Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition

Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the line...

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Main Author: Jukka Kemppainen
Format: Article
Language:English
Published: Hindawi Limited 2011-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2011/321903
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spelling doaj-5f49810924e6498ab9b5ad38aba33fe82020-11-24T23:16:30ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/321903321903Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary ConditionJukka Kemppainen0Mathematics Division, Department of Electrical and Information Engineering, Faculty of Technology, University of Oulu, PL 4500, 90014 Oulu, FinlandExistence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data is continuous, the initial data is continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.http://dx.doi.org/10.1155/2011/321903
collection DOAJ
language English
format Article
sources DOAJ
author Jukka Kemppainen
spellingShingle Jukka Kemppainen
Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition
Abstract and Applied Analysis
author_facet Jukka Kemppainen
author_sort Jukka Kemppainen
title Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition
title_short Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition
title_full Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition
title_fullStr Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition
title_full_unstemmed Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition
title_sort existence and uniqueness of the solution for a time-fractional diffusion equation with robin boundary condition
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2011-01-01
description Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data is continuous, the initial data is continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.
url http://dx.doi.org/10.1155/2011/321903
work_keys_str_mv AT jukkakemppainen existenceanduniquenessofthesolutionforatimefractionaldiffusionequationwithrobinboundarycondition
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