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

Singularly Perturbed Problems with Kernes Depending on Fundamental Solutions of Integro-differential Equations

Author(s):

Abduhafiz Abdurasulovich Bobodzhanov

Professor, Doctor of Physical and Mathematical Sciences, Professor of the Department of Higher Mathematics, National Research University «Moscow Power Engineering Institute» Moscow, Russia

bobojonova@mpei.ru

Mashkhura Abduhafizovna Bobodzhanova

Associate Professor, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Higher Mathematics, National Research University «Moscow Power Engineering Institute» Moscow, Russia

bobojonovaMA@mpei.ru

Valeriy Fedorovich Safonov

Professor, Doctor of Physical and Mathematical Sciences, Professor of the Department of Higher Mathematics, National Research University «Moscow Power Engineering Institute» Moscow, Russia

Singsaf@yandex.ru

Abstract:

The paper considers a system of two singularly perturbed integro-differential equations (IDEs), the first of which is a homogeneous equation, and the second is an inhomogeneous one, with an integral operator whose kernel contains the fundamental solution of the first IDE. The classical case, when the kernel depends on a rapidly changing scalar exponential, is the subject of a large number of papers (see bibliography at the end of the article). The case of the dependence of the kernel on the fundamental solutions of differential systems was studied in detail in the monograph by A.A. Bobodzhanov and V.F. Safonov “Singularly perturbed integral and integro-differential equations with rapidly changing kernels and equations with digonal degeneration of the kernel”, published by Sputnik+ in 2017. As shown in this paper, the difficulty of constructing a regularized (in the sense of Lomov) asymptotics of IDEs is due to the complex structure of asymptotic solutions of fundamental solutions of homogeneous differential equations. The problem of constructing the asymptotics of the fundamental solution of a homogeneous IDE and its influence through the kernel on the regularized asymptotics of a nonhomogeneous IDE has not been studied so far. In the present work, this gap is filled. It first constructs a regularized asymptotics of the fundamental solution of a homogeneous IDE, and then develops an algorithm for constructing an asymptotic solution of a nonhomogeneous IDE. It is shown that (in contrast to the asymptotics with a kernel depending on the fundamental solution of a homogeneous differential equation), the asymptotics of the solution of an inhomogeneous IDE will contain, in addition to rapidly changing terms, also slowly changing components induced by the asymptotics of the fundamental solution.

Keywords

• fundamental solution asymptotics
• regularization
• singularly perturbed integro-differential equations

References:

1. S. A. Lomov, Vvedenie v obshyyj teorij singuljrnix vozmysheniy [Introduction to the general theory of singular perturbations], Nauka, Moskva, 1981
2. A. A. Bobodzhanov, V. F. Safonov, Volterra integral equations with rapidly varying kernels and their asymptotic integration, Math. Sb., 192:8 (2001), 53-78, Sb. Math., 192:8 (2001), 1139-1164. (In Russian)
3. V. F. Safonov, B. T. Kalimbetov, Regularization method for systems with unstable spectral the value of the kernel of the integral operator, Diff. Uravn., 31:4 (1995), Diff. Equ., 31:4 (1995), 647-656. (In Russian)
4. S. A. Lomov, Single-valued solvability of some matrix partial differential equations, Matem. Zametki (1977), 525-530, 21:4 (1977), Math. Notes, 21:4 (1977), 293-296. (In Russian)
5. M. I. Imanaliyev, Kolebaniya i ustoychivost' resheniy singulyarno vozmush-chennykh integro-differentsial'nykh system [Fluctuations and stability of solutions of singularly perturbed integro-differential systems], Frunze: «Ilim», 1974
6. A. A. Bobodzhanov, V. F. Safonov, Singularly perturbed nonlinear integro-differential systems with rapidly changing kernel, Matem. zametki, 72:5 (2002), 654-664, Math. Notes, 72:5 (2002), 605-614. (In Russian)
7. S. A. Lomov, I. S. Lomov, Osnovy matematicheskoy teorii pogranichnogo sloya, [Foundations of the mathematical theory of the cognitive layer], Moskva, Izdatel'stvo Moskovskogo universiteta , 2011
8. V. F. Safonov, O. D., Tuychiyev, Regularization of singularly perturbed integral equations with rapidly varying kernels and their asymptotic, Diff. Uravn., 33:9(1997), 1199-1210; Diff. Equ., 33:9 (1997), 1203-1215. (In Russian)
9. A. A. Bobodzhanov, V. F. Safonov , Asymptotic solutions of an integro-differential system with rapidly changing kernels of a special type, Bulletin of the MEI., Vestnik MEI, №6(2011), 47-56 (In Russian)
10. V. F. Safonov, A. A. Bobodzhanov, Kurs vysshey matematiki. Singulyarno vozmushchennyye zadachi i metod regulyarizatsii [A course in higher mathematics. Singularly perturbedproblems and regularization method], uchebnoye posobiye, Izdatel'skiy dom MEI, Moskva, 2012

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