New Implementation of Legendre Polynomials for Solving Partial Differential Equations


In this paper we present a proposal using Legendre polynomials approximation for the solution of the second order linear partial differential equations. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The performance of presented method has been compared with other methods, namely Sinc-Galerkin, quadratic spline collocation and LiuLin method. Numerical examples show better accuracy of the proposed method. Moreover, the computation cost decreases at least by a factor of 6 in this method.

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Davari, A. and Ahmadi, A. (2013) New Implementation of Legendre Polynomials for Solving Partial Differential Equations. Applied Mathematics, 4, 1647-1650. doi: 10.4236/am.2013.412224.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Q. Wang, “Numerical Solutions for Fractional KdV-Burgers Equatin by Adomian Decomposition Method,” Applied Mathematics and Computation, Vol. 182, No. 2, 2006, pp. 1048-1055.
[2] X. Chen and J. Xiang, “Solving Diffusion Equation Using Wavelet Method,” Applied Mathematics and Computation, Vol. 217, No. 13, 2011, pp. 6426-6432.
[3] Q. N. Yang, J. J. Zheng, Y. Miao and Y. Z. Sima, “An Improved Hybrid Boundary Node Method for Solving Steady Fluid Flow Problems,” Engineering Analysis with Boundary Elements, Vol. 35, No. 1, 2011, pp. 18-24.
[4] K. W. Morton and D. F. Mayers, “Numerical Solution of Partial Differential Equations,” Cambridge University Press, Cambridge, 2005.
[5] N. Alonso and K. L. Bowers, “An Alternating-Direction Sinc-Galerkin Method for Elliptic Problems,” Journal of Complexity, Vol. 25, No. 3, 2009, pp. 237-252.
[6] G. Fairweather, A. Karageorghis and J. Maack, “Compact Optimal Quadratic Spline Collocation Methods for the Helmholtz Equation,” Journal of Computational Physics, Vol. 230, No. 8, 2011, pp. 2880-2895.
[7] N. Liu and E. B. Lin, “Legendre Wavelet Method for Numerical Solutions of Partial Differential Equations,” Numerical Methods for Partial Differential Equations, Vol. 26, No. 1, 2010, pp. 81-94.
[8] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, “Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamiccs,” Springer, New York, 2007.
[9] A. Saadatmandi and M. Dehghan, “Numerical Solution of the Higher-Order Linear Fredholm Integro Differential Difference Equation with Variable Coefficients,” Computers and Mathematics with Applications, Vol. 59, No. 8, 2010, pp. 2996-3004.

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