Reconstruction of a space-dependent source in the inexact order time-fractional diffusion equation

Abstract

Abstract An inverse problem to recover a space-dependent factor of a source term in the inexact order time-fractional diffusion equation from final data is considered. The problem arises in many applications, but it is in general ill-posed. The ill-posedness is since small errors in the input data cause large errors in the output solution. To overcome this instability we propose the stable approximation solution via a general modified quasi-boundary value regularization method. Order optimal convergence rates for the worst case error of the method are derived under the usual source condition by using an a-priori and an a-posteriori regularization parameter choice rules, respectively. Finally, several numerical examples are provided to illustrate the effectiveness of the proposed method.