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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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Citation and License

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


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/62


Received:2 April 2012
Accepted:30 May 2012
Published:22 June 2012

© 2012 Bhrawy and Alghamdi; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1 Introduction

Many practical problems arising in science and engineering require solving initial and boundary value problems of fractional order differential equations (FDEs), see [1,2] and references therein. Several methods have also been proposed in the literature to solve FDEs (see, for instance, [3-7]). Spectral methods are relatively new approaches to provide an accurate approximation to FDEs (see, for instance, [8-11]).

In this work, 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M1">View MathML</a>

(1)

subject to the three-point boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M2">View MathML</a>

(2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M3">View MathML</a> denotes the Caputo fractional derivative of order ν for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M4">View MathML</a>, λ is a real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M7">View MathML</a> are given constants and f is a given nonlinear source function.

The existence and uniqueness of solution of Langevin equation involving two fractional orders in different intervals (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M8">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M9">View MathML</a>) have been studied in [12], and for other choices of ν and μ, see [13,14].

Fractional Langevin equation is one of the basic equations in the theory of the evolution of physical phenomena in fluctuating environments and provides a more flexible model for fractal processes as compared with the usual ordinary Langevin equation. Moreover, fractional generalized Langevin equation with external force is used to model single-file diffusion. This equation has been the focus of many studies, see, for instance, [15-18].

Due to high order accuracy, spectral methods have gained increasing popularity for several decades, especially in the field of computational fluid dynamics (see, e.g., [19] and the references therein). Collocation methods have become increasingly popular for solving differential equations; also, they are very useful in providing highly accurate solutions to nonlinear differential equations [20-22]. Bhrawy and Alofi [20] proposed the spectral shifted Jacobi-Gauss collocation method to find the solution of the Lane-Emden type equation. Moreover, Doha et al. [23] developed the shifted Jacobi-Gauss collocation method for solving nonlinear high-order multi-point boundary value problems. To the best of our knowledge, there are no results on Jacobi-Gauss-Lobatto collocation method for three-point nonlinear Langevin equation arising in mathematical physics. This partially motivated our interest in such a method.

The advantage of using Jacobi polynomials for solving differential equations is obtaining the solution in terms of the Jacobi parameters α and β (see [24-27]). Some special cases of Jacobi parameters α and β are used for numerically solving various types of differential equations (see [28-31]).

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. It would be very useful to carry out a systematic study on Jacobi-Gauss-Lobatto collocation method with general indexes (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M10">View MathML</a>). The fractional Langevin equation is collocated only at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M11">View MathML</a> points; for suitable collocation points, we use the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M12">View MathML</a> nodes of the shifted Jacobi-Gauss-Lobatto interpolation (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M10">View MathML</a>). These equations together with the three-point boundary conditions generate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14">View MathML</a> nonlinear algebraic equations which can be solved using Newton’s iterative method. Finally, the accuracy of the proposed method is demonstrated by test problems.

The remainder of the paper is organized as follows. In the next section, we introduce some notations and summarize a few mathematical facts used in the remainder of the paper. In Section 3, the way of constructing the Gauss-Lobatto collocation technique for fractional Langevin equation is described using the shifted Jacobi polynomials; and in Section 4 the proposed method is applied to some types of Langevin equations. Finally, some concluding remarks are given in Section 5.

2 Preliminaries

In this section, we give some definitions and properties of the fractional calculus (see, e.g., [1,2,32]) and Jacobi polynomials (see, e.g., [33-35]).

Definition 2.1 The Riemann-Liouville fractional integral operator of order μ (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M15">View MathML</a>) is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M16">View MathML</a>

(3)

Definition 2.2 The Caputo fractional derivative of order μ is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M17">View MathML</a>

(4)

where m is an integer number and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M18">View MathML</a> is the classical differential operator of order m.

For the Caputo derivative, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M19">View MathML</a>

(5)

We use the ceiling function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M20">View MathML</a> to denote the smallest integer greater than or equal to μ and the floor function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M21">View MathML</a> to denote the largest integer less than or equal to μ. Also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M23">View MathML</a>. Recall that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M24">View MathML</a>, the Caputo differential operator coincides with the usual differential operator of an integer order.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M27">View MathML</a> be the standard Jacobi polynomial of degree k. We have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M28">View MathML</a>

(6)

Besides,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M29">View MathML</a>

(7)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M30">View MathML</a>, then we define the weighted space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M31">View MathML</a> as usual, equipped with the following inner product and norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M32">View MathML</a>

The set of Jacobi polynomials forms a complete <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M33">View MathML</a>-orthogonal system, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M34">View MathML</a>

(8)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M35">View MathML</a>, then the shifted Jacobi polynomial of degree k on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36">View MathML</a> is defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M37">View MathML</a>.

