A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation

Abstract

A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomialsPL,n(x)PL,m(y), for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier.

Numerical method for solving two‐dimensional of the space and space–time fractional coupled reaction‐diffusion equations

“…This section introduced a numerical algorithm for solving two-dimensional coupled space fractional reaction-diffusion equations based on the Sl-Gl-C method. 25,47 The core of the proposed method is to reduce the two-dimensional SFCRDEs to a system of ordinary differential equations, which can solve by the Runge-Kutta method or any other method. Consider the following system:…”

“…There are many numerical techniques to solve space‐fractional reaction–diffusion equations (SFRDEs) as the shifted Legendre collocation method, 25 Chebyshev collocation method, 26 shifted Jacobi polynomials, 27 second‐order operator splitting spectral element method, 28 surface finite element method in combination with the matrix transfer technique, 29 weighted and shifted Grünwald–Letnikov difference formula, 30 a combination of the matrix transfer technique with the linearly implicit predictor‐corrector methods, 31 finite‐difference Fibonacci collocation method, 32 and operator splitting methods 33 . Huang et al 34 constructed the ADI technique to solve two‐dimensional multiterm time–space fractional nonlinear diffusion‐wave equations.…”

A Reliable Way to Deal with Fractional-Order Equations That Describe the Unsteady Flow of a Polytropic Gas

“…Due to the complexity of the nonlinear equations, there is no united method to find every solution to the nonlinear FPDEs. Some of the numerical and analytical methods are the homotopy perturbation method (HPM) [21], Adomian decomposition method (ADM) [22], variational iteration method (VIM) [23], Jacobi spectral collocation method [24], G/G-expansion method [25], tau method [26], meshless method [27], the Haar wavelet method [28], Bernstein polynomials [29], the Legendre base method [30], the Laplace transform method [31], fractional complex transform method [32], Laplace variational iteration method [33], spectral Legendre-Gauss-Lobatto collocation method [34], and cylindrical-coordinate method [35] and so on [36][37][38].…”

Comparison of Numerical Approximations of One-Dimensional Space Fractional Diffusion Equation Using Different Types of Collocation Points in Spectral Method Based on Lagrange’s Basis Polynomials

“…Xie et al [4] also used Chebyshev polynomials to express the solution both in space and time but used Tau method to transform the fractional convection diffusion equation into a system of linear algebraic equations. Bhrawy [5] used shifted Legendre polynomials with Gauss-Lobatto nodes in space for a 2D space fractional diffusion equation and hence reduced it to a system of ODE which is then solved by fourth-order implicit Runge-Kutta method. Lin and Xu [6] introduced a method where they applied Legendre spectral scheme in space and finite difference in time to approximate the solution of time fractional diffusion equation.…”