n-foldlimits累次极限 Repeatedderivative偏导数 Partialdifferential偏微分 Partialdifferential；Total differentiation全微分 Total高阶偏导数Higher-order partial derivative混合偏导数Mixed partial derivativecondition约束条件 Constraintequation拉格朗日方程 Lagrange拉格朗日乘数 Lagrangemultiplier第九章二重积分——Chapter 9. Double integral积分区域Domain of integrationelement；Differential of area面积元素 Areaintegral二重积分 Doubleintegral多重积分 Multipleintegral；Repeated integral累次积分 Iterated雅可比行列式 Jacobian广义二重积分 Improperintegraldouble第十章微分方程——Chapter 10. Differential equationequation微分方程 Differential常微分 Ordinarydifferentialequationdifferential常微分方程 Ordinary偏微分方程Partial differential equation微分方程的阶Order of a differential equation一阶微分方程Differential equation of first orderequationdifferential线性微分方程 Linearcondition初始条件 Initialvalue；Original value；Starting value初始值 Initialsolution特解 Particularequation齐次方程 Homogeneousequation 齐次微分方程 Homogeneousdifferential齐次线性微分方程Homogeneous linear differential equation非齐次的 Non-homogeneousequation非齐次微分方程 Non-homogeneousdifferential非齐次线性微分方程Non-homogeneous linear differential equation非线性的 Nonlinearequation非线性方程 Nonlinear常数变异法Method of variation of constantsequation；Differential equation of higher orderdifferential高阶微分方程 Higher-order高阶常微分方程Higher-order ordinary differential equation高阶偏微分方程Higher-order partial differential equationequation特征方程 Characteristicroot特征根 Characteristicequationintegral微分积分方程 Differention第十一章差分方程——Chapter 11. Difference equation差、差分 Differencedifference高阶差分 Higherequation差分方程 Difference。
数学学院硕士研究生课程内容简介
数学与统计学院硕士研究生课程内容简介学科基础课-------------------- 泛函分析--------------------课程编号：121020202001 课程类别：学科基础课课程名称：泛函分析英文译名：Functional Analysis学时：60学时学分：3学分开课学期：1 开课形式：课堂讲授考核形式：闭卷考试适用学科：基础数学、应用数学、运筹与控制论、课程与教学论授课单位及教师梯队：数学与统计学院，基础数学系教师。
内容简介：本课程介绍紧算子与Fredholm算子、抽象函数简介、Banach代数的基本知识、C*代数、Hilbert 空间上的正常算子、无界正常算子的谱分解、自伴扩张、无界算子序列的收敛性、算子半群、抽象空间常微分方程。
主要教材：张恭庆、郭懋正：《泛函分析讲义》(下册)，北京大学出版社，1990年版。
参考书目（文献）：1．定光桂：《巴拿赫空间引论》，科学出版社，1984年版。
2．M. Reed, B. Simon, Methods of Modern Mathematical Physics I, Functional Analysis, 1972.3．K. Y osida, Functional Analysis, Sixth Edition, 1980.4．张恭庆、林源渠：《泛函分析讲义》(上册)，北京大学出版社，1987。
5．V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, 1976.6．A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, 1983.-------------------- 非线性泛函分析--------------------课程编号：121020502002 课程类别：学科基础课课程名称：非线性泛函分析英文译名：Nonlinear Functional Analysis学时：60学时学分：3学分开课学期：2 开课形式：课堂讲授考核形式：闭卷考试适用学科：应用数学、基础数学、运筹学与控制论授课单位及教师梯队：数学与统计学院，应用数学系教师。
Backward stochastic differential equations in finance
where f is the generator and ξ is the terminal condition. Actually, this type of equation appears in numerous problems in finance (as pointed out in Quenez’s doctorate 1993). First, the theory of contingent claim valuation in a complete market studied by Black and Scholes (1973), Merton (1973, 1991), Harrison and Kreps (1979), Harrison and Pliska (1981), Duffie (1988), and Karatzas (1989), among others, can be expressed in terms of BSDEs. Indeed, the problem is to determine the price of a contingent claim ξ ≥ 0 of maturity T , which is a contract that pays an amount ξ at time T . In a complete market it is possible to construct a portfolio which attains as final wealth the amount ξ . Thus, the dynamics of the value of the replicating portfolio Y are given by a BSDE with linear generator f , with Z corresponding to the hedging portfolio. Then the price at time t is associated naturally with the value at time t of the hedging portfolio. However, there exists an infinite number of replicating portfolios and consequently the
随机偏微分方程(英文版-Lorenzo)
LORENZO ZAMBOTTI
For questions, corrections and remarks, please write to lorenzo.zambotti@upmc.fr 1. A prototype: The Ornstein-Uhlenbeck process Let λ > 0 and x ∈ R and let us consider the SDE in R dXt = −λ Xt dt + dWt where (Wt )t≥0 is a standard BM. By Itˆ o’s formula d(eλt Xt ) = eλt (λ Xt dt − λ Xt dt + dWt ) = eλt dWt so that eλt Xt = x +
s t
where we consider two independent standard BMs (W 1 , W 2 ) and define t ≥ 0, Wt1 , ˆ Wt := 2 , W− t < 0. t
(1.2)
ˆ u, e−λ(t−u) dW
−∞ < s ≤ t < +∞.
t→+∞
1 2
α ∈ R. 1 2λ
lim E(Xt (x)) = 0,
t→+∞
lim Var(Xt (x)) =
and therefore we obtain that N e−λt x, 1 (1 − e−2λt ) 2λ
1
N
0,
1 2λ
=: µλ .
