A problem with inverse time for a~singularly perturbed integro-differential

A. A. Bobodzhanov ; V. F. Safonov

Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 285-298.

Résumé

We consider an algorithm for constructing asymptotic solutions regularized in the sense of Lomov (see [1], [2]). We show that such problems can be reduced to integro-differential equations with inverse time. But in contrast to known papers devoted to this topic (see, for example, [3]), in this paper we study a fundamentally new case, which is characterized by the absence, in the differential part, of a linear operator that isolates, in the asymptotics of the solution, constituents described by boundary functions and by the fact that the integral operator has kernel with diagonal degeneration of high order. Furthermore, the spectrum of the regularization operator $A(t)$ (see below) may contain purely imaginary eigenvalues, which causes difficulties in the application of the methods of construction of asymptotic solutions proposed in the monograph [3]. Based on an analysis of the principal term of the asymptotics, we isolate a class of inhomogeneities and initial data for which the exact solution of the original problem tends to the limit solution (as $\varepsilon\to+0$) on the entire time interval under consideration, also including a boundary-layer zone (that is, we solve the so-called initialization problem). The paper is of a theoretical nature and is designed to lead to a greater understanding of the problems in the theory of singular perturbations. There may be applications in various applied areas where models described by integro-differential equations are used (for example, in elasticity theory, the theory of electrical circuits, and so on).

Détail

Keywords: diagonal degeneration of the kernel, integro-differential equation, initialization.
Mots-clés : singular perturbation

Bibliographie

[1] S. A. Lomov, Introduction to the general theory of singular perturbations, Transl. Math. Monogr., 112, Amer. Math. Soc., Providence, RI, 1992, xviii+375 pp. | MR | MR | Zbl | Zbl

[2] S. A. Lomov, I. S. Lomov, Osnovy matematicheskoi teorii pogranichnogo sloya, Izd-vo Mosk. un-ta, M., 2011, 456 pp.

[3] M. I. Imanaliev, Metody resheniya nelineinykh obratnykh zadach i ikh prilozhenie, Ilim, Frunze, 1977, 347 pp. | MR | Zbl

[4] A. A. Bobodzhanov, V. F. Safonov, “Asimptoticheskoe integrirovanie singulyarno-vozmuschennykh integro-differentsialnykh uravnenii s diagonalnym vyrozhdeniem yadra proizvolnogo poryadka”, Vestnik MEI, 2008, no. 6, 16–25

[5] A. A. Bobodzhanov, F. E. Goryunov, V. F. Safonov, “Predelnyi perekhod v singulyarno vozmuschennykh sistemakh s neobratimym predelnym operatorom i zadacha initsializatsii”, Matematicheskie metody i prilozheniya: Trudy devyatnadtsatykh matematicheskikh chtenii RGSU, Chast I, APKiPPRO, M., 2010, 3–9

[6] A. A. Bobodzhanov, V. F. Safonov, “Singulyarno vozmuschennye integro-differentsialnye uravneniya s diagonalnym vyrozhdeniem yadra vysokogo poryadka”, Matematicheskie metody i prilozheniya: Trudy dvadtsatykh matematicheskikh chtenii RGSU, APKiPPRO, M., 2011, 3–11

[7] A. S. Omuraliev, Spektralnyi podkhod dlya asimptoticheskogo izucheniya integro-differentsialnykh uravnenii, Avtoref. diss. ... kand. fiz.-matem. nauk, Alma-Ata, 1979, 15 pp.