1960_Singular-perturbations-of-two-pointordinary-differential-equations_Arch.-Rational-Mech.-Anal

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6. B o u n d a r y C o n d i t i o n s Satisfied b y t h e L i m i t i n g Solution . . . . . . . . . . 7. R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
d xl (t, e) = A l l (t, e) xx(t, e) + . . . + Alt,(t, e).xt,(t, e) dt
(t.1)
**, d x, (t, ,) = A ~ (t, e) x~ (t, ~) + . . . + A~p (t, ,) xp (t, ~)
**, d % (t, ,) = Ap~ (t, e) xl(t, e) + . . . + Apt,(t, e) xt,(t, e),
dXl _ All(t, ~) x~ (t,~) + ....+ A~p(t, 8) xp(t, 8) dt eh, dx~ = A~(t, 8) x~(t,e) +... + Asp(t, e) xp(t, 8)
a'~p ." 8h,--gT = A ~ ( t , 8) x~(t, ~) + ... + App(t, 8) xp(t, 8)
2t4
W.A. HARRIS, JR.:
then E. R. RANG [8] has shown there exists a non-singular transformation, O~e<=eo, a<--t~b, of the form (2A)
x(t, e) = (TI(t) + e Q~(t)) (T2(t) + e 2 Qz(t)) ... (TL(t) + ez QL(t)) y(t, e)
(2.2)
h, d e ~ i y* (t, e) : C21(~, e) yl(t, e) + . - . C~p(t, e) yp(t, e)
e*, ff-i YP (t, e) = Cp~(t, e) yt(t, ~) + . . . C,p(t, e) yp(t, e),
d ~7 y~ = c i i (t, o) y, + . . . + t i p (t, o) y, 0 = C~l(t, o) y~ + . . . + c2p(t, o) yp 0 = Cp~(t,O) yl + . . . + Cpt,(t, O) yp
Singular Perturbations of TuJo-PointBoundaryProblemsfor Systems of Ordinary DifferentialEquations
W. A. HARRIS, JR.
Communicated by C. C. LIN
Contents Abstract t. 2. 3. 4. 5. Introduction . . . . . . . . . . . Preliminffry Transformations . . . The Canonical Problem . . . . . . A-a(*) for a Canonical P r o b l e m . . . E v a l u a t i o n of l i m X ( t , e) A -1 (8) C (e) . . . . . . . .
R (o) x (a) + S (o) x (b) = c (o~,
where the h i are integers, O < h ~ < h 3 < . . . < h ~ , = h , x i is a vector of dimension p hi, m = ~. hi, A(t, e)ij are matrices of appropriate orders with asymptotic expansions, x is the vector , R and S are square matrices of order in 1+ m) and e > 0 . p Under three hypotheses, H t, H 2, and H 3, we shall prove Theorem t (Section 6), which embodies our results for the problem indicated above. We begin b y reducing the problem (tA), (t.2) to a canonical form (2A5), (6.3). We show that the solution of the canonical boundary problem has a limit as e--->0§ which satisfies the corresponding degenerate differential system and rh of the degenerate boundary conditions. A discussion of the hypotheses is given in Section 7. The author wishes to express his deep appreciation to Professor H. L. TCRRITTIN for his guidance and encouragement during the preparation of this paper. 2. P r e l i m i n a r y T r a n s f o r m a t i o n s If App(t, 0) is non-singular, a ~ t ~ b , the last equation of (t.3) m a y be solved for~xp in terms of x 1. . . . . xp-1 and substituted into the preceeding equations of (1.3) to give a system of the same form as (t9 in x 1. . . . . xp_ 1. The last equation of this form is
222 224 225
Abstract
Asymptotic solutions of linear systems of ordinary differential equations are employed to discuss the relationship of the solution of a certain "complete" boundary problem.
(~.2)
d
R (e) x (a, e) + S (e) x (b, e) = c (e),
0.3) 0.4)
d--[x~ = A ~ (t, O) x~ + . . . + A~t,(t, O) x t, 0 = A21(t, 0 ) x~+ ... +A~p(t,O) xt, 0 = Ap~(t, O) x~ + . . . + Apt,(t, O! xp,
which changes the differential system (tA) into the form (29 with corresponding ~tegenerate differential system (2.3).
d ~ i Yl (t, e) = c l , (,, e) yl(t, e) + . . . cl p (t, e) yp (t, e)
where Cii(i, 0) is non-singular a<--t<_b, i = 2 . . . . , p, and in particular
I t is s h o w n t h a t u n d e r c e r t a i n c o n d i t i o n s t h e s o l u t i o n of t h e " c o m p l e t e " p r o b l e m as e - ~ 0 § a p p r o a c h e s a solution of t h e " d e g e n e r a t e " differential s y s t e m a n d satisfies n~ a p p r o p r i a t e " d e g e n e r a t e " b o u n d a r y conditions.
R(e) x(a, 8)+ S(e) x(b,
8)=c(8)
Leabharlann Baidu
as e-+O + a n d t h e - e i k t e d " d e g e n e r a t e " p r o b l e m o b t a i n e d b y s e t t i n g e = O. H e r e t h e h i a r e integers, O<'h,<...< hp, x i is a v e c t o r of d i m e n s i o n hi, Aii(t, ,) are m a t r i c e s of a p p r o p r i a t e o r d e r s w i t h a s y m p t o t i c e x p a n s i o n s , x is t h e v e c t o r P s q u a r e m a t r i c e s of o r d e r ~ n i a n d e > O. , R a n d S are
-1 0 = (Ap_l, 1 -- Ap_~,pA~,~Ap~) x 1 + . . . + (Ap_l,p_ 1 -- Ap_l, pAppAp, p_l) xp_~.
Thus, if ( A p _ l , p _ l - - A p _ l , l , A ~ A p , p_l) is non-singular, a ~ t ~ b , we m a y solve this equation for xp_ 1 in terms of xl, ..., xp_,' and repeat the process until we solve the first equation for x 1, which is a differential equation. If this process can be carried out completely, step b y step, and xp, xp_l, ..., xa all eliminated,
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213 213 217 2t 7
Singular Perturbations of Differential Systems 1. I n t r o d u c t i o n
2~3
We are concerned with showing the relationship of the solution of a boundary problem (~A), 0.2) as e-->0 + to the solutions of a related degenerate problem (t.3), (t 9 The problems are