By virtue of (6), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M38">View MathML</a>

(9)

Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M39">View MathML</a>, then we define the weighted space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M40">View MathML</a> in the usual way, with the following inner product and norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M41">View MathML</a>

The set of shifted Jacobi polynomials is a complete <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M42">View MathML</a>-orthogonal system. Moreover, due to (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M43">View MathML</a>

(10)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M44">View MathML</a> one recovers the shifted ultraspherical polynomials (symmetric shifted Jacobi polynomials) and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M46">View MathML</a>, the shifted Chebyshev of the first and second kinds and shifted Legendre polynomials respectively; and for the nonsymmetric shifted Jacobi polynomials, the two important special cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M47">View MathML</a> (shifted Chebyshev polynomials of the third and fourth kinds) are also recovered.

3 Shifted Jacobi-Gauss-Lobatto collocation method

In this section, we derive the SJ-GL-C method to solve numerically the following model problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M48">View MathML</a>

(11)

subject to the three-point boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M49">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M3">View MathML</a> denotes the Caputo fractional derivative of order ν for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M4">View MathML</a>λ is a real number, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M5">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M6">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M7">View MathML</a> are given constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M55">View MathML</a> is a given nonlinear source function. For the existence and uniqueness of solution of (11)-(12), see [12].

The choice of collocation points is important for the convergence and efficiency of the collocation method. For boundary value problems, the Gauss-Lobatto points are commonly used. It should be noted that for a differential equation with the singularity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M56">View MathML</a> in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M57">View MathML</a> one is unable to apply the collocation method with Jacobi-Gauss-Lobatto points because the two assigned abscissas 0 and L are necessary to use as a two points from the collocation nodes. Also, a Jacobi-Gauss-Radau nodes with the fixed node <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M58">View MathML</a> cannot be used in this case. In fact, we use the collocation method with Jacobi-Gauss-Lobatto nodes to treat the nonlinear Langevin differential equation; i.e., we collocate this equation only at the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M12">View MathML</a> Jacobi-Gauss-Lobatto points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36">View MathML</a>. These equations together with three-point boundary conditions generate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14">View MathML</a> nonlinear algebraic equations which can be solved.

Let us first introduce some basic notation that will be used in the sequel. We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M62">View MathML</a>

(13)

We next recall the Jacobi-Gauss-Lobatto interpolation. For any positive integer N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M63">View MathML</a> stands for the set of all algebraic polynomials of degree at most N. If we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M65">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M66">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M67">View MathML</a>), to the nodes and Christoffel numbers of the standard (shifted) Jacobi-Gauss-Lobatto quadratures on the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36">View MathML</a> respectively. Then one can easily show that

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M71">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M72">View MathML</a>

(14)

We introduce the following discrete inner product and norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M73">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M75">View MathML</a> are the nodes and the corresponding weights of the shifted Jacobi-Gauss-quadrature formula on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M36">View MathML</a> respectively.

Due to (14), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M77">View MathML</a>

(16)

Thus, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M78">View MathML</a>, the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M80">View MathML</a> coincide.

Associating with this quadrature rule, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M81">View MathML</a> the shifted Jacobi-Gauss interpolation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M82">View MathML</a>

The shifted Jacobi-Gauss collocation method for solving (11)-(12) is to seek <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M83">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M84">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M85">View MathML</a>

(18)

We now derive an efficient algorithm for solving (17)-(18). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M86">View MathML</a>

(19)

We first approximate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M88">View MathML</a>, as Eq. (19). By substituting these approximations in Eq. (11), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M89">View MathML</a>

(20)

Here, the fractional derivative of order μ in the Caputo sense for the shifted Jacobi polynomials expanded in terms of shifted Jacobi polynomials themselves can be represented formally in the following theorem.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M90">View MathML</a>be a shifted Jacobi polynomial of degreej, then the fractional derivative of orderνin the Caputo sense for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M90">View MathML</a>is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M92">View MathML</a>

(21)
where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M93">View MathML</a>

Proof This theorem can be easily proved (see Doha et al. [36]).

In practice, only the first <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14">View MathML</a>-terms shifted Jacobi polynomials are considered, with the aid of Theorem 3.1 (Eq. (21)), we obtain from (20) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M95">View MathML</a>

(22)

Also, by substituting Eq. (19) in Eq. (12) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M96">View MathML</a>

(23)

To find the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M97">View MathML</a>, we first collocate Eq. (22) at the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M12">View MathML</a> shifted Jacobi-Gauss-Lobatto notes, yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M99">View MathML</a>

(24)

Next, Eq. (23), after using (9) and (6), can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M100">View MathML</a>

(25)

The scheme (24)-(25) can be rewritten as a compact matrix form. To do this, we introduce the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M101">View MathML</a> matrix A with the entries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M102">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M103">View MathML</a>

