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Open Access Research

A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals

Ali H Bhrawy12* and Mohammed A Alghamdi1

Author Affiliations

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

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Boundary Value Problems 2012, 2012:62  doi:10.1186/1687-2770-2012-62

Published: 22 June 2012

Abstract

In this paper, we develop a Jacobi-Gauss-Lobatto collocation method for solving the nonlinear fractional Langevin equation with three-point boundary conditions. The fractional derivative is described in the Caputo sense. The shifted Jacobi-Gauss-Lobatto points are used as collocation nodes. The main characteristic behind the Jacobi-Gauss-Lobatto collocation approach is that it reduces such a problem to those of solving a system of algebraic equations. This system is written in a compact matrix form. Through several numerical examples, we evaluate the accuracy and performance of the proposed method. The method is easy to implement and yields very accurate results.

Keywords:
fractional Langevin equation; three-point boundary conditions; collocation method; Jacobi-Gauss-Lobatto quadrature; shifted Jacobi polynomials