An efficient fractional polynomial method for space fractional diffusion equations

Abstract

In this paper, we develop a new approximation technique for solving space fractional diffusion equation. The method of solution is based on fractional order Legendre function with the concept of Caputo's definition of fractional derivatives. Convergence analysis and error bound of the method are discussed. Several Illustrative examples are included to demonstrate the validity and applicability of the proposed method. The obtained results reveal that the method is more accurate and efficient than the methods such as Chebyshev finite difference method and Tau approach method discussed in the literature. Ó 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Lagrange’s Spectral Collocation Method for Numerical Approximations of Two-Dimensional Space Fractional Diffusion Equation

“…where To determine the expansion coefficients we chose collocation technique and for this we force the residual equation at Equation (16) to be zero at ( First by setting the residual at Equation (16) to be zero at the interior points of the spatial domain, we have the equation , , , ,…”

“…Zayernouri and Karniadakis [15] introduced fractional Lagrange interpolants and developed a new fractional spectral collocation method for 1D fractional differential equations. Krishnaveni et al [16] …”

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.…”