Identifying an unknown coefficient in the fractional parabolic differential equation

Abstract

In this study, we considered a fractional parabolic equation for identifying the unknown diffusion coefficient from the noisy measurement of the ultimate time solution. It is an inverse problem involving a nonlocal operator that is nonlinear and poorly formulated. By demonstrating the existence of this inverse problem singular solution with regard to the final observed data, we demonstrate the identifiability of this problem. The inverse problem is expressed as a regularized optimization problem that minimizes a cost function of the least-squares kind. We have covered various theoretical and practical difficulties pertaining to the problem under consideration. It has been demonstrated that a unique stable solution to the optimization problem exists. The Morozov discrepancy principle and the conjugate gradient approach are used to develop an iterative reconstruction procedure. The accuracy and efficiency of the suggested method are illustrated using a few numerical examples.

Keywords

• Fractional parabolic PDE
• Inverse potential problem
• Tikhonov’s regularization
• Conjugate gradient method
• Finit element method