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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 924956, 11 pages
The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations
1School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
2College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
3Department of Mathematics, National Institute of Technology, Jamshedpur, Jharkhand 831014, India
Received 23 October 2012; Revised 18 November 2012; Accepted 18 November 2012
Academic Editor: Igor Andrianov
Copyright © 2012 Jianping Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.
In recent years, nonlinear fractional differential equations (NFDEs) have been attracted great interest. It is caused by both the development of the theory of fractional calculus itself and by the applications of such constructions in various sciences such as physics, engineering, and biology [1–7]. For better understanding the mechanisms of the complicated nonlinear physical phenomena as well as further applying them in practical life, the solution of fractional differential equation [8–15] is much involved. In the past, many analytical and numerical methods have been proposed to obtain solutions of NFDEs, such as finite difference method , finite element method , differential transform method [18, 19], Adomian decomposition method [20–22], variational iteration method [23–25], and homotopy perturbation method [26–28].
Recently, He  introduced a new method called fractional sub-equation method to look for traveling wave solutions of NFDEs. The method is based on the homogeneous balance principle  and Jumarie's modified Riemann-Liouville derivative [31–33]. By using fractional sub-equation method, Zhang et al. successfully obtained traveling wave solutions of nonlinear time fractional biological population model and -dimensional space-time fractional Fokas equation. More recently, Guo et al.  and Lu  improved Zhang et al.’s work  and obtained exact solutions of some nonlinear fractional differential equations.
The present paper is motivated by the desire to improve the work made in [30, 34, 35] by proposing a new and more general ansätz so that it can be used to construct more general exact solutions which contain not only the results obtained by using the method in [30, 34, 35] as special cases but also a series of new and more general exact solutions. To illustrate the validity and advantages of the method, we will apply it to the space-time fractional coupled Burgers' equations and coupled MKdV equations.
The rest of this paper is organized as follows. In Section 2, we will describe the Modified Riemann-Liouville derivative and give the main steps of the method here. In Section 3, we illustrate the method in detail with space-time fractional coupled Burgers' equations and coupled MKdV equations. In Section 4, some conclusions are given.
2. Description of Modified Riemann-Liouville Derivative and the Proposed Method
The Jumarie's modified Riemann-Liouville derivative is defined as
Remark 2.1. In the above formulas (2.3)–(2.5), is nondifferentiable function in (2.3) and (2.4) and differentiable in (2.5), is non-differentiable, and is differentiable in (2.4) and non-differentiable in (2.5). We should use the formulas (2.3)–(2.5) carefully. In this paper, we use formulas (2.3) and (2.5) to obtain the solutions of fractional differential equations. By using the formulas (2.3) and (2.4) to search solutions, one can refer to [37, 38]. In [37, 38], He et al. introduced the fractional complex transform to convert an FDE into its differential partner easily. This transform is accessible to those who know advanced calculus.
We present the main steps of the extended fractional sub-equation method as follows.
Step 1. For a given NFDEs with independent variables and dependent variable , where , , , and are the modified Riemann-Liouville derivatives of with respect to , , , and , and is a polynomial in and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.
Step 2. By means of the traveling wave transformation, where and are constants to be determined later; (2.6) becomes the following form:
Step 3. We suppose that (2.6) has the following solution: where are constants to be determined later, and
Here , are arbitrary parameters, and satisfies the following fractional Riccati equation: where is a constant. Recently, Zhang et al.’s  first obtained the following solutions of (2.11): where the generalized hyperbolic and trigonometric functions are defined as Here, is the Mittag-Leffler function in one parameter.
Remark 2.2. As we know, the choice of an appropriate ansätz is very important when using the direct method to look for exact solutions. It can be easily found that the transformation (2.9) is more general than that introduced in [30, 34, 35]. To be more precise, if , then (2.9) becomes in . If we set , then (2.9) becomes in . It shows that taking full advantage of the transformation (2.9), we may obtain new and more general exact solutions including not only all solutions obtained by the methods [30, 34, 35] but also other new solutions. It should be noted that the method can also be extended to other similar sub-equations  easily.
3. Application of the Proposed Method
Using the traveling wave transformations , (3.1) can be reduced to the following nonlinear fractional ODEs:
Substituting (3.3) into (3.2), and setting each coefficients of to zero, we get a system of underdetermined equations for , and . To avoid tediousness, we omit the overdetermined nonlinear equations. Solving the system, we get the following solution sets.