Also, we define the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M101">View MathML</a> matrix B with the entries:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M105">View MathML</a>

and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M106">View MathML</a> matrix C with the entries:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M107">View MathML</a>

Further, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M108">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M109">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M110">View MathML</a> is the kth component of Ca. Then we obtain from (24)-(25) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M111">View MathML</a>

or equivalently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M112">View MathML</a>

(26)

Finally, from (26), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M14">View MathML</a> nonlinear algebraic equations which can be solved for the unknown coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M114">View MathML</a> by using any standard iteration technique, like Newton’s iteration method. Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M97">View MathML</a> given in Eq. (19) can be evaluated. □

Remark 3.2 In actual computation for fixed μ, ν and λ, it is required to compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M116">View MathML</a> only once. This allows us to save a significant amount of computational time.

4 Numerical results

To illustrate the effectiveness of the proposed method in the present paper, two test examples are carried out in this section. Comparison of the results obtained by various choices of Jacobi parameters α and β reveal that the present method is very effective and convenient for all choices of α and β.

We consider the following two examples.

Example 1 Consider the nonlinear fractional Langevin equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M117">View MathML</a>

(27)

subject to three-point boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M118">View MathML</a>

(28)

The analytic solution for this problem is not known. In Table 1 we introduce the approximate solution for (27)-(28) using SJ-GL-C method at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M120">View MathML</a>. The approximate solutions at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M121">View MathML</a> and a few collocation points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M122">View MathML</a> of this problem are depicted in Figure 1. The approximate solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M123">View MathML</a> agrees very well with the approximate solution at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M124">View MathML</a>; this means the numerical solution converges fast as N increases.

thumbnailFigure 1. Comparing the approximate solutions at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M125">View MathML</a>, for Example 1.

Table 1. Approximate solution of (27)-(28) using SJ-GL-C method for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M120">View MathML</a>

Example 2 In this example we consider the following nonlinear fractional Langevin differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M127">View MathML</a>

(29)

subject to the following three-point boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M128">View MathML</a>

(30)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M129">View MathML</a>

The exact solution of this problem is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M130">View MathML</a>.

Numerical results are obtained for different choices of ν, μ, α, β, and N. In Tables 2 and 3 we introduce the maximum absolute error, using the shifted Jacobi collocation method based on Gauss-Lobatto points, with two choices of α, β, and various choices of ν, μ, and N.

Table 2. Maximum absolute error of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M131">View MathML</a>using SJ-GL-C method for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M46">View MathML</a>

Table 3. Maximum absolute error of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M131">View MathML</a>using SJ-GL-C method for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M140">View MathML</a>

The approximate solutions are evaluated for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M151">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152">View MathML</a>. The results of the numerical simulations are plotted in Figure 2. In Figure 3, we plotted the approximate solutions at fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153">View MathML</a>, and various choices of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M154">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152">View MathML</a>. It is evident from Figure 2 and Figure 3 that, as ν and μ approach close to 2 and 1, the numerical solution by shifted Jacobi-Gauss-Lobatto collocation method with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M151">View MathML</a> for fractional order differential equation approaches to the solution of integer order differential equation.

thumbnailFigure 2. Approximate solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M158">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M159">View MathML</a>with 14 nodes and the exact solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150">View MathML</a>, for Example 2.

thumbnailFigure 3. Approximate solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M162">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153">View MathML</a>with 14 nodes and the exact solution at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150">View MathML</a>, for Example 2.

In the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M168">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152">View MathML</a>, the results of the numerical simulations are shown in Figure 4. In Figure 5, we plotted the approximate solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M171">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M168">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152">View MathML</a>. In fact, the approximate solutions obtained by the present method at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M175">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M152">View MathML</a> are shown in Figure 4 and Figure 5 to make it easier to show that; as ν and μ approach to their integer values, the solution of fractional order Langevin equation approaches to the solution of integer order Langevin differential equation.

thumbnailFigure 4. Approximate solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M177">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M150">View MathML</a>with 12 nodes, for Example 2.

thumbnailFigure 5. Approximate solution for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M179">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/62/mathml/M153">View MathML</a>with 12 nodes, for Example 2.

5 Conclusion

An efficient and accurate numerical scheme based on the Jacobi-Gauss-Lobatto collocation spectral method is proposed for solving the nonlinear fractional Langevin equation. The problem is reduced to the solution of nonlinear algebraic equations. Numerical examples were given to demonstrate the validity and applicability of the method. The results show that the SJ-GL-C method is simple and accurate. In fact, by selecting a few collocation points, excellent numerical results are obtained.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have equal contributions to each part of this article. All the authors read and approved the final manuscript.

Acknowledgements

This study was supported by the Deanship of Scientific Research of King Abdulaziz University. The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the article.

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