Applicability of Laplace transform for fractional PDE with variable rate of derivatives

Abstract

Method of Laplace transform is standard in mathematical physics. One of its advantages is that it is still applicable in the case of PDE with fractional derivatives. But the situation becomes much more complicated if the rate of differentiation can vary with time, even if these variations have very small amplitude. In this paper, Theorem is proved which gives the expression for the Laplace transform of the diffusion equation with time-dependent rate of time derivative. Such equations appear in biology and financial mathematics. It follows that in case of variable fractional derivatives the method of Laplace transform can be used only under certain conditions as approximate method.