Boundary Integral Solution of the Time-Fractional Diffusion Equation

Abstract

In this chapter we discuss the boundary integral solution of the fractional diffusion equation

$$\begin{array}{c}\partial^\alpha_t \Phi - \Delta\Phi = 0, {\rm in} \ Q_T = \Omega \times (0, T),\\ \Phi = g, {\rm on} \sum_T = \Gamma \times (0, T),\\ \Phi(x, 0) = 0, \ x \in \Omega,\end{array}$$

(20.1)

where $\Omega \subset \mathbb{R}^n$ is a smooth, bounded domain and ∂^α _t is the Caputo time derivative of the fractional order 0 < α ≤ 1. For α = 1 we get the ordinary diffusion equation and for ? = 0 we have the Helmholtz equation.

We present the fundamental solution by means of the Fox H-functions, and represent the solution of (1) as a single-layer potential. By the jump relations of the potential we derive the appropriate boundary integral operator.We give spaces, which yields the unique solution of the boundary integral equation and thus the unique solution of the initial boundary value problem.