Inverse Space-dependent Source Problem For A Time Fractional Diffusion Equation And A Diffusion Wave Equation

This paper divide two parts.In the first part,we consider an inverse spacedependent source problem for a time-fractional diffusion equation by an adjoint problem approach.Based on the series expression of the solution for the direct problem,we improve the regularity of the weak solution for the direct problem under strong conditions.And we provide the existence and uniqueness for the adjoint problem.Further,we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional.Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.The second part is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data.Based on the series expression of solution for the direct problem,we improve the regularity of the weak solution for the direct problem under strong conditions.And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle.Further,we use an optimal perturbation algorithm to find an approximation to the source term.Numerical examples in one-dimensional case are provided to show the effectiveness of the proposed method.