Variational Method For Solving An Inverse Initial Value Problem For A Time-fractional Diffusion Equation

In this paper,we consider an inverse initial value problem for a time-fractional diffusion equation.By using the Fourier separation of variables,the regularity of a weak solution for the direct problem and the existence and uniqueness of a weak solution for the adjoint problem are proved.We transform the inverse initial value problem into a variational problem by using the Tikhonov regularization method,and obtain an approximation to the minimizer of the variational problem by using a conjugate gradient method.Four numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.