An Inverse Time-dependent Source Term Problem For A Time-space Fractional Diffusion Wave Equation

This paper studies time source term inversion problem in a time-space fractional diffusion wave equation.Firstly,we analyze the existence,uniqueness and regularity for the solution of direct problem.Secondly the inverse time-dependent source term problem is considered,the integral equation satisfied by the time source term is derived and the ill-posedness of the inverse problem is illustrated.A stability of the solution of inverse problem is proved,and the continuous problem is discretized using the piecewise constants to obtain the ill-conditioned linear algebraic equations.To solve this ill-posed problem,we select the split Bregman iterative method to solve.Finally,through several numerical examples for twodimensional case,it is shown that the split Bregman iterative method is more effective for non-smooth examples than the classic Tikhonov method.It can be seen from the main conclusions of the paper that the solution of the direct problem exists and is unique,and the regularity is relatively high;the inverse problem is seriously ill-posed,and an effective regularization method is needed to solve it.