On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equations

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The variational iteration method and the homotopy analysis method, as alternative methods, have been widely used to handle linear and nonlinear models. The main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently. This paper deals with the numerical solutions of nonlinear fractional differential equations, where the fractional derivatives are considered in Caputo sense. The main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations. In these algorithms, Legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work. The proposed algorithms are capable of reducing the size of calculations, improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals. Numerical examples are examined to show the efficiency of the algorithms.

论文关键词:Variational iteration method,Homotopy analysis method,Legendre polynomials,Fractional differential equations,Caputo derivative

论文评审过程:Received 17 September 2009, Revised 14 June 2010, Available online 22 December 2010.

论文官网地址:https://doi.org/10.1016/j.cam.2010.12.013