# Asymptotic Analysis: From Theory to Application by J. Mauss (auth.), Ferdinand Verhulst (eds.)

By J. Mauss (auth.), Ferdinand Verhulst (eds.)

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Extra info for Asymptotic Analysis: From Theory to Application

Example text

A unique solution V E E ( 0 , a 0] this solution the estimate suplu(',t)-Jo(-,t)i 0 ~ Q~ t~ where Q denotes a constant > 0 independent For the p r o o f of this LEMMA with of a. t h e o r e m we shall use t h e following lemma. l) APPROXIMATION CASE. ,t)l 0 ~< Q(8)e -st. ,t) and z is a g e n e r a l i z e d solution ( 5 . 5 ) ~~z = £ z - ( c . 2. 8) z(x,t) (BC) (IC) be shown C([0,L]×[0,~)\{(x,t)[t Our m o t i v a t i o n --- 0 that C i is a w e l l - d e f i n e d = 0 7 x = 0 •r x = L}) and that to look at these formula holds = ~(x,t) - nft Ci's is, that ~(T) )dT of C i is hounded.

F O R M A L ASYMPTOTIC OF THE The TIME-DEPENDENT construction solution of approximation, with With which J0(x,t,~) c + 0 OF THE SOLUTION asymptotic analogous as b e f o r e we call = U0(x,t) J0' approximation for to the c o n s t r u c t i o n to the 0-th order term ~ + 0 of the given in s e c t i o n of such a formal we now put: + X0(~,t) + X0(~,t) U 0 the 0-th o r d e r term of the X 0 the 0-th o r d e r term of the b o u n d a r y layer at x = 0, ~ = x / ~ 2 0 the 0-th term of the b o u n d a r y layer at x = L, ~ = ( L - x ) / ~ .

24-46, part II, Degeneration to first order operator~, J. Math. Anal. Appl. 49 Ackerberg, (1975), p. 324-346. , Singular perturbation problems for linear elliptic boundary value problems, part III, Counterexamples, Nieuw Archief voor Wiskunde, ser. 3, 25 (1977), p. 1-39. [4] Eckhaus, W. M. de Jager, Asymptotic solutions of singular perturba- tion problems for linear differential equations of elliptic type, Arch. Rat. Mech. Anal. 23 (1966), p. 26-86. , 1964. [6] Gardlng, Lars, On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators, Math.