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

By Rudolf Gorenflo, Yuri Luchko, Mirjana Stojanovic. 2013

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AbstractIn 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 Dtβ 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.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/8d74565b67ec0f56a19feff5962cd9ef1ffe080e