Existence and uniqueness of the solution for a time-fractional diffusion equation

In the paper existence and uniqueness of the solution for a time-fractional diffusion equation on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a fundamental solution of the problem is known, we may seek the solution as the double layer potential. This approach leads to a Volterra integral equation of the second kind associated with a compact operator. Then classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary datum is continuous and satisfies a compatibility condition. This proves that the original problem has a unique solution and the solution is given by the double layer potential.