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The Scientific World Journal
Volume 2014, Article ID 489495, 10 pages
http://dx.doi.org/10.1155/2014/489495
Research Article

A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

1Department of Management Information Systems, School of Applied Sciences, Dumlupinar University, 43100 Kutahya, Turkey
2Department of Mathematics Education, Education Faculty, Yildiz Technical University, 34220 Istanbul, Turkey

Received 2 October 2013; Accepted 11 December 2013; Published 10 March 2014

Academic Editors: A. Kılıçman, S. S. Ray, and A. Secer

Copyright © 2014 Özkan Güner and Adem C. Cevikel. 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.

Abstract

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

1. Introduction

Fractional differential equations (FDEs) are generalizations of classical differential equations of integer order. Recently, fractional differential equations have gained much attention as they are widely used to describe various complex phenomena in various applications such as the fluid flow, signal processing, control theory, systems identification, finance and fractional dynamics, and physics. The fractional differential equations have been investigated by many researchers [13]. In recent decades, a large amount of literature has been provided to construct the exact solutions of fractional ordinary differential equations and fractional partial differential equations of physical interest. Many powerful and efficient methods have been proposed to obtain approximate solutions of fractional differential equations, such as the Adomian decomposition method [4, 5], the variational iteration method [6, 7], the homotopy analysis method [8, 9], the homotopy perturbation method [10, 11], and the differential transformation method [1214]. The fractional subequation method [1517], the first integral method [18], the Exp-function method [19, 20], and the (G′/G)-expansion method [2123] can be used to construct the exact solutions for some time and space fractional differential equations.

He and Wu [24] systematically proposed a new method in 2006, called the Exp-function method, to obtain exact solutions of nonlinear differential equations. The Exp-function method has been successfully applied to many kinds of nonlinear differential equations [2528], such as high-dimensional equations [2931], variable-coefficient equations [32, 33], differential-difference equations [34, 35], and stochastic equations [36, 37].

The present paper investigates for the first time the applicability and effectiveness of the Exp-function method on fractional nonlinear partial differential equations.

2. The Modified Riemann-Liouville Derivative

Jumarie proposed a modified Riemann-Liouville derivative. With this kind of fractional derivative and some useful formulas, we can convert fractional differential equations into integer-order differential equations by variable transformation in [38].

In this section, we firstly give some properties and definitions of the modified Riemann-Liouville derivative which are used further in this paper.

Assume that , denote a continuous (but not necessarily differentiable) function. The Jumarie modified Riemann-Liouville derivative of order is defined by the expression

A few properties of the fractional modified Riemann-Liouville derivative were summarized and three famous formulas of them are which are direct consequences of the equality

Secondly, let us consider the time fractional differential equation with independent variables and a dependent variable :

Using the fractional variable transformation where and are constants to be determined later. Similarly, let us consider the space fractional differential equation with independent variables and a dependent variable :

Next, using the fractional variable transformation where and are constants to be determined later.

The fractional differential equation (6) is reduced to a nonlinear ordinary differential equation where .

3. Description of the Exp-Function Method

We consider the general nonlinear ordinary differential equation in (8). According to Exp-function method, we assume that the wave solution can be expressed in the following form [24]: where , , , and are positive integers which are known to be further determined and and are unknown constants. We can rewrite (9) in the following equivalent form:

This equivalent formulation plays an important and fundamental part for finding the analytic solution of problems. To determine the value of and , we balance the linear term of the highest order of (8) with the highest degree nonlinear term. Similarly, to determine the value of and , we balance the linear term of the lowest order of (8) with the lowest degree nonlinear term.

We suppose that the solution in (8) can be expressed as where is the solution of the auxiliary equation . In a similar way, can be expressed in (11).

Theorem 1. Suppose that and are, respectively, the highest order linear term and the highest degree nonlinear term of a nonlinear ODE, where and are both positive integers. Then the balancing procedure using the Exp-function ansatz leads to and and , [39].

To show the efficiency of the method described in the previous part, we present some FDEs examples.

