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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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/321903 |
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| _version_ | 1832568289174224896 |
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| author | Jukka Kemppainen |
| author_facet | Jukka Kemppainen |
| author_sort | Jukka Kemppainen |
| collection | DOAJ |
| 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. |
| format | Article |
| id | doaj-art-5f49810924e6498ab9b5ad38aba33fe8 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-5f49810924e6498ab9b5ad38aba33fe82025-02-03T00:59:21ZengWileyAbstract 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 |
| spellingShingle | Jukka Kemppainen Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition Abstract and Applied Analysis |
| title | 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_short | 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 |
| url | http://dx.doi.org/10.1155/2011/321903 |
| work_keys_str_mv | AT jukkakemppainen existenceanduniquenessofthesolutionforatimefractionaldiffusionequationwithrobinboundarycondition |