Inverse Initial Value Problem For A Space-time Fractional Diffusion Equation

Fractional diffusion equations are deduced by replacing the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion,especially in physics.In this paper,we consider a backward problem for a space-time fractional diffusion equation in a general bounded domain.That is to determine the initial data from a noisy final data.Based on a series expression of the solution,a conditional stability for the initial data is given.Further,we use a modified quasi-boundary value regularization method to deal with the backward problem and obtain two kinds of convergence rates by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule.Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the used methods.