Boundary Element Collocation Method for Time-Fractional Diffusion Equations

Chapter

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.

Keywords

Approximation Space Spline Space Boundary Integral Operator Anisotropic Sobolev Space Spline Collocation Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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