A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals
© Bhrawy and Alghamdi; licensee Springer 2012
Received: 2 April 2012
Accepted: 30 May 2012
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.
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]).
where denotes the Caputo fractional derivative of order ν for , λ is a real number, , , are given constants and f is a given nonlinear source function.
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.,  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  proposed the spectral shifted Jacobi-Gauss collocation method to find the solution of the Lane-Emden type equation. Moreover, Doha et al.  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 (). The fractional Langevin equation is collocated only at points; for suitable collocation points, we use the nodes of the shifted Jacobi-Gauss-Lobatto interpolation (). These equations together with the three-point boundary conditions generate 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.
where m is an integer number and is the classical differential operator of order m.
We use the ceiling function to denote the smallest integer greater than or equal to μ and the floor function to denote the largest integer less than or equal to μ. Also and . Recall that for , the Caputo differential operator coincides with the usual differential operator of an integer order.
Let , then the shifted Jacobi polynomial of degree k on the interval is defined by .
For one recovers the shifted ultraspherical polynomials (symmetric shifted Jacobi polynomials) and for , , 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 (shifted Chebyshev polynomials of the third and fourth kinds) are also recovered.
3 Shifted Jacobi-Gauss-Lobatto collocation method
where denotes the Caputo fractional derivative of order ν for λ is a real number, are given constants and is a given nonlinear source function. For the existence and uniqueness of solution of (11)-(12), see .
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 in the interval 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 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 Jacobi-Gauss-Lobatto points . These equations together with three-point boundary conditions generate nonlinear algebraic equations which can be solved.
where and are the nodes and the corresponding weights of the shifted Jacobi-Gauss-quadrature formula on the interval respectively.
Thus, for any , the norms and coincide.
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 3.1 Letbe a shifted Jacobi polynomial of degree j, then the fractional derivative of order ν in the Caputo sense foris given by
Proof This theorem can be easily proved (see Doha et al. ).
Finally, from (26), we obtain nonlinear algebraic equations which can be solved for the unknown coefficients by using any standard iteration technique, like Newton’s iteration method. Consequently, given in Eq. (19) can be evaluated. □
Remark 3.2 In actual computation for fixed μ, ν and λ, it is required to compute 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.
Approximate solution of (27)-(28) using SJ-GL-C method for
−9.994 × 10−20
6.098 × 10−19
The exact solution of this problem is .
Maximum absolute error of using SJ-GL-C method for
2.09 × 10−4
4.91 × 10−5
1.07 × 10−7
1.39 × 10−5
4.02 × 10−7
3.99 × 10−10
3.25 × 10−6
5.87 × 10−8
2.33 × 10−11
Maximum absolute error of using SJ-GL-C method for
3.64 × 10−4
1.15 × 10−4
2.83 × 10−7
9.66 × 10−6
1.16 × 10−6
1.01 × 10−9
1.99 × 10−6
8.35 × 10−8
7.15 × 10−11
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.
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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