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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


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.

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