Filter regularization method for a nonlinear Riesz-Feller space-fractional backward diffusion problem with temporally dependent thermal conductivity

Abstract

Abstract This paper concerns a backward problem for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity. Such a problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2), which is usually used to model the anomalous diffusion. We show that the problem is severely ill-posed. Using the Fourier transform and a filter function, we construct a regularized solution from the data given inexactly and explicitly derive the convergence estimate in the case of the local Lipschitz reaction term. Special cases of the regularized solution are also presented. These results extend some earlier works on the space-fractional backward diffusion problem.