Boundary Element Collocation Method for Time-Fractional Diffusion Equations

Abstract

In this chapter, we discuss the numerical solution of the space-time boundary integral equation

$$S_{\Gamma u \Gamma} (x, t) = \int^t_0 \int_\Gamma u_\Gamma (y, \tau)E(x - y, t - \tau){\rm d}s_y {\rm d}\tau = f(x, t),\ \ x \in \Gamma, \quad 0 < t < T,$$

where Γ is a smooth plane curve. The kernel of the integral operator,

$$E(x,t) = \frac{1} {\pi }t^{\alpha - 1} |x|^{ - 2} H_{12}^{20} \left( {\frac{1} {4}|x|^2 t^{ - \alpha } |\begin{array}{*{20}l} {(\alpha ,\alpha )} \\ {(1,1),(1,1)} \\ \end{array} } \right),\,\,\,\,\,\,0 < \alpha \leq 1,$$

is the fundamental solution of the time-fractional diffusion equation (see [KiSa04] and [PBM90]). We consider the problem

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

where the boundary operator B(Φ) = Φ|_ΣT and ∂^α _t is the Caputo time derivative of the fractional order 0 < α ≤ 1.

We shall consider the spline collocation method for the numerical approxproductimation of the solution on quasi-uniform meshes with piecewise linear tensor splines as the approximation space. We will show that the spline collocation method is stable in a suitable anisotropic Sobolev space, and it furnishes quasi-optimal error estimates.