(1.1)
数字信号处理(英文版)课后习题答案2
(Partial) Solutions to Assignment 2pp.73-761.16In each of the following systems, let or be the input and or be the output. Determine whether each systems is (1) linear, (2) time invariant, (3) causal, (4) BIBO stable(g).(i).ans: omitted----------------------------------------------------1.17 A linear time invariant system has impulse response Determine the output sequence for each of the followign input signals:(b)(f)(b) ans:h n is given byThe z-transform of []where ROC1:x n is given byz-transform of []where ROC2:h n is given byTherefore, the z-transform of the output []y nPerform inverse z to get [](f) ans: using the same method as in (b) (details omitted )----------------------------------------------------1.18. A linear time invariant system is defined by the difference equationb. Determine the output of the system when the intpu isc. Determine the output of the system when the input isans: omitted----------------------------------------------------1.19 The following expressions define linear time invariant systems. For each one determine the impulse respnose(a)(e)(a) ans: the impulse response is(e) ans: the impulse response is----------------------------------------------------1.20 Each of the following expressions defines a linear time invariant system. For each one determine whether it is BIBO stable or not(g)(k)BIBO: Bounded input and bounded output(g) ans: omitted(k) ans: omitted----------------------------------------------------1.21. Using the geometric series, for each of the following sequence determine the z-transform and its ROC(d)(g)(i)(d) ans:where ROC:(g) ans:The first part is equal towhere ROC1 isThe second part is equal towhere ROC2 isTherefore combining both parts:where ROC={ROC1 and ROC2}:(i) ans:where ROC: whole complex domain----------------------------------------------------1.22. You know what the and are. Using theproperties only (do not reuse the definition of the z-transform.) determine the z-transform of the following signals(c)(g)where ROC1:where ROC2:(c) ans: using z-transform property:We have:where ROC:(g) ans:details omitted. The final answer isTherefore combining both parts:where ROC={ROC1 and ROC2}:----------------------------------------------------1.23 Using partial fraction expansion, determine the inverse z-transform of the following functions:(c) ,(e) ,(c) ans:(e) ans:procedures are the same as above. details omitted.----------------------------------------------------1.24. For each of the followign linear difference equations, determine the impulse response, and indicate whether the system is BIBO stable or not(a)(c)(a) ans:Take z-transform on both sideswhere ROC:Because is finiteTherefore, the system is BIBO stable(c) ans: omitted (the same as (a))----------------------------------------------------1.25. Although most of the time we assume causality, a linear difference equation can be interpreted in a number of ways. Consider the linear difference equation(a) Determine the transfer function and the impulse response. Is the system causal ? BIBO stable ?(a) omitted.----------------------------------------------------1.26. 1.26 Consider the linear difference equation(a) Determine the transfer function . Do you have enough information to determine theregion of convergenceans:Don't have enough information to determine ROC.----------------------------------------------------1.27. Given the system described by the linear difference equationDetermine the output for each of the following input signals(a)(e)(a) ans:Take z-transform on both sides:----------------------------------------------------1.28. Repeat Problem 1.27 when the system is given in terms of the impulse responseBefore you do anything, is the system stable ? Does the frequency responseexist ?ans: omitted.----------------------------------------------------1.29. Repeat Problem 1.27 when the system isgiven by the linear difference equationBefore you do anything, is the system stable ? Does the frequency response exist ?Ans: omitted.----------------------------------------------------。
