Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient

Abstract

We consider numerical methods for solving inverse problems for time fractional diffusion equation (TFDE) with the variable generalized diffusion coefficient q(x). Inverse problems are to find q(x) and the order α of the time derivative according to an additional information about a solution of TFDE. The weighted difference scheme is constructed via integro-interpolation method and the generalized factorization method is developed; results of stability analysis of the difference scheme are presented and properties of numerical solutions of forward problems are investigated. Inverse problems for TFDE with the variable coefficient are formulated as residual function minimization problems and properties of corresponding residual functions are discussed. The Levenberg–Marquardt algorithm for residual function minimization is used and numerical results are presented.