Abstract
In this paper, we develop a JacobiGaussLobatto collocation method for solving the nonlinear fractional Langevin equation with threepoint boundary conditions. The fractional derivative is described in the Caputo sense. The shifted JacobiGaussLobatto points are used as collocation nodes. The main characteristic behind the JacobiGaussLobatto 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; threepoint boundary conditions; collocation method; JacobiGaussLobatto quadrature; shifted Jacobi polynomials1 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, [37]). Spectral methods are relatively new approaches to provide an accurate approximation to FDEs (see, for instance, [811]).
In this work, we propose the shifted JacobiGaussLobatto collocation (SJGLC) method to solve numerically the following nonlinear Langevin equation involving two fractional orders in different intervals:
subject to the threepoint boundary conditions
where
The existence and uniqueness of solution of Langevin equation involving two fractional
orders in different intervals (
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 singlefile diffusion. This equation has been the focus of many studies, see, for instance, [1518].
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 [2022]. Bhrawy and Alofi [20] proposed the spectral shifted JacobiGauss collocation method to find the solution of the LaneEmden type equation. Moreover, Doha et al. [23] developed the shifted JacobiGauss collocation method for solving nonlinear highorder multipoint boundary value problems. To the best of our knowledge, there are no results on JacobiGaussLobatto collocation method for threepoint 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 [2427]). Some special cases of Jacobi parameters α and β are used for numerically solving various types of differential equations (see [2831]).
The main concern of this paper is to extend the application of collocation method
to solve the threepoint nonlinear Langevin equation involving two fractional orders
in different intervals. It would be very useful to carry out a systematic study on
JacobiGaussLobatto collocation method with general indexes (
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 GaussLobatto 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., [3335]).
Definition 2.1 The RiemannLiouville fractional integral operator of order μ (
Definition 2.2 The Caputo fractional derivative of order μ is defined as
where m is an integer number and
For the Caputo derivative, we have
We use the ceiling function
Let
Besides,
Let
The set of Jacobi polynomials forms a complete
Let
By virtue of (6), we have that
Next, let
The set of shifted Jacobi polynomials is a complete
For
3 Shifted JacobiGaussLobatto collocation method
In this section, we derive the SJGLC method to solve numerically the following model problem:
subject to the threepoint boundary conditions
where
The choice of collocation points is important for the convergence and efficiency of
the collocation method. For boundary value problems, the GaussLobatto points are
commonly used. It should be noted that for a differential equation with the singularity
at
Let us first introduce some basic notation that will be used in the sequel. We set
We next recall the JacobiGaussLobatto interpolation. For any positive integer N,
For any
We introduce the following discrete inner product and norm:
where
Due to (14), we have
Thus, for any
Associating with this quadrature rule, we denote by
The shifted JacobiGauss collocation method for solving (11)(12) is to seek
We now derive an efficient algorithm for solving (17)(18). Let
We first approximate
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
Proof This theorem can be easily proved (see Doha et al. [36]).
In practice, only the first
Also, by substituting Eq. (19) in Eq. (12) we obtain
To find the solution
Next, Eq. (23), after using (9) and (6), can be written as
The scheme (24)(25) can be rewritten as a compact matrix form. To do this, we introduce
the
Also, we define the
and the
Further, let
where
or equivalently
Finally, from (26), we obtain
Remark 3.2 In actual computation for fixed μ, ν and λ, it is required to compute
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
subject to threepoint boundary conditions:
The analytic solution for this problem is not known. In Table 1 we introduce the approximate solution for (27)(28) using SJGLC method at
Figure 1. Comparing the approximate solutions at
Table 1. Approximate solution of (27)(28) using SJGLC method for
Example 2 In this example we consider the following nonlinear fractional Langevin differential equation
subject to the following threepoint boundary conditions:
where
The exact solution of this problem is
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 GaussLobatto points, with two choices of α, β, and various choices of ν, μ, and N.
Table 2. Maximum absolute error of
Table 3. Maximum absolute error of
The approximate solutions are evaluated for
Figure 2. Approximate solution for
Figure 3. Approximate solution for
In the case of
5 Conclusion
An efficient and accurate numerical scheme based on the JacobiGaussLobatto 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 SJGLC 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.
References

Magin, RL: Fractional Calculus in Bioengineering, Begell House Publishers, New York (2006)

Das, S: Functional Fractional Calculus for System Identification and Controls, Springer, New York (2008)

Jafari, H, Yousefi, SA, Firoozjaee, MA, Momanic, S, Khalique, CM: Application of Legendre wavelets for solving fractional differential equations. Comput. Math. Appl.. 62, 1038–1045 (2011). Publisher Full Text

Bhrawy, AH, Alofi, AS: The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. (2012)

Lotfi, A, Dehghan, M, Yousefi, SA: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl.. 62, 1055–1067 (2011). Publisher Full Text

Lakestani, M, Dehghan, M, Irandoustpakchin, S: The construction of operational matrix of fractional derivatives using Bspline functions. Commun. Nonlinear Sci. Numer. Simul.. 17, 1149–1162 (2012). Publisher Full Text