HandbookofLinearPartialDifferential EquationsforEngineersRef
REFERENCESAbramowitz,M.and Stegun,I.A.(Editors),Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,National Bureau of Standards Applied Mathematics, Washington,1964.Acrivos,A.,A note of the rate of heat or mass transfer from a small sphere freely suspended in linear shearfield,J.Fluid Mech.,V ol.98,No.2,pp.299–304,1980.Aksenov,A.V.,Linear differential relations between solutions of the equations of Euler–Poisson–Darboux class,Mechanics of Solids,V ol.36,No.1,pp.11–15,2001.Akulenko,L.D.and Nesterov,S.V.,Determination of the frequencies and forms of oscillations of non-uniform distributed systems with boundary conditions of the third kind,Appl.Math.Mech.(PMM),V ol.61,No.4,p.531–538,1997.Akulenko,L.D.and Nesterov,S.V.,Vibration of an nonhomogeneous membrane,Mechanics of Solids,V ol.34,No.6,pp.112–121,1999.Akulenko,L.D.and Nesterov,S.V.,Free vibrations of a homogeneous elliptic membrane, Mechanics of Solids,V ol.35,No.1,pp.153–162,2000.Akulenko,L.D.,Nesterov,S.V.,and Popov,A.L.,Natural frequencies of an elliptic plate with clamped edge,Mechanics of Solids,V ol.36,No.1,pp.143–148,2001.Andreev,V.K.,Kaptsov,O.V.,Pukhnachov,V.V.,and Rodionov,A.A.,Applications of Group-Theoretical Methods in Hydrodynamics,Nauka,Moscow,1994.(English translation:Kluwer, Dordrecht,1999.)Appell,P.,Trait´e de M´e canique Rationnelle,T.1:Statique.Dinamyque du Point(Ed.6),Gauthier-Villars,Paris,1953.Arscott,F.,Periodic Differential Equations,Macmillan(Pergamon),New York,1964. Arscott,F.,The Whittaker–Hill equation and the wave equation in paraboloidal coordinates,Proc.Roy.Soc.Edinburg,V ol.A67,pp.265–276,1967.Babich,V.M.,Kapilevich,M.B.,Mikhlin,S.G.,et al.,Linear Equations of Mathematical Physics [in Russian],Nauka,Moscow,1964.Batchelor,G.K.,Mass transfer from a particle suspended influid with a steady linear ambient velocity distribution,J.Fluid Mech.,V ol.95,No.2,pp.369–400,1979.Bateman,H.and Erd´e lyi,A.,Higher Transcendental Functions,Vol.1and Vol.2,McGraw-Hill, New York,1953.Bateman,H.and Erd´e lyi,A.,Higher Transcendental Functions,Vol.3,McGraw-Hill,New York, 1955.Bateman,H.and Erd´e lyi,A.,Tables of Integral Transforms,Vol.1and Vol.2,McGraw-Hill,New York,1954.Belotserkovskii,O.M.,and Oparin,A.A.,Numerical Experiment in Turbulence,Nauka,Moscow, 2000.Beyer,W.H.,CRC Standard Mathematical Tables and Formulae,CRC Press,Boca Raton,1991. Bitsadze,A.V.and Kalinichenko,D.F.,Collection of Problems on Mathematical Physics Equations [in Russian],Nauka,Moscow,1985.Bolotin V.V.(Editor),Vibration in Engineering:a Handbook.Vol.1.Vibration of Linear Systems [in Russian],Mashinostroenie,Moscow,1978.Borzykh,A.A.and Cherepanov,G.P.,A plane problem of the theory of convective heat transfer and mass exchange,PMM[Applied Mathematics and Mechanics],V ol.42,No.5,pp.848–855, 1978.Boyer,C.,The maximal kinematical invariance group for an arbitrary potential,Helv.Phys.Acta, V ol.47,pp.589–605,1974.Boyer,C.,Lie theory and separation of variables for equation+2−(21+22)=0, SIAM J.Math.Anal.,V ol.7,pp.230–263,1976.Bˆo cher,M.,Die Reihenentwickelungen der Potentialtheory,Leipzig,1894.Brenner,H.,Forced convection-heat and mass transfer at small Peclet numbers from particle of arbitrary shape,Chem.Eng.Sci.,V ol.18,No.2,pp.109–122,1963.Brychkov,Yu.A.and Prudnikov,A.P.,Integral Transforms of Generalized Functions,Gordon& Breach Sci.Publ.,New York,1989.Budak,B.M.,Samarskii,A.A.,and Tikhonov,A.N.,Collection of Problems on Mathematical Physics[in Russian],Nauka,Moscow,1980.Burde,G.I.,The construction of special explicit solutions of the boundary-layer equations.Steady flows,Q.J.Mech.Appl.Math.,V ol.47,No.2,pp.247–260,1994.Butkov,E.,Mathematical Physics,Addison-Wesley,Reading,Mass.,1968.Butkovskiy,A.G.,Characteristics of Systems with Distributed Parameters[in