Existence and calculation of the solution to the time distributed order diffusion equation

Abstract

The aim of this paper is to prove the existence of the solution to the Cauchy problem for the time distributed order diffusion equation as well as to calculate it. The existence is proved in this paper by reducing the Cauchy problem to an abstract Volterra equation in the case where the weight distribution in the distributed order derivative is a finite sum of Dirac distributions. Calculation of the solution is done by the use of Fourier and Laplace transformations in the case where the weight distribution (or function) is not specified. The solution is expressed in terms of heat potential kernel. The solutions for several special cases of the weight distribution, including the case of a finite sum of Dirac distributions, are presented as well.