4. The Time Fractional Sharma-Tasso-Olver Equation

We consider the nonlinear fractional Sharma-Tasso-Olver equation [40] subject to the initial condition where and are arbitrary constants and is a parameter describing the order of the fractional time derivative. The function is assumed to be a causal function of time.

For our purpose, we introduce the following transformations: where is a constant.

Substituting (14) into (12), we can know that (12) reduced into an ODE where .

Integrating (15) with respect to yields where is a constant of integration.

Here take notice of the nonlinear term in (16), and we can balance and by the idea of the Exp-function method [24] to determine the values of , , , and . By simple calculation, we have where are determined coefficients only for simplicity. Balancing the highest order of Exp-function in (17) we have which leads to the result Similarly to determine values of and , we balance the linear term of the lowest order in (16): where are determined coefficients only for simplicity. From (20), we obtain and this gives

For simplicity, we set and , so (10) reduces to

Substituting (23) into (16) and by the help of symbolic computation, we have where

Solving this system of algebraic equations by using symbolic computation, we obtain the following results.

Case 1. We have where and are free parameters. Substituting these results into (23), we obtain the following exact solution:

If we set and , (27) becomes which is the other exact solution of the fractional Sharma-Tasso-Olver equation.

If we set , (27) becomes which is the other exact solution of the fractional Sharma-Tasso-Olver equation.

Case 2. We have where and are free parameters. Substituting these results into (23), we obtain the following exact solution:

Comparing our results with the results [18, 19], it can be seen that our results are new to our best knowledge.

5. The Space Fractional Burgers Equation

We consider the space fractional Burgers equation [41] with the following initial value problem: where and are arbitrary constants and is a parameter describing the order of the fractional space derivative. The function is assumed to be a causal function of time.

For our purpose, we introduce the following transformations: where is a constant.

Substituting (34) into (32), we can know that (32) reduced into an ODE where .

Integrating (35) with respect to yields where is a constant of integration.

Here take notice of the nonlinear term in (36), and we can balance and by the idea of the Exp-function method [24] to determine the values of , , , and . By simple calculation, we have where are determined coefficients only for simplicity. Balancing the highest order of Exp-function in (37) we have which leads to the result Similarly to determine values of and , we balance the linear term of the lowest order in (36): where are determined coefficients only for simplicity. From (40), we obtain and this gives

For simplicity, we set and , so (10) reduces to

Substituting (43) into (36) and by the help of computation, we have where

Solving this system of algebraic equations by using symbolic computation, we obtain the following results.

Case 1. We have where , , and are free parameters. Substituting these results into (43), we get the following exact solution: which is the exact solution of the space fractional Burgers equation.

Case 2. We have where , , and are free parameters. Substituting these results into (43), we obtain the following exact solution: which is the exact solution of the space fractional Burgers equation.

Case 3. We have where , , , and are free parameters. Substituting these results into (43), we get the following exact solution: which is the exact solution of the space fractional Burgers equation.

The obtained solutions for the space fractional Burgers equation are new to our best knowledge.

6. The Time Fractional fmKdV Equation

We consider the following fractional time fractional fmKdV equation [42]: with the initial conditions as where is an arbitrary constant and is a parameter describing the order of the fractional time derivative.

For our purpose, we introduce the following transformations where and are constants.

Substituting (54) into (52), we can know that (52) reduced into an ODE where .

By using the ansatz (55), for the linear term of highest order with the highest order and the nonlinear term , balancing with in (55) gives where are determined coefficients only for simplicity. Balancing the highest order of Exp-function in (56) we have which leads to the result Similarly to determine values of and , we balance the linear term of the lowest order in (55): where are determined coefficients only for simplicity. From (59), we obtain and this gives For simplicity, we set and , so (10) reduces to

Substituting (62) into (55) and by the help of computation, we have where Solving this system of algebraic equations by using symbolic computation, we obtain the following results.

Case 1. We have where , , and are free parameters. Substituting these results into (62), we obtain the following exact solution:

Case 2. We have where is a free parameter. Substituting these results into (62), we obtain the following exact solution:
If we set and , (68) becomes which is the exact solution of the time fractional fmKdV equation.