Russian],Nauka, Moscow,1979.Butkovskiy,A.G.,Green’s Functions and Transfer Functions Handbook,Halstead Press–John Wiley&Sons,New York,1982.Carslaw,H.S.and Jaeger,J.C.,Conduction of Heat in Solids,Clarendon Press,Oxford,1984. Clarkson,P.A and Kruskal,M.D.,New similarity reductions of the Boussinesq equation,J.Math.Phys.,V ol.30,No.10,pp.2201–2213,1989.Colton,D.,Partial Differential Equations.An Introduction,Random House,New York,1988. Courant,R.and Hilbert,D.,Methods of Mathematical Physics,Vol.2,Wiley–Interscience Publ., New York,1989.Crank,J.,The Mathematics of Diffusion,Clarendon Press,Oxford,1975.Davis,B.,Integral Transforms and Their Applications,Springer-Verlag,New York,1978. Davis,E.J.,Exact solutions for a class of heat and mass transfer problems,Can.J.Chem.Eng., V ol.51,No.5,pp.562–572,1973.Deavours,C.A.,An exact solution for the temperature distribution in parallel plate Poiseuilleflow, Trans.ASME,J.Heat Transfer,V ol.96,No.4,1974.Dezin,A.A.,Partial Differential Equations.An Introduction to a General Theory of Linear Boundary Value Problems,Springer-Verlag,Berlin-New York,1987.Ditkin,V.A.and Prudnikov,A.P.,Integral Transforms and Operational Calculus,Pergamon Press,New York,1965.Doyle,Ph.W.,Separation of variables for scalar evolution equations in one space dimension,J.Phys.A:Math.Gen.,V ol.29,pp.7581–7595,1996.Doyle,Ph.W.and Vassiliou,P.J.,Separation of variables for the1-dimensional non-linear diffusion equation,Int.J.Non-Linear Mech.,V ol.33,No.2,pp.315–326,1998.Elrick,D.E.,Source functions for diffusion in uniform shearflows,Australian J.Phys.,V ol.15, No.3,p.283–288,1962.Faddeev,L.D.(Editor),Mathematical Physics.Encyclopedia[in Russian],Bol’shaya Rossiiskaya Entsyklopediya,Moscow,1998.Faminskii,A.V.,On mixed problems for the Corteveg–de Vries equation with irregular boundary data,Doklady Mathematics,V ol.59,No.3,pp.366–367,1999.Farlow,S.J.,Partial Differential Equations for Scientists and Engineers,John Wiley&Sons,New York,1982.Galaktionov,V.A.,Invariant subspace and new explicit solutions to evolution equations with quadratic nonlinearities,Proc.Roy.Soc.Edinburgh,V ol.125A,No.2,pp.225–448,1995. Galaktionov,V.A.and Posashkov,S.A.,On new exact solutions of parabolic equations with quadratic nonlinearities,Zh.Vych.Matem.i Mat.Fiziki,V ol.29,No.4,pp.497–506,1989. Galaktionov,V.A.and Posashkov,S.A.,Exact solutions and invariant subspace for nonlinear gradient-diffusion equations,Zh.Vych.Matem.i Mat.Fiziki,V ol.34,No.3,pp.374–383,1994. Galaktionov,V.A.,Posashkov,S.A.,and Svirshchevskii,S.R.,Generalized separation of variables for differential equations with polynomial right-hand sides,Dif.Uravneniya,V ol.31, No.2,pp.253–261,1995.Gel’fand,I.M.and Shilov,G.E.,Distributions and Operations on Them[in Russian],Fizmatlit, Moscow,1959.Gradshteyn,I.S.and Ryzhik,I.M.,Tables of Integrals,Series,and Products,Academic Press, Orlando,2000.Graetz,L.,¨Uber die Warmeleitungsf¨a higkeit von Fl¨u ssigkeiten,Annln.Phys.,Bd.18,S.79–84, 1883.Grundland,A.M.and Infeld,E.,A family of nonlinear Klein–Gordon equations and their solutions, J.Math.Phys.,V ol.33,No.7,pp.2498–2503,1992.Guenther,R.B.and Lee,J.W.,Partial Differential Equations of Mathematical Physics and Integral Equations,Dover Publ.,Mineola,1996.Gupalo,Yu.P.,Polyanin,A.D.,and Ryazantsev,Yu.S.,Mass Exchange of Reacting Particles with Flow[in Russian],Nauka,Moscow,1985.Gupalo,Yu.P.and Ryazantsev,Yu.S.,Mass and heat transfer from a sphere in a laminarflow, Chem.Eng.Sci.,V ol.27,pp.61–68,1972.Haberman,R.,Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems,Prentice-Hall,Englewood Cliffs,1987.Hanna,J.R.and Rowland,J.H.Fourier Series,Transforms,and Boundary Value Problems, Wiley-Interscience Publ.,New York,1990.Happel,J.and Brenner,H.,Low Reynolds Number Hydrodynamics,Prentice-Hall,Englewood Cliffs,1965.H¨o rmander,L.,The Analysis of Linear Partial Differential Operators.II.Differential Operators with Constant Coefficients,Springer-Verlag,Berlin-New York,1983.H¨o rmander,L.,The Analysis of Linear Partial Differential Operators.I.Distribution Theory and Fourier Analysis,Springer-Verlag,Berlin,1990.Ibragimov N.H.