Inhomogeneous Fractional Diffusion Equations

Abstract

Fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. An initial value problem involving a space-fractional diffusion equation is an abstract Cauchy problem, whose analytic solution can be written in terms of the semigroup whose generator gives the space-fractional derivative operator. The corresponding time-fractional initial value problem is called a fractional Cauchy problem. Recently, it was shown that the solution of a fractional Cauchy problem can be expressed as an integral transform of the solution to the corresponding Cauchy problem. In this paper, we extend that results to inhomogeneous fractional diffusion equations, in which a forcing function is included to model sources and sinks. Existence and uniqueness is established by considering an equivalent (non-local) integral equation. Finally, we illustrate the practical application of these results with an example from groundwater hydrology, to show the effect of the fractional time derivative on plume evolution, and the proper specification of a forcing function in a time-fractional evolution equation.