Solving a final value fractional diffusion problem by boundary condition regularization
The authors consider the following backward evolution problem \[ \begin{alignedat}{2} {2} \partial^\gamma u/ \partial t^\gamma&= \Delta u &&\text{ for } (x,t) \in \Omega \times (0,T),\\ u(x,t)&= 0 &&\text{ for } x \in \partial \Omega,~ t \in [0,T], \\ u(x,T)&= u_T(x) &&\text{ for } x \in \Omega \end{alignedat} \] in a domain \(\Omega \subset \mathbb{R}^d\), where \(\gamma \in (0,1)\) and \(\partial^\gamma u/ \partial t^\gamma\) denotes the fractional Caputo derivative of \(u\). In the end condition only noisy data \(u_T^\delta\) are supposed to be available satisfying \[ \| u_T^\delta-u_T \|_{L^2(\Omega)} \leq \delta. \] Approximate solutions are obtained by replacing the end condition by the regularizing condition \[ \alpha u(x,0)+u(x,T)= u_T(x) \text{ for } x \in \Omega, \] where \(\alpha>0\) is a regularizing parameter. The regularized problem can be solved exactly in the form of an infinite series with products of Mittag-Leffler functions and eigenfunctions of the \(\Delta\)-operator. The authors prove the convergence of the method as \(\delta \to 0\) if \(\alpha=\alpha(\delta)\) is chosen suitably. An optimal choice of \(\alpha(\delta)\) is discussed. Some nice numerical experiments with \(\gamma=1/2\) and \(\Omega\) as a rectangular domain in \(\mathbb{R}^2\) are reported.
scientific article; zbMATH DE number 6140836