(Editor),CRC Handbook of Lie Group to Differential Equations,Vol.1,CRC Press,Boca Raton,1994.Ignatovich,N.V.,Invariant irreducible,partially invariant solutions of stationary boundary layer equations,Mat.Zametki,V ol.53,No.1,pp.140–143,1993.Ivanov,V.I.,and Trubetskov,M.K.,Handbook of Conformal Mapping with Computer-Aided Visualization,Boca Raton,CRC Press,1994.John,F.,Partial Differential Equations,Springer-Verlag,New York,1982.Kalnins,E.,On the separation of variables for the Laplace equation in two-and three-dimensional Minkowski space,SIAM J.Math.Anal.,Hung.,V ol.6,pp.340–373,1975.Kalnins,E.and Miller,W.(Jr.),Lie theory and separation of variables,5:The equations +=0and+−Kalnins,E.and Miller,W.(Jr.),Lie theory and separation of variables,8:Semisubgroup coordinates for−2=0,J.Math.Phys.,V ol.16,pp.2507–2516,1975.Kalnins,E.and Miller,W.(Jr.),Lie theory and separation of variables,9:Orthogonal-separable coordinate systems for the wave equation−2=0,J.Math.Phys.,V ol.17,pp.331–335, 1976.Kalnins,E.and Miller,W.(Jr.),Lie theory and separation of variables,10:Nonorthogonal-separable solutions of the wave equation−2=0,J.Math.Phys.,V ol.17,pp.356–368, 1976.Kamke,E.,Differentialgleichungen:L¨o sungsmethoden und L¨o sungen,I,Gew¨o hnliche Differential-gleichungen,B.G.Teubner,Leipzig,1977.Kamke,E.,Differentialgleichungen:L¨o sungsmethoden und L¨o sungen,II,Partielle Differential-gleichungen Erster Ordnung f¨u r eine gesuchte Funktion,Akad.Verlagsgesellschaft Geest& Portig,Leipzig,1965.Kanwal,R.P.,Generalized Functions.Theory and Technique,Academic Press,Orlando,1983. Korn,G.A.and Korn,T.M.,Mathematical Handbook for Scientists and Engineers,McGraw-Hill, New York,1968.Koshlyakov,N.S.,Gliner,E.B.,and Smirnov,M.M.,Partial Differential Equations of Mathematical Physics[in Russian],Vysshaya Shkola,Moscow,1970.Krein,S.G.(Editor),Functional Analysis[in Russian],Nauka,Moscow,1972.Krylov,A.N.,Collected Works:III Mathematics,Pt.2[in Russian],Izd-vo AN SSSR,Moscow, 1949.Lamb,H.,Hydrodynamics,Dover Publ.,New York,1945.Lavrent’ev,M.A.and Shabat B.V.,Methods of Complex Variable Theory[in Russian],Nauka, Moscow,1973.Lavrik,V.I.and Savenkov,V.N.,Handbook of Conformal Mappings[in Russian],Naukova Dumka,Kiev,1970.Landau,L.D.and Lifshits,E.M.,Quantum Mechanics.Nonrelativistic Theory[in Russian], Nauka,Moscow,1974.Lebedev,N.N.,Skal’skaya,I.P.,and Uflyand,Ya.S.,Collection of Problems on Mathematical Physics[in Russian],Gostekhizdat,Moscow,1955.Leis,R.,Initial-Boundary Value Problems in Mathematical Physics,John Wiley&Sons,Chichester, 1986.Levich,V.G.,Physicochemical Hydrodynamics,Prentice-Hall,Englewood Cliffs,New Jersey,1962. Levitan,B.M.and Sargsyan,I.S.,Sturm–Liouville and Dirac Operators[in Russian],Nauka, Moscow,1988.Loitsyanskiy,L.G.,Mechanics of Liquids and Gases,Begell House,New York,1996. Lykov,A.V.,Theory of Heat Conduction[in Russian],Vysshaya Shkola,Moscow,1967. Mackie,A.G.,Boundary Value Problems,Scottish Academic Press,Edinburgh,1989. Makarov,A.,Smorodinsky,J.,Valiev,K.,and Winternitz,P.,A systematic search for nonrelativistic systems with dynamical symmetries.Part I:The integrals of motion,Nuovo Cimento,V ol.52A,pp.1061–1084,1967.Marchenko,V.A.,Sturm–Liouville Operators and Applications,Birkhauser Verlag,Basel-Boston, 1986.Markeev,A.P.,Theoretical Mechanics[in Russian],Nauka,Moscow,1990.Mathematical Encyclopedia[in Russian],Sovetskaya Entsiklopediya,Moscow,1977. McLachlan,N.W.,Theory and Application of Mathieu Functions,Clarendon Press,Oxford,1947.Meixner,J.and Sch¨a fke,F.,Mathieusche Funktionen und Sph¨a roidfunktionnen,Springer-Verlag, Berlin,1965.Mikhlin,S.G.,Variational Methods in Mathematical Physics[in Russian],Nauka,Moscow,1970. Miles,J.W.,Integral Transforms in Applied Mathematics,Cambridge Univ.Press,Cambridge, 1971.Miller,W.