Numerical solution of inverse source problem for one-dimensional integer/fractional order diffusion equation by one point observation data

Sen Zhang, Zhousheng Ruan

In this paper, we consider reconstructing the space-dependent source of a one-dimensional fractional diffusion equation by observing the data from the left endpoint. First, we analyze the ill-posedness of the problem, and then use the Laplace transform and analytical continuation techniques to prove the uniqueness of the inverse source problem. Then, the inverse source problem is transformed into a variational optimization problem by Tikhonov regularization method. The gradient of the functional is derived based on the idea of variational adjoint method, and then the conjugate gradient method is used to solve the problem. Finally, we give several numerical examples to show the effectiveness of the proposed method.

https://dx.doi.org/10.31873/IJEAS.6.11.06