Case 3. We have where and are free parameters. Substituting these results into (62), we obtain the following exact solution:

Case 4. We have where and are free parameters. Substituting these results into (62), we obtain the following exact solution:
If we set and , (73) becomes which is the exact solution of the time fractional fmKdV equation.

Case 5. We have where and are free parameters. Substituting these results into (62), we have the following exact solution:
If we take , (76) becomes which is the exact solution of the time fractional fmKdV equation.

The established solutions have been checked by putting them back into the original equation (52). To the best of our knowledge, they have not been obtained in literature.

7. Conclusion

In this paper, we use the Exp-function method to calculate the exact solutions for the time and space fractional nonlinear partial differential equations. When the parameters take certain values, the solitary wave solutions are derived from the exponential form. Since this method is very efficient, reliable, simple, and powerful in finding the exact solutions for the nonlinear fractional differential equations, the proposed method can be extended to solve many systems of nonlinear fractional partial differential equations. We hope that the present solutions may be useful in further numerical analysis and these results are going to be very useful in further future research.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
  2. I. Podlubny, Fractional Differential Equations, Academic Press, California, Calif, USA, 1999.
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  4. 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. View at Publisher · View at Google Scholar · View at Scopus
  5. M. Safari, D. D. Ganji, and M. Moslemi, “Application of He's variational iteration method and Adomian's decomposition method to the fractional KdV-Burgers-Kuramoto equation,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2091–2097, 2009. View at Publisher · View at Google Scholar · View at Scopus
  6. N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007. View at Publisher · View at Google Scholar · View at Scopus
  7. 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. View at Publisher · View at Google Scholar · View at Scopus
  8. L. N. Song and H. Q. Zhang, “Solving the fractional BBM-Burgers equation using the homotopy analysis method,” Chaos, Solitons and Fractals, vol. 40, no. 4, pp. 1616–1622, 2009. View at Publisher · View at Google Scholar · View at Scopus
  9. A. A. M. Arafa, S. Z. Rida, and H. Mohamed, “Homotopy analysis method for solving biological population model,” Communications in Theoretical Physics, vol. 56, no. 5, pp. 797–800, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. K. A. Gepreel, “The homotopy perturbation method applied to the nonlinear fractional KolmogorovPetrovskiiPiskunov equations,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1428–1434, 2011. View at Publisher · View at Google Scholar · View at Scopus
  11. P. K. Gupta and M. Singh, “Homotopy perturbation method for fractional Fornberg-Whitham equation,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 250–254, 2011. View at Publisher · View at Google Scholar · View at Scopus
  12. 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. View at Publisher · View at Google Scholar · View at Scopus
  13. V. S. Ertürk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642–1654, 2008. View at Publisher · View at Google Scholar · View at Scopus
  14. A. Secer, M. Akinlar, and A. C. Cevikel, “Efficient solutions of systems of fractional PDEs by the differential transform method,” Advances in Difference Equations, vol. 2012, p. 188, 2012. View at Publisher · View at Google Scholar
  15. S. Zhang and H. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. B. Tong, Y. He, L. Wei, and X. Zhang, “A generalized fractional sub-equation method for fractional differential equations with variable coefficients,” Physics Letters A, vol. 376, no. 38-39, pp. 2588–2590, 2012. View at Publisher · View at Google Scholar
  17. 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. View at Publisher · View at Google Scholar · View at Scopus
  18. B. Lu, “The first integral method for some time fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, pp. 684–693, 2012. View at Publisher · View at Google Scholar
  19. B. Zheng, “Exp-function method for solving fractional partial differential equations,” The Scientific World Journal, vol. 2013, Article ID 465723, 8 pages, 2013. View at Publisher · View at Google Scholar