(Jr.),Symmetry and Separation of Variables,Addison-Wesley,London,1977. Miller,J.(Jr.)and Rubel,L.A.,Functional separation of variables for Laplace equations in two dimensions,J.Phys.A,V ol.26,No.8,pp.1901–1913,1993.Moon,P.and Spencer,D.,Field Theory Handbook,Springer-Verlag,Berlin,1961.Morse,P.M.and Feshbach,H.,Methods of Theoretical Physics,Vols.1–2,McGraw-Hill,New York,1953.Murphy,G.M.,Ordinary Differential Equations and Their Solutions,D.Van Nostrand,New York, 1960.Myint-U,T.and Debnath,L.,Partial differential equations for scientists and engineers,North-Holland Publ.,New York,1987.Naimark,M.A.,Linear Differential Operators[in Russian],Nauka,Moscow,1969. Niederer,U.,The maximal kinematical invariance group of the harmonic oscillator,Helv.Phys.Acta,V ol.46,pp.191–200,1973.Nikiforov,A.F.and Uvarov,V.B.,Special Functions of Mathematical Physics.A Unified Introduction with Applications,Birkhauser Verlag,Basel-Boston,1988.Novikov,E.A.,Concerning turbulent diffusion in a stream with a transverse gradient of velosity, Appl.Math.Mech.(PMM),V ol.22,No.3,p.412–414,1958.Nusselt,W.,Abh¨a ngigkeit der W¨a rme¨u bergangzahl con der Rohr¨a nge,VDI Zeitschrift,Bd.54, No.28,S.1154–1158,1910.Olver,P.J.,Application of Lie Groups to Differential Equations,Springer-Verlag,New York,1986. Ovsiannikov,L.V.,Group Analysis of Differential Equations,Academic Press,New York,1982. Pavlovskii,Yu.N.,Analysis of some invariant solutions of boundary layer equations,Zh.Bych.Matem.i Mat.Fiziki,V ol.1,No.2,pp.280–294,1961.Petrovsky,I.G.,Lectures on Partial Differential Equations,Dover Publ.,New York,1991. Pinsky,M.A.,Introduction to Partial Differential Equations with Applications,McGraw-Hill,New York,1984.Polozhii,G.N.,Mathematical Physics Equations[in Russian],Vysshaya Shkola,Moscow,1964. Polyanin,A.D.,The structure of solutions of linear nonstationary boundary-value problems of mechanics and mathematical physics,Doklady Physics,V ol.45,No.8,pp.415–418,2000a. Polyanin,A.D.,Partial separation of variables in unsteady problems of mechanics and mathematical physics,Doklady Physics,V ol.45,No.12,pp.680–684,2000b.Polyanin,A.D.,Linear problems of heat and mass transfer:general relations and results,Theor.Found.Chem.Eng.,V ol.34,No.6,pp.509–520,2000c.Polyanin,A.D.,Handbook of Linear Mathematical Physics Equations[in Russian],Fizmatlit, Moscow,2001a.Polyanin,A.D.,Transformations and exact solutions of boundary layer equations with arbitrary functions,Doklady AN,V ol.379,No.3,2001b.Polyanin,A.D.,Exact solutions and transformations of the equations of a stationary laminar boundary layer,Theor.Found.Chem.Eng.,V ol.35,No.4,pp.319–328,2001c. Polyanin,A.D.,Generalized separable solutions of Navier–Stokes equations,Doklady AN,V ol.380, No.4,2001d.Polyanin,A.D.and Dilman,V.V.,Methods of Modeling Equations and Analogies in Chemical Engineering,CRC Press,Boca Raton,1994.Polyanin,A.D.,Kutepov,A.M.,Vyazmin,A.V.,and Kazenin,D.A.,Hydrodynamics,Mass and Heat Transfer in Chemical Engineering,Gordon&Breach Sci.Publ.,London,2001. Polyanin,A.D.and Manzhirov,A.V.,Handbook of Integral Equations,CRC Press,Boca Raton, 1998.Polyanin,A.D.,Vyazmin,A.V.,Zhurov,A.I.,and Kazenin,D.A.,Handbook of Exact Solutions of Heat and Mass Transfer Equations[in Russian],Faktorial,Moscow,1998.Polyanin,A.D.and Zaitsev,V.F.,Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press,Boca Raton,1995.Polyanin,A.D.,Zaitsev,V.F.,and Moussiaux,A.,Handbook of First Order Partial Differential Equations,Gordon&Breach,London,2001.Polyanin,A.D.and Zhurov,A.I.,Exact solutions to nonlinear equations of mechanics and mathematical physics,Doklady Physics,V ol.43,No.6,pp.381–385,1998.Polyanin,A.D.,Zhurov,A.I.,and Vyazmin,A.V.,Generalized separation of variables in nonlinear heat and mass transfer equations,J.Non-Equilibrium Thermodynamics,V ol.25, No.3/4,pp.251–267,2000.Prudnikov,A.P.,Brychkov,Yu.A.,and Marichev,O.I.,Integrals and Series,Vol.1,Elementary Functions,Gordon&Breach