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Construction of solutions to parabolic and hyperbolic initial–boundary value problems

William G. Litvinov, Eugene Lytvynov Orcid Logo

Applicable Analysis, Volume: 99, Issue: 14, Pages: 2381 - 2413

Swansea University Author: Eugene Lytvynov Orcid Logo

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DOI (Published version): 10.1080/00036811.2019.1566528

Abstract

Assume that, in a parabolic or hyperbolic equation, the right-hand side is analytic in time and the coefficients are analytic in time at each fixed point of the space. We show that the infinitely differentiable solution to this equation is also analytic in time at each fixed point of the space. This...

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Published in: Applicable Analysis
ISSN: 0003-6811 1563-504X
Published: 2019
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URI: https://cronfa.swan.ac.uk/Record/cronfa48055
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first_indexed 2019-01-07T14:02:03Z
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spelling 2021-11-30T15:48:16.4399215 v2 48055 2019-01-07 Construction of solutions to parabolic and hyperbolic initial–boundary value problems e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2019-01-07 SMA Assume that, in a parabolic or hyperbolic equation, the right-hand side is analytic in time and the coefficients are analytic in time at each fixed point of the space. We show that the infinitely differentiable solution to this equation is also analytic in time at each fixed point of the space. This solution is given in the form of the Taylor expansion with respect to time $t$ with coefficients depending on $x$. The coefficients of the expansion are defined by recursion relations, which are obtained from the condition of compatibility of order $k=\infty$. The value of the solution on the boundary is defined by the right-hand side and initial data, so that it is not prescribed. We show that exact regular and weak solutions to the initial-boundary value problems for parabolic and hyperbolic equations can be determined as the sum of a function that satisfies the boundary conditions and the limit of the infinitely differentiable solutions for smooth approximations of the data of the corresponding problem with zero boundary conditions. These solutions are represented in the form of the Taylor expansion with respect to $t$. The suggested method can be considered as an alternative to numerical methods of solution of parabolic and hyperbolic equations. Journal Article Applicable Analysis 99 14 2381 2413 0003-6811 1563-504X Parabolic equation, hyperbolic equation, smooth solution, regular solution, Taylor expansion 21 1 2019 2019-01-21 10.1080/00036811.2019.1566528 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2021-11-30T15:48:16.4399215 2019-01-07T11:43:38.6838569 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics William G. Litvinov 1 Eugene Lytvynov 0000-0001-9685-7727 2 0048055-07012019121952.pdf finalversionv2.pdf 2019-01-07T12:19:52.3530000 Output 367773 application/pdf Accepted Manuscript true 2020-01-21T00:00:00.0000000 true eng
title Construction of solutions to parabolic and hyperbolic initial–boundary value problems
spellingShingle Construction of solutions to parabolic and hyperbolic initial–boundary value problems
Eugene Lytvynov
title_short Construction of solutions to parabolic and hyperbolic initial–boundary value problems
title_full Construction of solutions to parabolic and hyperbolic initial–boundary value problems
title_fullStr Construction of solutions to parabolic and hyperbolic initial–boundary value problems
title_full_unstemmed Construction of solutions to parabolic and hyperbolic initial–boundary value problems
title_sort Construction of solutions to parabolic and hyperbolic initial–boundary value problems
author_id_str_mv e5b4fef159d90a480b1961cef89a17b7
author_id_fullname_str_mv e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov
author Eugene Lytvynov
author2 William G. Litvinov
Eugene Lytvynov
format Journal article
container_title Applicable Analysis
container_volume 99
container_issue 14
container_start_page 2381
publishDate 2019
institution Swansea University
issn 0003-6811
1563-504X
doi_str_mv 10.1080/00036811.2019.1566528
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description Assume that, in a parabolic or hyperbolic equation, the right-hand side is analytic in time and the coefficients are analytic in time at each fixed point of the space. We show that the infinitely differentiable solution to this equation is also analytic in time at each fixed point of the space. This solution is given in the form of the Taylor expansion with respect to time $t$ with coefficients depending on $x$. The coefficients of the expansion are defined by recursion relations, which are obtained from the condition of compatibility of order $k=\infty$. The value of the solution on the boundary is defined by the right-hand side and initial data, so that it is not prescribed. We show that exact regular and weak solutions to the initial-boundary value problems for parabolic and hyperbolic equations can be determined as the sum of a function that satisfies the boundary conditions and the limit of the infinitely differentiable solutions for smooth approximations of the data of the corresponding problem with zero boundary conditions. These solutions are represented in the form of the Taylor expansion with respect to $t$. The suggested method can be considered as an alternative to numerical methods of solution of parabolic and hyperbolic equations.
published_date 2019-01-21T03:58:21Z
_version_ 1763752963301441536
score 11.036706

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