Abstract and Applied Analysis
Volume 2011 (2011), Article ID 321903, 11 pages
doi:10.1155/2011/321903
Research Article

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

Mathematics Division, Department of Electrical and Information Engineering, Faculty of Technology, University of Oulu, PL 4500, 90014 Oulu, Finland

Received 10 January 2011; Revised 1 March 2011; Accepted 8 March 2011

Academic Editor: W. A. Kirk

Copyright © 2011 Jukka Kemppainen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.