By S. A. Lomov
This publication is geared toward researchers and scholars in physics, arithmetic, and engineering. It includes the 1st systematic presentation of a normal method of the mixing of singularly perturbed differential equations describing nonuniform transitions, resembling the prevalence of a boundary layer, discontinuities, boundary results, etc. the tactic of regularization of singular perturbations offered right here should be utilized to the asymptotic integration of platforms of standard and partial differential equations.
Readership: Researchers and scholars in physics, arithmetic and engineering.
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Sample text
We have thus described the possibility of obtaining resonance-free solutions (these are elements of the space U) for the general case where the exponentials have oscillatory character or nonoscillatory character or both simultaneously. §4. Formal regularized series 1. Determination of the coefficients of the series of perturbation theory. 9). We introduce the following DEFINITION. We call each problem of some series of problems asymptotically well posed in some function space if the solution of the corresponding partial differential equation is uniquely determined in the space indicated on the basis of point data on successive solution of the entire series of problems.
9) is uniquely determined in U. 9). 1) on §4. 1) for linear ordinary system in the case of simple spectrum of the matrix A(x) if the conditions for stability of the spectrum are satisfied. 9). We denote the restriction of the space U for t = p(x, e) by U and we introduce a norm in this space. We write an arbitrary element u(x, t) E U in the form u(x, t) = ui(x)e" + + un(x)e", + un+I (x), u;(x) E C°°([0, a]; "). For an element u(x, (x, E)) E U we define the norm as follows: llu(x, (x, E))IIU _ ai = max Re xE o,a] Ilu(x)II, :e:h/e fx lk,(x)I Ilu(x)II k=l XE[O'a] (J Ai(i)dil i = 1, ...
3) We extend the operator L. to a certain operator TE with the help of the spectrum of the limit operator A(x) according to the following scheme. 1) we study a certain "extended" function u(x, t, e) . We have introduced the notation t = {tl , ... , t } , I, anwe further introduce p(x, e) = Moreover, we observe that all I ti I -> oo as e -+ 0 , x > 0 . 6) U t=P ( (x, e) = u(x, e). Under these conditions u '(x, e) - (8u/8x + ,E), where the defines differentiation in the direction operator D. 1i (x)- + +A,,(x) of the spectral vector (x) = (fi(x), ...