Exact solutions to some models of distributed-order time fractional diffusion equations via the Fox H functions

Abstract

In this article, three types of time-fractional diffusion equation of distributed order are introduced and some aspects of these equations are discussed. Using the appropriate joint integral transforms, fundamental solutions of these equations are obtained through the Fox H functions. The Mellin transform is an approach to change the fundamental solutions into the Fox H functions.

Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations

“…Then another higher-order difference scheme is also developed with the second-order convergence in time and fourth-order convergence in both space and distributed order. The numerical approach presented here can also be easily applied to solve the time-distributed Klein-Gordon equations [34]. The high-dimensional problems will be our future concern.…”

Analytic Solution of a System of Linear Distributed Order Differential Equations in the Reimann-Liouville Sense

“…The Riemann-Liouville fractional derivative of the power function is given by [12]: RL 0 D t t = ( + 1)t = ( + 1); t > 0; > 1: (7) According to Eq. ( 6), the representation of fractional integral of a function in the Laplace domain is given by [12]: L t!s f 0 I t f(t)g = F (s)=s ;…”

“…The typical approach to obtaining a solution to such equations in the time domain is based on direct inverse Laplace transform using the Fourier-Mellin formula or Titchmarsh theorem [5]. This approach, which is considered in most papers [4,[6][7][8][9] in the literature, generates a solution expressed by a Laplace-type integral in both Riemann-Liouville and Caputo cases. Since the link between this representation and the Fox-Wright functions used in fractional order di erential equations is not clear, an alternative representation of the solution, which incorporates Fox-Wright functions, is proposed in [9] in which the Laplace-type integral still lingers.…”

Time-fractional telegraph equation of distributed order in higher dimensions

“…Fractional partial differential equations (FPDE) with distributed order have been studied over the past decades (see e.g. [2,3,9,20,21,25,31]). One reason of the interest is the relation of these equations with physical processes involving times-scales, for example, fractional kinetics, the Cauchy problem of time-fractional diffusion-wave, generalized time-fractional diffusion, time-fractional reaction-diffusion, fractional sub-diffusion equations, and continuous random walk processes (see [16,24] and references therein indicated).…”

“…Chechkin et al [10,12] discussed the properties of diffusion equations with fractional derivatives of distributed order for the description of anomalous relaxation and diffusion phenomena getting less anomalous in the course of time, called, respectively, accelerating subdiffusion and decelerating superdiffusion. Fundamental solutions for time-fractional diffusion equations of distributed order were presented in [3] and [15] for the one-dimensional case that is, one space variable. It was proved in [15] that in the cases of the time-fractional diffusion and wave equations of distributed order the first fundamental solution can be interpreted as a spatial probability density function evolving in time.…”