title: Singularly Perturbed Problems with Kernes Depending on Fundamental Solutions of Integro-differential Equations
metadata.name: dc.contributor.author
metadata.value: Bobodzhanov, Abduhafiz Abdurasulovich
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Bobodzhanova, Mashkhura Abduhafizovna
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Safonov, Valeriy Fedorovich
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2024-12-02T13:28:10Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2024-12-02T13:28:10Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2024-06
metadata.lang: -
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/11701/spbu35.2024.202
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/48046
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: 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.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: The work was partially supported by Russian Science foundation (project
   23-21-00496).
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Differential Equations and Control Processes;Issue 1
metadata.lang: -
metadata.name: dc.subject
metadata.value: singularly perturbed
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: integro-differential equations
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: regularization
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: fundamental solution asymptotics
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Singularly Perturbed Problems with Kernes Depending on Fundamental Solutions of Integro-differential Equations
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 24202-jdecp-babodzhanov.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/48046/3/24202-jdecp-babodzhanov.pdf
resources.format: Adobe PDF
resources.size: 297,9 kB