%0 Journal Article
%T Tau-Path Following Method for Solving the Riccati Equation with Fractional Order
%A Muhammed I. Syam
%A Hani I. Siyyam
%A Ibrahim Al-Subaihi
%J Journal of Computational Methods in Physics
%D 2014
%R 10.1155/2014/207916
%X A formulation for the fractional Legendre functions is constructed to find the solution of the fractional Riccati equation. The fractional derivative is described in the Caputo sense. The method is based on the Tau Legendre and path following methods. Theoretical and numerical results are presented. Analysis for the presented method is given. 1. Introduction Recently, many papers on fractional boundary value problems have been studied extensively. Several forms of them have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solutions. Several numerical techniques are used to solve such problems such as Laplace and Fourier transforms [1, 2], Adomian decomposition and variational iteration methods [3, 4], eigenvector expansion [5], differential transform and finite differences methods [6, 7], power series method [8], collocation method [9], and wavelet method [10, 11]. Many applications of fractional calculus on various branches of science such as engineering, physics, and economics can be found in [12, 13]. Considerable attention has been given to the theory of fractional ordinary differential equations and integral equations [14, 15]. Additionally, the existence of solutions of ordinary and fractional boundary value problems using monotone iterative sequences has been investigated by several authors [16¨C20]. We consider the Riccati equation with fractional orders of the form where , and are continuous functions on and£¿£¿ is a constant. Riccati equation with fractional order has been discussed by many researchers using different techniques such as collocation method based on Muntz polynomials [21], homotopy perturbation method [22], and series solution method [23]. In this paper we study the Tau-path following method for solving the Riccati equation with fractional order. We organize this paper as follows. In Section 2, we present basic definitions and results of fractional derivatives. We extend basic results to path following method for the fractional case. In Section 3, we introduce the fractional-order Legendre Tau method with path following method for solving the Riccati equation with fractional order. In Section 4, we present some numerical results to illustrate the efficiency of the presented method. Finally, we conclude with some comments in the last section. 2. Preliminaries In this section, we review the definition and some preliminary results of the Caputo fractional derivatives, as well as, the definition of the fractional-order Legendre functions and their properties. Definition
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