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

Ali H Bhrawy,Mohammed A Alghamdi

doi:10.1186/1687-2770-2012-62

Ali H Bhrawy, Mohammed A Alghamdi

Open Access

https://doi.org/10.1186/1687-2770-2012-62

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Journal: Boundary Value Problems	Publication Date: Jun 22, 2012
Citations: 42	License type: CC BY 2.0

Affiliation: King Abdulaziz University

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.

Highlights

Many practical problems arising in science and engineering require solving initial and boundary value problems of fractional order differential equations (FDEs), see [, ] and references therein
We propose the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) method to solve numerically the following nonlinear Langevin equation involving two fractional orders in different intervals: Dν Dμ + λ u(x) = f x, u(x), < μ ≤, < ν ≤, x ∈ I = [, L], ( )
The main concern of this paper is to extend the application of collocation method to solve the three-point nonlinear Langevin equation involving two fractional orders in different intervals

Summary

Introduction

Many practical problems arising in science and engineering require solving initial and boundary value problems of fractional order differential equations (FDEs), see [ , ] and references therein. We propose the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C) method to solve numerically the following nonlinear Langevin equation involving two fractional orders in different intervals: Dν Dμ + λ u(x) = f x, u(x) , < μ ≤ , < ν ≤ , x ∈ I = [ , L], ( ) Doha et al [ ] developed the shifted Jacobi-Gauss collocation method for solving nonlinear high-order multi-point boundary value problems.

Results

Conclusion