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