An Inverse Problem of Recovering the Variable Order of the Derivative in a Fractional Diffusion Equation

Abstract

We considera fractional diffusion equation with variable space-dependent order of the derivativein a bounded multidimensional domain.The initial data are homogeneous and the right-hand side and its time derivativesatisfy some monotonicity conditions.Addressing the inverse problem with final overdetermination, we establishthe uniqueness of a solution as well as some necessary and sufficient solvability conditionsin terms of a certain constructive operator $ A $ .Moreover, we give a simple sufficient solvability condition for the inverse problem.The arguments rely on the Birkhoff–Tarski theorem.