Behaviour of the boundary potentials and boundary integral solution of the time fractional diffusion equation

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>The dissertation considers the time fractional diffusion equation (TFDE) with the Dirichlet boundary condition in the sub-diffusion case, i.e. the order of the time derivative is α ∈ (0,1). In the thesis we have studied the solvability of TFDE by the method of layer potentials. We have shown that both the single layer potential and the double layer potential approaches lead to integral equations which are uniquely solvable.</p><p>The dissertation consists of four articles and a summary section. The first article presents the solution for the time fractional diffusion equation in terms of the single layer potential. In the second and third article we have studied the boundary behaviour of the layer potentials for TFDE. The fourth paper considers the spline collocation method to solve the boundary integral equation related to TFDE.</p><p>In the summary part we have proved that TFDE has a unique solution and the solution is given by the double layer potential when the lateral boundary of a bounded domain admits <em>C</em>¹ regularity. Also, we have proved that the solution depends continuously on the datum in the sense that a nontangential maximal function of the solution is norm bounded from above by the datum in <em>L</em>²(Σ<em>_T</em>). If the datum belongs to the space <em>H</em>^1,α/2(Σ<em>_T</em>), we have proved that the nontangential function of the gradient of the solution is norm bounded from above by the datum in <em>H</em>^1,α/2(Σ<em>_T</em>).</p></div>