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Русская версия

**Abduhafiz A. Bobodzhanov**

Department of mathematics,

National research University MPEI,

Moscow

**Maushkura A. Bobodzhanova**

Department of mathematics,

National research University MPEI,

Moscow

**Valeriiy F. Safonov**

Department of mathematics,

National research University MPEI,

Moscow

For nonlinear singularly perturbed integro-differential problem with diagonal degeneration of the kernel the algorithm for constructing regularized by Lomov asymptotic solutions is developed. Theorems of normal and unique solvability of the corresponding iterative tasks both in non-resonance and resonance case are proved. Asymptotic convergence of formal solutions to the exact one can be justified by using Newton's method in the Srubshchik-Yudovich form. Basing on the analysis of the leading term of the asymptotic behavior we describe the class of right-hand parts and initial data for which the exact solution of the original problem tends to limit function (when the small parameter tends to zero ) on all the considered period of time (i.e. the so called initialization problem is solved).

- boundary layer
- diagonal degeneration of the kernel
- integro-differential equation
- singular perturbation
- the initialization problem

**Bobodzhanov A. A., Safonov V. F.**Regularized asymptotic solutions of singularly perturbed integral systems with diagonal degeneration of the kernel*, Diff. Yravn.*,**37 (**2001**)**, no 10 , 1330-1341 (in Russian)**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)**Bobodzhanov A. A., Safonov V. F.**Singularly perturbed nonlinear integro-differential systems with fast varying kernels,*Math. Zametki*,**72**(2002), no 5, 654-664 (in Russian)**Srubshchik L. S, Judovich V. I.**[About application of a method of Newton in problems of asymptotic integration of the nonlinear equations].*Tezisi dokladov Vsesojznoi mezhvvzovskoi konferencii po primeneniy metodov funkzionalnogo analiza k resheniy nelineinikh zadach*. [Abstracts of All-Union interuniversity conference on application of functional analysis methods to nonlinear problems solving]. Baku ,**1975**. p. 103 (in Russian)**S. A. Lomov, I. S. Lomov.***Osnovi mathematicheckoi teorii pogranichnogo cloy*[Fundamentals of the mathematical theory of the boundary layer]. Moscow, Publishing house of MSU, 2011. 456 p. (in Russian)**Bobodzhanov A. A., Safonov V. F.**Method of a regularization for integro-differential systems of the Fredholm type with fast varying kernels,*Diff. Yravn.*,**51**(2015), no 2, 251-262 (in Russian)**Bobodzhanova M. A, Safonov V. F.**The asymptotic analysis of singularly perturbed integro-differential systems with a zero operator of a differential part,*Diff. Yravn.*,**47**(2011), no 4, 519-536 (in Russian).**Bobodzhanov A. A., Safonov V. F.**Regularized asymptotic solutions of singularly perturbed integral systems with diagonal degeneration of the kernel*, Diff. Yravn.*,**37 (**2001**)**, no 10 , 1330-1341 (in Russian)**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)**Bobodzhanov A. A., Safonov V. F.**Singularly perturbed nonlinear integro-differential systems with fast varying kernels,*Math. Zametki*,**72**(2002), no 5, 654-664 (in Russian)**Srubshchik L. S, Judovich V. I.**[About application of a method of Newton in problems of asymptotic integration of the nonlinear equations].*Tezisi dokladov Vsesojznoi mezhvvzovskoi konferencii po primeneniy metodov funkzionalnogo analiza k resheniy nelineinikh zadach*. [Abstracts of All-Union interuniversity conference on application of functional analysis methods to nonlinear problems solving]. Baku ,**1975**. p. 103 (in Russian)**S. A. Lomov, I. S. Lomov.***Osnovi mathematicheckoi teorii pogranichnogo cloy*[Fundamentals of the mathematical theory of the boundary layer]. Moscow, Publishing house of MSU, 2011. 456 p. (in Russian)**Bobodzhanov A. A., Safonov V. F.**Method of a regularization for integro-differential systems of the Fredholm type with fast varying kernels,*Diff. Yravn.*,**51**(2015), no 2, 251-262 (in Russian)**Bobodzhanova M. A, Safonov V. F.**The asymptotic analysis of singularly perturbed integro-differential systems with a zero operator of a differential part,*Diff. Yravn.*,**47**(2011), no 4, 519-536 (in Russian)