On the Numerical Solution of Fractional Partial Differential Equations

Abstract

In this paper, a technique generally known as meshless method is presented for solving fractional partial differential equations (FPDEs). Some physical linear and nonlinear experiments such as time-fractional convective-diffusion equation, timefractional wave equation and nonlinear space-fractional Fisher's equation are considered. We present the advantages of using the radial basis functions (RBFs) especially wherein the data points are scattered. Comparing between the numerical results obtained from our method and the other methods confirms the good accuracy of the presented scheme.

Computational method for solving space fractional Fisher's nonlinear equation

“…From Tables , we note that the obtained approximate numerical solutions are in good agreement with the approximate solutions obtained using methods for all values of x and t .…”

“…And boundary conditions u.0.0125, t/ 0.0125.1 C t/ C 0.00609375t 2 0.082176t 3 0.0210541t 4 7.16634 10 6 t 5 Table I. The comparison between present method and other existing methods [15,18] when t D 0.1, k D 0.0005, and h D 0.025.…”

“…Consider the following nonlinear space fractional Fisher's equation subject to the initial condition …”

“…The solution will be approximated using the proposed method, with k = 0.0005 and h = 0.025. Tables illustrate the comparison between present method, developed in Section 2 and other existing methods .…”

An efficient approach based on radial basis functions for solving stochastic fractional differential equations

“…Also RBFs is applied in mechanics [25], Kdv equation [26], Klein-Gordon equation [27], then in 2012 Vanani et al used RBF for solving fractional partial differential equations [28]. GonzalezGaxiola and Gonzalez-Perez used Multi-Quadratic RBF for approximating the solution of the Black-Scholes equation in 2014 [29].…”