One issue of future work is to develop the theory analysis of the method for the proposed fractional differential equation. The authors declare that there is no conflict of interests regarding the publication of this paper. National Center for Biotechnology Information , U. Journal List ScientificWorldJournal v. Published online Apr 1. Author information Article notes Copyright and License information Disclaimer.

Received Jan 9; Accepted Feb This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article has been cited by other articles in PMC.

Abstract This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. Introduction Fractional models have been increasingly shown by many scientists to describe adequately the problems with memory and nonlocal properties in fluid mechanics, viscoelasticity, physics, biology, chemistry, finance, and other areas of applications [ 1 — 6 ].

Notations and Some Preliminary Results In this section, we introduce some basic definitions and derive several preliminary results for developing the presented method.

The Composite Translated Sinc Functions The sinc functions and their properties are discussed in [ 23 , 24 ]. Numerical Examples To validate the effectiveness of the proposed method for problem 1 with 2 and 3 , we consider the example given in [ 16 ]. Open in a separate window. Figure 1. Figure 2. Figure 3. Figure 4.

Figure 5. Conclusion In this paper, we develop and analyze the efficient numerical methods for the fractional diffusion-wave equation.

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Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References 1. Podlubny I. Fractional Differential Equations. Mathematics in Science and Engineering. Zaslavsky GM. Chaos, fractional kinetics, and anomalous transport.

### 1. Introduction

Physics Reports. Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Protein conformational dynamics probed by single-molecule electron transfer. Liu F, Burrage K. Novel techniques in parameter estimation for fractional dynamical models arising from biological systems. On the fractional signals and systems. Signal Processing.

The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General. Mainardi F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. An explicit difference method for solving fractional diffusion and diffusion-wave equations in the caputo form.

Journal of Computational and Nonlinear Dynamics.

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## Partial Differential Equations Book

Magin RL. Fractional Calculus in Bioengineering. Time fractional diffusion: a discrete random walk approach.

Nonlinear Dynamics. Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. Yuste SB, Acedo L. An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations. New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. Fractional diffusion equations by the Kansa method. Sun Z-Z, Wu X. A fully discrete difference scheme for a diffusion-wave system.

Applied Numerical Mathematics. Two finite difference schemes for time fractional diffusion-wave equation. Numerical Algorithms. Ding H, Li C. Numerical algorithms for the fractional diffusion-wave equation with reaction term. Abstract and Applied Analysis. Li X, Xu C.

## Partial Differential Equations Book

A space-time spectral method for the time fractional diffusion equation. Agrawal OP. Solution for a fractional diffusion-wave equation defined in a bounded domain. The sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Communications in Nonlinear Science and Numerical Simulation.

Weilbeer M. Efficient numerical methods for fractional differential equations and their analytical background [Ph.

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Meerschaert, D. Benson, H. Scheffler, and B. Baeumer, "Stochastic solution of space-time fractional diffusion equations," Physical Review E, vol. Eidelman and A. Kochubei, "Cauchy problem for fractional diffusion equations," Journal of Differential Equations, vol.