Download Singular Perturbations and Asymptotics. Proceedings of an by Richard E. Meyer, Seymour V. Parter PDF

By Richard E. Meyer, Seymour V. Parter

ISBN-10: 0124932606

ISBN-13: 9780124932609

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Read or Download Singular Perturbations and Asymptotics. Proceedings of an Advanced Seminar Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, May 28–30, 1980 PDF

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Additional resources for Singular Perturbations and Asymptotics. Proceedings of an Advanced Seminar Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, May 28–30, 1980

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U^ t^ because The resolution of the difficulty, as for the anal­ ogous situation in §2, consists in adding to the function layer term which smooths out the irregularity of u at u an interior t^. ^) than it was for (P ) since we must now distinguish two more or less 1 SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS distinct cases. 53 Before discussing the general theory we consider two model problems which illustrate the difficulties. The first example is f ey" = -ty , -1 < t < 1, y(-l,€) = -1, y(l,e) = 1, whose solution is (to transcendentally small terms) 2 y(t,e) = -1 + exp[-t /(2e)] on [-1,0] and 2 y(t,e) = 1 - exp[-t /(2e)] on [0,1], that is, lim, y(t,e) = -1(1) on [-1,0) ((0,1]).

REMARKS. It has been shown how different limits can be used to obtain expan­ sions valid in different regions of physical or parameter space. This is a general procedure when there are several independent variables and parameters and can be applied with good effect to various other physical prob lems. 40 JULIAN D. COLE Direct limits may fail however when modulated waves are an intrinsic part of the problem so that multi-variable or WKB type expansions must be used. Finally, several unsolved problems are noted: the detailed connec­ tion of transonic and linearized theories of Section 2 details about the passage through F = 1 and further in Section 3« Department of Mathematics University of California Los Angeles, California Some Old and New Results on Singularly Perturbed Boundary Value Problems Fred A.

11], [13], 1 [25]) / 2 can be used1 to / 2show that on [0,1] y(t,e) = max {0,t+B-l} H e e x P[ - | t - t 0| / e ] for t Q = 1-B, that is, lim e-O y(t,e) = 0 on [0,1-B] and lira y(t,e) = t+B-1 on [1-B,1]. e-0 We note that the function u = 0 is a singular solution of the reduced equation uu' - u = 0 (cf. [25]). With these examples as motivation, we can develop a corresponding theory for such problems. Let us only outline the details and ask the interested reader to complete the analysis or consult [28] and [32].

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