  20. A. Bekir, Ö. Güner, and A. C. Cevikel, “Fractional complex transform and exp-function methods for fractional differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 426462, 8 pages, 2013. View at Publisher · View at Google Scholar
  21. B. Zheng, “G'/G-expansion method for solving fractional partial differential equations in the theory of mathematical physics,” Communications in Theoretical Physics, vol. 58, no. 5, pp. 623–630, 2012. View at Publisher · View at Google Scholar
  22. K. A. Gepreel and S. Omran, “Exact solutions for nonlinear partial fractional differential equations,” Chinese Physics B, vol. 21, no. 11, pp. 110–204, 2012. View at Google Scholar
  23. A. Bekir and Ö. Güner, “Exact solutions of nonlinear fractional differential equations by G'/G-expansion method,” Chinese Physics B, vol. 22, no. 11, pp. 110–202, 2013. View at Google Scholar
  24. J. H. He and X. H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Scopus
  25. J. H. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Exp-function method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1421–1429, 2007. View at Publisher · View at Google Scholar · View at Scopus
  26. A. Ebaid, “Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method,” Physics Letters A, vol. 365, no. 3, pp. 213–219, 2007. View at Publisher · View at Google Scholar · View at Scopus
  27. S. Kutluay and A. Esen, “Exp-function method for solving the general improved KdV equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 6, pp. 717–725, 2009. View at Google Scholar · View at Scopus
  28. A. Bekir, “The exp-function method for Ostrovsky equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 6, pp. 735–739, 2009. View at Google Scholar · View at Scopus
  29. S. Zhang, “Application of Exp-function method to high-dimensional nonlinear evolution equation,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 270–276, 2008. View at Publisher · View at Google Scholar · View at Scopus
  30. A. Bekir and A. Boz, “Application of Exp-function method for 2+1-dimensional nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 458–465, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. A. C. Cevikel and A. Bekir, “New solitons and periodic solutions for 2+1- dimensional Davey-Stewartson equations,” Chinese Journal of Physics, vol. 51, pp. 1–13, 2013. View at Google Scholar
  32. S. Zhang, “Application of Exp-function method to a KdV equation with variable coefficients,” Physics Letters A, vol. 365, no. 5-6, pp. 448–453, 2007. View at Publisher · View at Google Scholar · View at Scopus
  33. S. A. El-Wakil, M. A. Madkour, and M. A. Abdou, “Application of Exp-function method for nonlinear evolution equations with variable coefficients,” Physics Letters A, vol. 369, no. 1-2, pp. 62–69, 2007. View at Publisher · View at Google Scholar · View at Scopus
  34. S. D. Zhu, “Exp-function method for the Hybrid-Lattice system,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 3, pp. 461–464, 2007. View at Google Scholar · View at Scopus
  35. A. Bekir, “Application of the Exp-function method for nonlinear differential-difference equations,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 4049–4053, 2010. View at Publisher · View at Google Scholar · View at Scopus
  36. C. Q. Dai and J. L. Chen, “New analytic solutions of stochastic coupled KdV equations,” Chaos, Solitons and Fractals, vol. 42, no. 4, pp. 2200–2207, 2009. View at Publisher · View at Google Scholar · View at Scopus
  37. C. Q. Dai and J. F. Zhang, “Application of he's EXP-function method to the stochastic mKdV equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 5, pp. 675–680, 2009. View at Google Scholar · View at Scopus
  38. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers and Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Scopus
  39. A. Ebaid, “An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms,” Journal of Mathematical Analysis and Applications, vol. 392, no. 1, pp. 1–5, 2012. View at Publisher · View at Google Scholar · View at Scopus
  40. L. N. Song, Q. Wang, and H. Q. Zhang, “Rational approximation solution of the fractional Sharma-Tasso-Olever equation,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 210–218, 2009. View at Publisher · View at Google Scholar · View at Scopus
  41. S. Momani, “Non-perturbative analytical solutions of the space- and time-fractional Burgers equations,” Chaos, Solitons and Fractals, vol. 28, no. 4, pp. 930–937, 2006. View at Publisher · View at Google Scholar · View at Scopus
  42. M. Kurulay and M. Bayram, “Approximate analytical solution for the fractional modified KdV by differential transform method,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1777–1782, 2010. View at Publisher · View at Google Scholar · View at Scopus