Abstract
This paper investigates a Dirichlet problem for a time fractional diffusion-wave equation (TFDWE) in Lipschitz domains. Since (TFDWE) is a reasonable interpolation of the heat equation and the wave equation, it is natural trying to adopt the techniques developed for solving the aforementioned problems. This paper continues the work done by the author for a time fractional diffusion equation in the subdiffusive case, i.e. the order of the time differentiation is 0 < α < 1. However, when compared to the subdiffusive case, the operator ∂ α t in (TFDWE) is no longer positive. Therefore we follow the approach applied to the hyperbolic counterpart for showing the existence and uniqueness of the solution.
We use the Laplace transform to obtain an equivalent problem on the space-Laplace domain. Use of the jump relations for the single layer potential with density in H −1/2(Γ) allows us to define a coercive and bounded sesquilinear form. The obtained variational form of the original problem has a unique solution, which implies that the original problem has a solution as well and the solution can be represented in terms of the single layer potential.
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Kemppainen, J. Solvability of a Dirichlet problem for a time fractional diffusion-wave equation in Lipschitz domains. fcaa 15, 195–206 (2012). https://doi.org/10.2478/s13540-012-0014-3
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DOI: https://doi.org/10.2478/s13540-012-0014-3