Case 1. Consider
Case 2. Consider
Case 3. Consider
Consider where and .
Consider where and .
Consider where and .
Consider where and .
Using the traveling wave transformations , (3.12) can be reduced to the following nonlinear fractional ODEs:
Substituting (3.14) into (3.12), and setting each coefficients of to zero, we get a system of underdetermined equations for , and . To avoid tediousness, we omit the overdetermined, highly nonlinear equations. Solving the system, we get the following solution sets.
Case 1. Consider where is an arbitrary constant.
Case 2. Consider where is an arbitrary constant.
Case 3. Consider where is an arbitrary constant.
Consider where and .
Consider where and .
Consider where and .
Consider where and .
Remark 3.3. It seems that the Exp-function method  is more general than the extend sub-equation method. The adopted sub-equation method actually uses the same idea as the one using an FDE with constants and . You can choose either of these methods for analyzing a new equation or a previously unstudied (or partially studied) problem.
In this paper, based on a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions, we propose a new method called extended fractional sub-equation method to construct exact solutions of fractional differential equations. In order to illustrate the validity and advantages of the algorithm, we apply it to space-time fractional coupled Burgers' equations and coupled MKdV equations. As a result, many exact solutions are obtained. The results show that the extended fractional sub-equation method is direct, effective, and can be used for many other fractional differential equations in mathematical physics.
This work is supported by the NSF of China (nos. 10971166, 11171269, 61163027).
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA, 2000.
- B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
- K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
- V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1994.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.
- J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.
- F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010.
- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Boston, Mass, USA, 2012.
- X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong.
- X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, NY, USA, 2012.
- A. H. A. Ali, “The modified extended tanh-function method for solving coupled MKdV and coupled Hirota-Satsuma coupled KdV equations,” Physics Letters. A, vol. 363, no. 5-6, pp. 420–425, 2007.
- C. Li, A. Chen, and J. Ye, “Numerical approaches to fractional calculus and fractional ordinary differential equation,” Journal of Computational Physics, vol. 230, no. 9, pp. 3352–3368, 2011.
- G. H. Gao, Z. Z. Sun, and Y. N. Zhang, “A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions,” Journal of Computational Physics, vol. 231, no. 7, pp. 2865–2879, 2012.
- W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAM Journal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008/09.
- S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters. A, vol. 370, no. 5-6, pp. 379–387, 2007.
- Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008.
- Y. Hu, Y. Luo, and Z. Lu, “Analytical solution of the linear fractional differential equation by Adomian decomposition method,” Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 220–229, 2008.
- A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters. A, vol. 359, no. 3, pp. 175–182, 2006.
- A. M. A. El-Sayed, S. H. Behiry, and W. E. Raslan, “Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1759–1765, 2010.
- Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2199–2208, 2009.
- M. Inc, “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476–484, 2008.
- G. C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters. A, vol. 374, no. 25, pp. 2506–2509, 2010.
- J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
- E. Fan, “Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation,” Physics Letters. A, vol. 282, no. 1-2, pp. 18–22, 2001.
- J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.
- S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters. A, vol. 375, no. 7, pp. 1069–1073, 2011.
- M. L. Wang, “Solitary wave solutions for variant Boussinesq equations,” Physics Letters. A, vol. 199, no. 3-4, pp. 169–172, 1995.
- G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
- G. Jumarie, “Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution,” Journal of Applied Mathematics & Computing, vol. 24, no. 1-2, pp. 31–48, 2007.
- S. Guo, L. Mei, Y. Li, and Y. Sun, “The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,” Physics Letters. A, vol. 376, no. 4, pp. 407–411, 2012.
- B. Lu, “Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations,” Physics Letters. A, vol. 376, no. 28-29, pp. 2045–2048, 2012.
- G. Jumarie, “Cauchy's integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1444–1450, 2010.
- J. H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters. A, vol. 376, no. 4, pp. 257–259, 2012.
- J. H. He, “Asymptotic methods for solitary solutions and compacts,” Abstract and applied analysis. In press.
- S. E. Esipov, “Coupled Burgers equations: a model of polydispersive sedimentation,” Physical Review E, vol. 52, no. 4, pp. 3711–3718, 1995.
- A. A. Soliman, “The modified extended tanh-function method for solving Burgers-type equations,” Physica A, vol. 361, no. 2, pp. 394–404, 2006.