Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity

Author:

Kalimbetov Burkhan T.1,Tuychiev Olim D.2

Affiliation:

1. Akhmed Yassawi University, B. Sattarkhanov 29 , Turkestan , 161200 , Kazakhstan

2. Khudjant State University named after B. Gafurov, Movlonbekov Ave. , 735700 , Khudjant , Tajikistan

Abstract

Abstract In this paper, the regularization method of S. A. Lomov is generalized to integro-differential equations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficient and a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observed in the problem. Other cases of the relationship between frequencies can lead to so-called non-identical resonances, the study of which is nontrivial and requires the development of a new approach. It is supposed to study these cases in our further work.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Reference46 articles.

1. A. B. Vasileva and V. F. Butuzov , Asymptotic Expansions of Solutions of Singularly Perturbed Equations, Nauka, Moscow, 1973.

2. A. B. Vasileva and V. F. Butuzov , Singularly Perturbed Equations in Critical Cases, Publishing House of MSU, Moscow, 1978.

3. A. B. Vasileva , V. F. Butuzov , and N. N. Nefedov , Contrast structures in singularly perturbed equations, Fundam. Appl. Math. 4 (1998), no. 3, 799–851.

4. M. Imanaliev , Asymptotic Methods in the Theory of Singularly Perturbed Integro-Differential Equations, Ilim, Frunze, 1972.

5. M. Imanaliev , Methods for Solving Nonlinear Inverse Problems and their Applications, Ilim, Frunze, 1977.

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