Special Functions in Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena

In this tutorial survey we recall the basic properties of the special function of the Mittag-Leffler and Wright type that are known to be relevant in processes dealt with the fractional calculus. We outline the major applications of these functions. For the Mittag-Leffler functions we analyze the Abel integral equation of the second kind and the fractional relaxation and oscillation phenomena. For the Wright functions we distinguish them in two kinds. We mainly stress the relevance of the Wright functions of the second kind in probability theory with particular regard to the so-called M-Wright functions that generalizes the Gaussian and is related with the time-fractional diffusion equation.

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On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equatio

10.20944/preprints201711.0072.v1 ◽

2017 ◽

Author(s):

Yuri Luchko

Keyword(s):

Wave Equation ◽

Fundamental Solution ◽

Open Problem ◽

Special Functions ◽

Dimensional Space ◽

Fractional Diffusion ◽

Two Dimensional ◽

Diffusion Wave ◽

Space And Time ◽

Mellin Convolution

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

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Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽

10.3390/math7100923 ◽

2019 ◽

Vol 7 (10) ◽

pp. 923 ◽

Cited By ~ 1

Author(s):

Abdul Ghafoor ◽

Sirajul Haq ◽

Manzoor Hussain ◽

Poom Kumam ◽

Muhammad Asif Jan

Keyword(s):

Finite Difference ◽

Wave Equations ◽

Quadrature Rule ◽

Approximate Solutions ◽

Fractional Diffusion ◽

Haar Wavelets ◽

Test Problems ◽

Algebraic Equations ◽

Diffusion Wave ◽

Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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