Pedas, A, Tamme, E: Piecewise polynomial collocation for linear boundary value problems of fractional differential equations. J. Comput. Appl. Math. (2012)

Bhrawy, AH, Alofi, AS, EzzEldien, SS: A quadrature tau method for variable coefficients fractional differential equations. Appl. Math. Lett.. 24, 2146–2152 (2011). Publisher Full Text

Bhrawy, AH, Alshomrani, M: A shifted Legendre spectral method for fractionalorder multipoint boundary value problems. Adv. Differ. Equ.. 2012, (2012)

Doha, EH, Bhrawy, AH, EzzEldien, SS: Efficient Chebyshev spectral methods for solving multiterm fractional orders differential equations. Appl. Math. Model.. 35, 5662–5672 (2011). Publisher Full Text

Doha, EH, Bhrawy, AH, EzzEldien, SS: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl.. 62, 2364–2373 (2011). Publisher Full Text

Ahmad, B, Nieto, JJ, Alsaedi, A, ElShahed, M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl.. 13, 599–606 (2012). Publisher Full Text

Ahmad, B, Nieto, JJ: Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions. Int. J. Differ. Equ.. 2010, (2010)

Chen, A, Chen, Y: Existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value conditions. Bound. Value Probl.. 2011, (2011)

Fa, KS: Fractional Langevin equation and RiemannLiouville fractional derivative. Eur. Phys. J. E. 24, 139–143 (2007). PubMed Abstract  Publisher Full Text

Picozzi, S, West, B: Fractional Langevin model of memory in financial markets. Phys. Rev. E. 66, 46–118 (2002)

Lim, SC, Li, M, Teo, LP: Langevin equation with two fractional orders. Phys. Lett. A. 372, 6309–6320 (2008). Publisher Full Text

Eab, CH, Lim, SC: Fractional generalized Langevin equation approach to singlefile diffusion. Physica A. 389, 2510–2521 (2010). Publisher Full Text

Canuto, C, Hussaini, MY, Quarteroni, A, Zang, TA: Spectral Methods in Fluid Dynamics, Springer, New York (1988)

Bhrawy, AH, Alofi, AS: A JacobiGauss collocation method for solving nonlinear LaneEmden type equations. Commun. Nonlinear Sci. Numer. Simul.. 17, 62–70 (2012). Publisher Full Text

Guo, BY, Yan, JP: LegendreGauss collocation method for initial value problems of second order ordinary differential equations. Appl. Numer. Math.. 59, 1386–1408 (2009). Publisher Full Text

Saadatmandi, A, Dehghan, M: The use of sinccollocation method for solving multipoint boundary value problems. Commun. Nonlinear Sci. Numer. Simul.. 17, 593–601 (2012). Publisher Full Text

Doha, EH, Bhrawy, AH, Hafez, RM: On shifted Jacobi spectral method for highorder multipoint boundary value problems. Commun. Nonlinear Sci. Numer. Simul.. 17, 3802–3810 (2012). Publisher Full Text

Doha, EH, Bhrawy, AH: Efficient spectralGalerkin algorithms for direct solution of fourthorder differential equations using Jacobi polynomials. Appl. Numer. Math.. 58, 1224–1244 (2008). Publisher Full Text

Doha, EH, Bhrawy, AH: A Jacobi spectral Galerkin method for the integrated forms of fourthorder elliptic differential equations. Numer. Methods Partial Differ. Equ.. 25, 712–739 (2009). Publisher Full Text

ElKady, M: Jacobi discrete approximation for solving optimal control problems. J. Korean Math. Soc.. 49, 99–112 (2012)

Doha, EH, AbdElhameed, WM, Youssri, YH: Efficient spectralPetrovGalerkin methods for the integrated forms of third and fifthorder elliptic differential equations using general parameters generalized Jacobi polynomials. Appl. Math. Comput.. 218, 7727–7740 (2012). Publisher Full Text

Xie, Z, Wang, LL, Zhao, X: On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math. Comput. (2012, in press)

Liu, F, Ye, X, Wang, X: Efficient Chebyshev spectral method for solving linear elliptic PDEs using quasiinverse technique. Numer. Math. Theor. Meth. Appl.. 4, 197–215 (2011)

Zhu, L, Fan, Q: Solving fractional nonlinear Fredholm integrodifferential equations by the second kind Chebyshev wavelet. Commun. Nonlinear Sci. Numer. Simul.. 17, 2333–2341 (2012). Publisher Full Text

Doha, EH, Bhrawy, AH: An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectralGalerkin method. Comput. Math. Appl. (2012)

Podlubny, I: Fractional Differential Equations, Academic Press, San Diego (1999)

Doha, EH: On the coefficients of differentiated expansions and derivatives of Jacobi polynomials. J. Phys. A, Math. Gen.. 35, 3467–3478 (2002). Publisher Full Text

Doha, EH: On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials. J. Phys. A, Math. Gen.. 37, 657–675 (2004). Publisher Full Text

Doha, EH, Bhrawy, AH, EzzEldien, SS: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. (2012)