Identifying the Order and a Space Source Term in a Time Fractional Diffusion-Wave Equation

Abstract

This paper is devoted to identifying the order of time fractional derivative and a space-dependent source term in a time fractional diffusion-wave equation from some additional measured data in a subdomain or on a subboundary with a small time period. The Lipschitz continuity of forward operators mapping the unknown order and source term into the given data are established based on the stability estimates of solution for the direct problem. We prove the uniqueness of the considered inverse problems by using the asymptotic behavior of the solution at $t$ = 0, the Titchmarsh convolution theorem and the Duhamel principle. Moreover, a Tikhonov-type regularization method is proposed with $H^1$-norm as a penalty term. The existence of the regularized solution and its convergence to the exact solution under a suitable regularization parameter choice are obtained. Then we employ a linearized iteration algorithm combined with the piecewise linear finite element approximation to find simultaneously the approximate order and space source term. Three numerical examples for one- and two-dimensional cases are tested and the numerical results demonstrate the effectiveness of the proposed method.