ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

# Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

## Regularization Method for Singularly Perturbed Integro-differential Systems with Two Independent Variables

### Author(s):

A. A. Bobodzhanov

Higher mathematics department
National Research University "Moscow Power Engineering Institute",
Address: Krasnokazarmennaya 14, Moscow, 111250 Russia

bobojanova@mpei.ru

V. F. Safonov

Higher mathematics department
National Research University "Moscow Power Engineering Institute",
Address: Krasnokazarmennaya 14, Moscow, 111250 Russia

### Abstract:

In this paper, the Lomov regularization method is generalized to Volterra-type integro-differential partial differential equations with a two-dimensional integral operator. We consider the case when the operator of the differential part depends only on the differentiation variable. Thus, in contrast Imanaliev, M. I. (where only the passage to the limit is studied, when a small parameter tends to zero), in this paper, the focus is on constructing a regularized asymptotic solution of any order (with respect to a small parameter). Note that the Lomov regularization method was used mainly for ordinary singularly perturbed integro-differential equations (see the detailed bibliography at the end of the article). In one of the authors’ works, the case of a partial differential equation with a one-dimensional integral operator was considered. The development of this method for systems of partial differential equations with a two-dimensional integral operator has not been carried out before. The paper also considers and solves the «initialization problem», i.e., the problem of choosing the initial data of the problem in which the limit transition in its solution (in a uniform metric over the entire considered set of independent variables, including the boundary layer zone) becomes possible.

### Keywords

• integro-differential equation
• regularization of the integral
• singular perturbations

### References:

1. Safonov, V. F., Bobodzhanov, A. A. Kurs vysshey matematiki. Singulyarno vozmushchennyye uravneniya i metod regulyarizatsii: uchebnoye posobiye [Course of higher mathematics. Singularly perturbed equations and the regularization method: a training manual]. Moscow, Izdatel'skiy dom MEI, 2012. 416 p. (in Russian)
2. Lomov, S. A. Vvedenie v obshyj teorij singuliajrnikh vozmyshenii [Introduction to the General Theory of Singular Perturbations]. Moscow, Nauka Publ., 1981. 400 p. (in Russian)
3. Lomov, S. A., Lomov, I. S. Osnovi mathematicheckoi teorii pogranichnogo sloya [Fundamentals of the mathematical theory of the boundary layer]. Moscow, Publishing house of MGU, 2011. 456 p. (in Russian)
4. Imanaliev, M. I. Metody resheniya obratnykh zadach i ikh prilozheniye [Methods for solving inverse problems and their application]. Frunze. ILIM Publ, 1977. 347 p. (in Russian)
5. Smirnov, V. I. Kurs vysshey matematiki, Tom 4 [Course in Higher Mathematics, Volume 4]. Moscow, GIFML Publ., 1953. 804 p. (in Russian)
6. Filatov, A.N., Sharova, L. V. Integral'nye neravenstva i teoriya nelineynykh kolebaniy [Integral Inequalities and the Theory of Nonlinear Oscillations]. Moscow, Nauka Publ., 1976. 152 p. (in Russian)
7. Bobodzhanov, A. A., Safonov, V. F. Asymptotic integration of integro-differential equation with two variables, Siberian Electronic Mathematical Repots, 15 (2018), 186-197(in Russian)
8. Bobodzhanov, A. A., Safonov, V. F. A generalization of the regularization method to a singularly perturbed partial integro-differential equations, Izv. Vyzov, (2018), no 3, 9-22 (in Russian)
9. Bobodzhanov, A. A., Safonov, V. F. Regularized asymptotic solutions of the initial problem for the system of partial integro-differential equations. Math Notes 102(2017), no 1, 28–38 (in Russian)