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https://open.uns.ac.rs/handle/123456789/8890
Title: | Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density | Authors: | Gorenflo R. Luchko Y. Stojanović, Mirjana |
Issue Date: | 1-Jun-2013 | Journal: | Fractional Calculus and Applied Analysis | 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. | URI: | https://open.uns.ac.rs/handle/123456789/8890 | ISSN: | 13110454 | DOI: | 10.2478/s13540-013-0019-6 |
Appears in Collections: | PMF Publikacije/Publications |
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