渐近方法

Asymptotic MethodsLecturer:Xuesong Wu,ProfessorPre-requisitesLinear algebra,calculus(including Fourier/Laplace transforms),analysis of complex variable, ordinary and partial differential equations.(Knowledge of classical mechanics andfluid dynamics would be helpful but not essential.) Basic competency in English language:familiarity with mathematic terminology in English. Useful referencesO.M.Bender&S.A.Orszag Advanced Mathematical Methods for Scientists and Engineers. M.H.Holmes Introduction to Perturbation Methods.E.J.Hinch Perturbation Methods.M.Van Dyke Perturbation Methods in Fluid Mechanics.J.Kevorkian&J.D.Cole Perturbation Methods in Applied Mathematics.A.H.Nayfeh Perturbation Methods.Note:This course may be called asymptotic analysis or perturbation methods. Teaching arrangement30lectures and plus5tutorials.Extensive printed notes;5Question Sheets plus solutions(no assessed coursework).General descriptionMany mathematical problems arising from sciences and engineering often consist of small or large parameters.In exceptional cases(e.g.certain types of linear ordinary/partial differential equations),the problem can be solved exactly,often by means of an integral transformation(place or Fourier transform),and the solution may then be expressed in the form of an integral,referred to as integral representation.Numerical evaluation by using a computer,though straightforward nowadays,may not be most efficient.Moreover,the interest is often in the analytical property of the solution when the parameter or variable is large,and it is then desirable to derive an ana-lytical formula which approximates the integral for asymptotically large values of the parameter of variable.Such a formula is referred to as asymptotic representation of the integral.In general,the mathematical problems formulated are so complex(e.g.nonlinear)thatfinding an exact analytical solution in closed form is impossible.However,because of the presence of a large or small parameter,approximate solutions may be constructed in the form of asymptotic expansions or asymptotic series,and the original system systematically reduced to a simpler one,which can be solved analytically or numerically.The resulting solution represents afirst-order approximation,which may be further improved by seeking higher-order corrections.A range of powerful techniques and methods have been developed,each pertaining to a specific mathematical feature of the problem.They represent some of the most effective and important tools,which are being widely used by researches in both natural and engineering sciences. Aims&objectivesThis advanced course presents a systematical introduction to asymptotic methods,which form one of cornerstones of modern applied mathematics.The main ideas and various techniquesfor deriving asymptotic representations of integrals,and for constructing appropriate solutions will be explained.This is primarily a method course,but some important applications to classical mechanics andfluid mechanics will be demonstrated.Historically these subjects directly motivated the development of several key methods.The intended learning outcomes for students are•To familiarise with major asymptotic methods available,and to acquire a thorough un-derstanding of key ideas involved and mathematical contexts to which these methods are applicable.•To acquire basic skills for solving problems of‘textbook level’.•To acquire the ability of applying/modifying/extending relevant methods to unfamiliar problems,such as those emerging from research topics or practical applications.Main topics1.Asymptotic approximationConventional converging series,divergent series and asymptotic series;asymptotic versus Taylor expansions.2.Asymptotic representation of integralsIntegrals of Laplace type:Laplace’s method,Watson’s Lemma.Integrals of Fourier type:Method of stationary phase.Integral in complex plane:Method of steepest descent.Applications.3.Matched asymptotic expansionOrdinary and partial differential equations with small parameters,regular expansions.Singular asymptotic perturbations:notion of‘boundary layer’,inner and outer expansions.Overlap region.Matching of the asymptotic expansions.Location of the boundary layer.Composite approximation.Singular perturbation problem with an inner layer.4.Methods of multiple scalesMethod of strained coordinates;Poincare-Lindstedt method,periodic solutions.Quasi-periodic solutions,secular terms&solvability condition.Multiple-scale methods,averaging method.Nonlinear oscillation:frequency response,sub/super-harmonic resonance,hysteresis.5.WKB MethodTurning point problems,and analysis of transition layers.Ray tracing,caustics.Importance of asymptotic methodsThe importance of asymptotic methods may be appreciated by looking at how they were originated.1.Classical analysis and special functions(Laplace,Legendre,Poisson,Cauchy)2.Specific applications in science and engineering,e.g.celestial mechanics(Poincare),wave mechanics(classical or quantum),fluid mechanics(Prandtl).Methods developed have become•A fundamental analytical tool that has been,and is still being,used to prove rigorous theorems in many branches of pure or applied mathematics(e.g.existence&regularity of solutions to differential equations,approximation theory).•Powerful methods for tackling many increasingly challenging problems in applied sciences and engineering,which often require a sophisticated combination/extension of existing methods.Most methods were developed before1970s,i.e.before computers became powerful and easily accessible.The availability of powerful computers and software raises the equation: what is the role of asymptotic methods in the era of computer?1.Analytical properties cannot be readily extracted from numerics.2.Guide numerical computations:mesh size,algorithms,validation,framework for interpret-ing numerical results.3.Asymptotic analysis and reduction allow us to identify the key physical factors,and pro-vides essential insights into complex problems.4.Many classical and emerging problems remain unsolvable even by the most powerful com-puters because they involve multi-scales and multi-physics(e.g.turbulentflows around aircrafts;weather forecast;combustion,aeroacoustics).Computers broaden the role of asymptotic methods:Before the era of computer:asymptotically reduced systems had to be analytically tractable;Now in the era of computer:reduced systems only need to be numerically tractable sys-tems.5.A general methodology in scienceWell-known theories viewed from the perspective of asymptotic reduction,e.g.•Einstein’s Special Relativity theory−−→Newton’s mechanics in the limit of V/c 1.•Boltzmann’s equation−−→Continuum model,pressible Navier-Stokes equa-tions when Knutsen number Kn=L/λ 1−−>Compressible Euler equation if the Reynolds number Re 1−−>Boundary layer theory−−>Incompressible Navier-Stokes equations the Mach number M=V/a 1.Often works in the opposite direction,i.e.extend a less comprehensive/complete theory in such a fashion that the original theory is an asymptotic limit.Examples(To corroborate the points,and to illustrate the‘flavour’of the problems tackled and the general‘philosophy’involved.)Gamma functionΓ(x)≡ ∞0t x−1e−t dt(x>0is real).If x=n(an integer),Γ(n)=(n−1)!,a factorial,suggesting thatΓmust be very large for x 1.But how large precisely?Can we express it in terms of elementary functions?Stirling’s formula for x 1:Γ(x)∼x x e−x(2π2.Remark:Computing the valueΓfor any given x is trivial,but results such as Stirling’s formula cannot be established purely by numerical means.Airy equation&functionw (x)−xw=0(x is real)Two linear independent solutions:Ai(x)and Bi(x).Ai(x)has the integral representationAi(x)=13+xt)dt.Evaluation is straightforward for any moderate values of x,but tricky for large x.(Why?) Asymptotic representation for x>0and x 1,Ai(x)∼e−ζ2x 1ζs,Ai(−x)∼12x 14) s=0(−1)s a2s4) s=0(−1)s a2s+1 3x3(216)s s!.Remarks:1.Evaluating Ai for large x;complementary to numerical means.2.Asymptote cannot readily be extracted from numerics;need analytic methods.3.The series: s=0(−1)s a s。