Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

Abstract

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent.

Parameter Estimation for Nonlinear Diffusion Problems by the Constrained Homotopy Method

“…For example, it can help reservoir engineers make important decisions about the type of the recovery method, fluid production and injection rates, and well locations. From then on, a variety of effective numerical methods have appeared in the literatures of the inverse problem for non-linear diffusion problems [1][2][3][4][5][6]. This inverse problem can be viewed as a parametric data-fitting problem.…”