Sci.Publ.,New York,1986.Prudnikov,A.P.,Brychkov,Yu.A.,and Marichev,O.I.,Integrals and Series,Vol.2,Special Functions,Gordon&Breach Sci.Publ.,New York,1986.Pukhnachev,V.V.,Group properties of the Navier–Stokes equations in the plane case,Zh.Prikl.Mekh.i Tekhn.Fiziki,No.1,pp.83–90,1960.Rimmer,P.L.,Heat transfer from a sphere in a stream of small Reynolds number,J.Fluid Mech., V ol.32,No.1,pp.1–7,1968.Rotem,Z.,and Neilson,J.E.,Exact solution for diffusion toflow down an incline,Can.J.Chem.Engng.,V ol.47,pp.341–346,1966.Schlichting,H.,Boundary Layer Theory,McGraw-Hill,New York,1981.Sedov,L.I.,Plane Problems of Hydrodynamics and Airdynamics[in Russian],Nauka,Moscow, 1980.Shilov,G.E.,Mathematical Analysis:A Second Special Course[in Russian],Nauka,Moscow,1965. Smirnov,V.I.,A Course of Higher Mathematics.Vols.2–3[in Russian],Nauka,Moscow,1974. Smirnov,M.M.,Second Order Partial Differential Equations[in Russian],Nauka,Moscow,1964. Smirnov,M.M.,Problems on Mathematical Physics Equations[in Russian],Nauka,Moscow, 1975.Sneddon,I.,Fourier Transformations,McGraw-Hill,New York,1951.Stakgold,I.,Boundary Value Problems of Mathematical Physics.Vols.I,II,SIAM,Philadelphia, 2000.Strauss,W.A.,Partial Differential Equations.An Introduction,John Wiley&Sons,New York, 1992.Sutton,W.G.L.,On the equation of diffusion in a turbulent medium,Proc.Poy.Soc.,Ser.A, V ol.138,No.988,pp.48–75,1943.Svirshchevskii,S.R.,Lie–B¨a cklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations,Phys.Letters A,V ol.199,pp.344–348,1995.Taylor,M.,Partial Differential Equations,Vol.3,Springer-Verlag,New York,1996. Temme,N.M.,Special Functions.An Introduction to the Classical Functions of Mathematical Physics,Wiley-Interscience Publ.,New York,1996.Tikhonov,A.N.and Samarskii,A.A.,Equations of Mathematical Physics,Dover Publ.,New York,1990.Thomas,H.C.,Heterogeneous ion exchange in aflowing system,J.Amer.Chem.Soc.,V ol.66, pp.1664–1666,1944.Tomotika,S.and Tamada,K.,Studies on two-dimensional transonicflows of compressiblefluid, Part1,Quart.Appl.Math.,V ol.7,p.381,1950.Urvin,K.and Arscott,F.,Theory of the Whittaker–Hill equation,Proc.Roy.Soc.,V ol.A69, pp.28–44,1970.Vereshchagina,L.I.,Groupfibering of the spatial nonstationary boundary layer equations,Vestnik LGU,V ol.13,No.3,pp.82–86,1973.Vladimirov,V.S.,Mikhailov,V.P.,Vasharin A.A.,et al.,Collection of Problems on Mathematical Physics Equations[in Russian],Nauka,Moscow,1974.Vladimirov,V.S.,Mathematical Physics Equations[in Russian],Nauka,Moscow,1988. Vvedensky,D.,Partial Differential Equations,Addison-Wesley,Wakingham,1993. Whittaker,E.T.and Watson,G.N.,A Course of Modern Analysis,Vols.1–2,Cambridge Univ.Press,Cambridge,1952.Zachmanoglou,E.C.and Thoe,D.W.,Introduction to Partial Differential Equations with Applications,Dover Publ.,New York,1986.Zaitsev,V.F.and Polyanin,A.D.,Handbook of Partial Differential Equations:Exact Solutions[in Russian],MP Obrazovaniya,Moscow,1996.Zauderer,E.,Partial Differential Equations of Applied Mathematics,Wiley–Interscience Publ., New York,1989.Zhdanov,R.Z.,Separation of variables in the non-linear wave equation,J.Phys.A,V ol.27, pp.L291–L297,1994.Zwillinger,D.,Handbook of Differential Equations,Academic Press,San Diego,1998.。
常微分方程中的英文单词和短语
operator polynomial
rse transform
第七章
Chapter7Power Series Solutions of Differential Equations
柯西定理
Cauchy theorem
解析解
analytic solution
常数矩阵
constant matrix
非齐次线性微分方程组
non-homogeneous linear differential equations
常数列向量
constant arrange vector
常数变异公式
formula of variation of constant
计算公式
computing formula
若尔当标准型
Jordan normal form
矩阵函数
matrix function
实值解
real-valued function
复值解
complex-value function
代数余子式
algebraic cofactor
特征方程
characteristic equation
算子式
operator method
一阶线性方程
first-order linear equation
齐次
homogeneous
非齐次
non-homogeneous
积分因子法
method of integration factor
常数变易法
method of constants variation
初等变换法
elementary method of transformation
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
2
−¥
§
= 0).
