Solving a backward problem for a distributed-order time fractional diffusion equation by a new adjoint technique

Abstract

Abstract This paper studies a backward problem for a time fractional diffusion equation, with the distributed order Caputo derivative, of determining the initial condition from a noisy final datum. The uniqueness, ill-posedness and a conditional stability for this backward problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization. Based on the series representation of the regularized solution, we give convergence rates under an a-priori and an a-posteriori regularization parameter choice rule. With a new adjoint technique to compute the gradient of the functional, the conjugate gradient method is applied to reconstruct the initial condition. Numerical examples in one- and two-dimensional cases illustrate the effectiveness of the proposed method.