## Abstract and Applied Analysis

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

A. H. Bhrawy

#### 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 polynomials ${P}_{L,n}(x){P}_{L,m}(y)$, 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 636191, 10 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412607137

Digital Object Identifier
doi:10.1155/2014/636191

Mathematical Reviews number (MathSciNet)
MR3208554

Zentralblatt MATH identifier
07022794

#### Citation

Bhrawy, A. H. A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 636191, 10 pages. doi:10.1155/2014/636191. https://projecteuclid.org/euclid.aaa/1412607137

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