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 equationis 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.

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

“…In contrast to the slow diffusion, which is characterized by the mean square displacement of the diffusing particles of the power type t α , the mean square displacement in the framework of the ultra slow diffusion is just of logarithmic growth (see e.g., [3]- [4], [15], [17], [21] and the references therein). Another direction of research regarding the distributed order fractional differential equations, which is important for potential applications was to investigate how they preserve the positivity of the initial conditions in time, e.g., to analyze if their fundamental solutions can be interpreted as some probability density functions (see e.g., [3], [6] and the references therein).…”

The mellin integral transform in fractional calculus

“…to show that fundamental solutions to some fractional differential equations can be interpreted as some probability densities (see e.g. [10], [25] and references therein). In this section we demonstrate how the Mellin integral transform can be employed to deduce some new completely monotone functions based on the known ones.…”

Probability Theory and Integrals of Distributed Order

“…Отметим, что концепция интегрирование и дифференцирования распределенного порядка была предложена М. Капуто в 1995 году [2], и затем развита в работах других авторов (например, см. [3,4,5,6,7]). …”