Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density

Abstract

In this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation is considered. Here, the time-fractional derivative D tβ is understood in the Caputo sense and p(β) is a non-negative weight function with support somewhere in the interval [0, 2]. By employing the technique of the Fourier and Laplace transforms, a representation of the fundamental solution of the Cauchy problem in the transform domain is obtained. The main focus is on the interpretation of the fundamental solution as a probability density function of the space variable x evolving in time t. In particular, the fundamental solution of the time-fractional distributed order wave equation (p(β) ≡ 0, 0 ≤ β < 1) is shown to be non-negative and normalized. In the proof, properties of the completely monotone functions, the Bernstein functions, and the Stieltjes functions are used. © 2013 Versita Warsaw and Springer-Verlag Wien.