In this case, equations (5) and (6) coincide and have a common general integral, ( , )=
.
By passing from ,
£
¤
to new independent variables , in accordance with the relations
¥ ¤ £
− −
¦
+ −
¦ 2
−¥ −¥
§
= 0, = 0,
(5) (6)
and and find their general integrals.
¥ ¤
¦
¦ 2
§
£
0.1.1-3. Canonical form of parabolic equations (case
* The right-hand side of equation (4) may be nonlinear. The classification and the procedure of reducing such equations to a canonical form are only determined by the left-hand side of the equation.
A ¥ B C B £
1 , 89898
¦
2
−¥
§
< 0).
In this case the general integrals of equations (5) and (6) are complex conjugate; these determine two families of complex characteristics. Let the general integral of equation (5) have the form
2 ¢
¡
2
−
2 £
¡
2
= 0,
(2)
where the variables and play the role of time and the spatial coordinate, respectively. Note that the highest derivative terms in equation (2) differ in sign. The simplest example of an elliptic equation is the Laplace equation
¦ ¦ ¦
£
¤
2 2 2
−¥
§ ¥ § ¥ §
= 0, > 0, <0
if if
− −
at this point. In order to reduce equation (4) to a canonical form, one should first write out the characteristic equation £ ¤ £ ¥ ¤ 2 − 2 ¦ + § 2 = 0, which splits into two equations
¡ ¢ ¢ £
−
2 £
¡
2
= 0,
(1)
where the variables and play the role of time and the spatial coordinate, respectively. Note that equation (1) contains only one highest derivative term. The simplest example of a hyperbolic equation is the wave equation
0.1.2. Equations with Many Independent Variables
Consider a second-order partial differential equation with 7 independent variables @ has the form © ¡ ¡ ¡
¥
( , )
£
¤
2 £
¡ £
2
+ 2¦ ( , )
¤
2
¡ £
£ ¤
+§ ( , )
¤
2 ¤
¡
©
2
=
¨
£
, , ,
¤
¡
¡ £
¡
,
¤
,
(4)
© 2002 by Chapman & Hall/CRC
where ¥ , ¦ , § are some functions of and that have continuous derivatives up to the second-order inclusive.* £ ¤ Given a point ( , ), equation (4) is said to be parabolic hyperbolic elliptic if
¦ ¤
2
−¥
§
> 0).
The general integrals
( , )=
£
¤
1, 4
( , )=
£
2
of equations (5) and (6) are real and different. These integrals determine two different families of real characteristics.
£ ¤
£
¤
2
¡
©
2
=
¨ 1
, , ,
¡
¡
¡
,
.
(7)
As , one can take = or = . It is apparent that, just as the heat equation (1), the transformed equation (7) has only one highest-derivative term. ¡ % &(' ) 021 3 In the degenerate case where the function ¨ 1 does not depend on the derivative ! , equation (7) is an ordinary differential equation for the variable , in which serves as a parameter. 0.1.1-4. Canonical form of hyperbolic equations (case
Introduction
Some Definitions, Formulas, Methods, and Solutions
0.1. Classification of Second-Order Partial Differential Equations
0.1.1. Equations with Two Independent Variables
© 2002 by Chapman & Hall/CRC
By passing from ,
£
¤
to new independent variables , in accordance with the relations
= ( , ),
£
¤
= 4 ( , ),
¡ ¡
£
¤
2 ¢
¡
2
−
2
¡
©
5 2
=
¨ 3
¢
,5 , ,
¡
¡ ¢
¡
,
5
,
where ¨ 3 = 4 ¨ 2 . This is the so-called second canonical form of a hyperbolic equation. Apart from notation, the left-hand side of the last equation coincides with that of the wave equation (2). 0.1.1-5. Canonical form of elliptic equations (case
we reduce equation (4) to
2
¡
©
=
¨ 2
, , ,
¡
,
.
This is the so-called first canonical form of a hyperbolic equation. The transformation ¢ ¢ = +5 , = −5 brings the above equation to another canonical form,
0.1.1-1. Examples of equations encountered in applications. Three basic types of partial differential equations are distinguished— parabolic, hyperbolic, and elliptic. The solutions of the equations pertaining to each of the types have their own characteristic qualitative differences. The simplest example of a parabolic equation is the heat equation