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The Scientific World Journal
Volume 2014, Article ID 706296, 7 pages
Research Article

A Legendre tau-Spectral Method for Solving Time-Fractional Heat Equation with Nonlocal Conditions

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni Suef University, Beni-Suef 62511, Egypt

Received 15 April 2014; Accepted 6 May 2014; Published 25 June 2014

Academic Editor: Dumitru Baleanu

Copyright © 2014 A. H. Bhrawy and M. A. Alghamdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We develop the tau-spectral method to solve the time-fractional heat equation (T-FHE) with nonlocal condition. In order to achieve highly accurate solution of this problem, the operational matrix of fractional integration (described in the Riemann-Liouville sense) for shifted Legendre polynomials is investigated in conjunction with tau-spectral scheme and the Legendre operational polynomials are used as the base function. The main advantage in using the presented scheme is that it converts the T-FHE with nonlocal condition to a system of algebraic equations that simplifies the problem. For demonstrating the validity and applicability of the developed spectral scheme, two numerical examples are presented. The logarithmic graphs of the maximum absolute errors is presented to achieve the exponential convergence of the proposed method. Comparing between our spectral method and other methods ensures that our method is more accurate than those solved similar problem.