Exact solutions for time-fractional diffusion-wave equations by decomposition method

Abstract

The time-fractional diffusion-wave equation is considered. The time-fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 2]. The fractional derivative is described in the Caputo sense. This paper presents the analytical solutions of the fractional diffusion equations by an Adomian decomposition method. By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their numerical solutions have been represented graphically. Four examples are presented to show the application of the present technique. The present method performs extremely well in terms of efficiency and simplicity.

Linking the viscous grain-shearing mechanism of wave propagation in marine sediments to fractional calculus

“…The time domain solutions of the equation suggest that as γ decreases from 1 to 0, the phenomenon of diffusion transforms into a lossless wave propagation. 28,29 Following the same approach as in the case of compressional wave equation, we can examine the dispersion characteristic of the fractional diffusion-wave equation. We Fourier transform Eq.…”

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

“…Many researchers have proposed various methods to solve the time-fractional diffusion-wave equations from the perspective of analytical solution and numerical solution. The method of separation of variables in [1], Sumudu transform method in [2], and decomposition method in [3] were used to construct analytical approximate solutions of fractional diffusion-wave equations, respectively. Finite difference schemes in [4][5][6][7] were widely used to solve the numerical solutions of the fractional diffusion-wave equations.…”

The Chebyshev collocation solution of the time fractional coupled Burgers' equation

“…These equations happen in a large number of physical problems such as the phenomena of turbulence flow through a shock wave traveling in a viscous fluid (see [6,24]). In recent years, many researchers have studied the fractional partial differential equations and dealt with the fractional Burgers' equation utilizing different techniques [9,16,17,19,20,27,28,32,35]. More recently, the authors in [15] applied the Chebyshev polynomials expansion method to find both analytical and numerical solutions of the fractional transport equation in the one dimensional geometry.…”