Fractional-order Legendre functions for solving fractional-order differential equations
Fractional-order Legendre functions for solving fractional-order differential equations
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DOI：
10.1016/j.apm.2012.10.026
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发表时间：
2013-04-01
影响因子：
5
通讯作者：
Kumar, Sunil
中科院分区：
文献类型：
--
作者：
Kazem, S.;Abbasbandy, S.;Kumar, Sunil
通过构造分数阶Legendre函数的一般形式，得到了分数阶微分方程的解.分数阶微积分已经被用于模拟物理和工程过程，发现分数阶微分方程能最好地描述这些过程。因此，一种高效可靠的方法来解决这些问题就显得尤为重要。对于分数阶导数的概念，我们将采用Caputo的定义，使用Riemann-Liouville分数阶积分算子。我们的主要目的是将基于Legendre多项式的新的正交函数推广到分数阶微积分中。此外，还推导了FLF分数阶导数和乘积运算矩阵的一般公式。然后利用这些矩阵和Tau方法将该问题的解简化为一个代数方程组的解。将该方法应用于求解线性和非线性分数阶微分方程。最后,给出数值算例证明该方法的有效性. (C)2012爱思唯尔公司出版
In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo's definition by using Riemann-Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. (C) 2012 Published by Elsevier Inc.