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

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Abstract

In this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation

$\int_0^2 {p(\beta )D_t^\beta u(x,t)d\beta } = \frac{{\partial ^2 }} {{\partial x^2 }}u(x,t) $

is considered. Here, the time-fractional derivative D β t 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.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Free University of Berlin, Arnimallee 3, D-14195, Berlin, Germany

    Rudolf Gorenflo

  2. Department of Mathematics, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, D-13353, Berlin, Germany

    Yuri Luchko

  3. Department of Mathematics and Informatics Faculty of Science, University of Novi Sad, Trg D.Obradovića 4, 21 000, Novi Sad, Serbia

    Mirjana Stojanović

Authors
  1. Rudolf Gorenflo
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  2. Yuri Luchko
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  3. Mirjana Stojanović
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Correspondence to Rudolf Gorenflo.

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Dedicated to Professor Francesco Mainardi on the occasion of his 70th anniversary

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Gorenflo, R., Luchko, Y. & Stojanović, M. Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. fcaa 16, 297–316 (2013). https://doi.org/10.2478/s13540-013-0019-6

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  • DOI: https://doi.org/10.2478/s13540-013-0019-6

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