About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 491359, 13 pages
http://dx.doi.org/10.1155/2013/491359
Research Article

The Extended Trial Equation Method for Some Time Fractional Differential Equations

1Department of Mathematics, Faculty of Science and Arts, Bozok University, 66100 Yozgat, Turkey
2Department of Mathematics, Faculty of Science, Ege University, 35100 Izmir, Turkey

Received 21 February 2013; Revised 25 March 2013; Accepted 14 April 2013

Academic Editor: Ferhan M. Atici

Copyright © 2013 Yusuf Pandir 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.

Abstract

Nonlinear fractional partial differential equations have been solved with the help of the extended trial equation method. Based on the fractional derivative in the sense of modified Riemann-Liouville derivative and traveling wave transformation, the fractional partial differential equation can be turned into the nonlinear nonfractional ordinary differential equation. For illustrating the reliability of this approach, we apply it to the generalized third order fractional KdV equation and the fractional equation according to the complete discrimination system for polynomial method. As a result, some new exact solutions to these nonlinear problems are successfully constructed such as elliptic integral function solutions, Jacobi elliptic function solutions, and soliton solutions.

1. Introduction

Many physical phenomena in mathematical biology, electromagnetic theory, fluid mechanics, signal processing, engineering, and other fields of science can be modeled using the fractional derivatives. For example, the use of fractional derivative is analyzed in the fields of controller tuning, redundant robots, legged robots, digital circuit synthesis, and heat diffusion [1]. Also, we refer the reader to [1, 2] for details concerning fractional derivative or calculus and their applications in science and engineering. In some sense fractional differential equations could represent various real-life problems [35]. However, an effective and general method for solving them cannot be found even in the most useful works. Also, remarkable progress has been made in the construction of the approximate solutions for fractional nonlinear partial differential equations. There are various mathematical methods which can constitute the exact solutions for some time fractional differential equations [6, 7]. Single kink soliton solutions, multiple-soliton solutions, compacton-like solutions, singular solitons, and other solutions have been found by use of these approaches. Apart from all these, some new exact solutions are obtained by using the trial equation methods. Some of them are elliptic integral , , and functions, and Jacobi elliptic function solutions.

In Section 2, primarily we give some definitions and properties of the fractional calculus theory and also produce the extended trial equation method for fractional nonlinear evolution equations with higher order nonlinearity. The power of this steerable method showed that this method can be applied to two nonlinear fractional problems. In Section 3, as applications, we solve the fractional nonlinear partial differential equations such as the generalized third order fractional KdV equation [8] where and model the subdiffusion and the subdispersion term, and the fractional equation [9] where . Equation (1) has a wide range of applications in plasma physics, fluid physics, and capillary-gravity waves. Although there are a lot of studies for the classical generalized third order KdV and the equation, this is the first study with respect to the fractional versions of these problems. The aim of this paper is to obtain exact solutions of the generalized third order fractional KdV and the fractional equation by extendend trial equation method and to determine the accuracy of extended trial equation method in solving these kinds of problems. The exact solutions obtained by the interesting wave transformation are used to determine some new phenomena of the fractional nonlinear evolution equations. Based on the derived solutions for (1) and (2), the phenomena of some solutions are shown graphically (see Figures 1, 2, and 3).

fig1
Figure 1: Graph of the solution (31) corresponding to the values and from (a) to (c) when , .
fig2
Figure 2: Graph of the solution (33) corresponding to the values and from (a) to (c) when , , and .
fig3
Figure 3: Graph of the solution (60) corresponding to the values and from (a) to (c) when , , and .

2. Preliminaries and the Extended Trial Equation Method

In this part of the paper, it would be helpful to give some definitions and properties of the fractional calculus theory. For an introduction to the classical fractional calculus we indicate the reader to [35]. Here, we shortly review the modified Riemann-Liouville derivative from the recent fractional calculus proposed by Jumarie [1012]. Let be a continuous function and . The Jumarie modified fractional derivative of order and may be defined by expression of [13] as follows:

In addition to this expression, we may give a summary of the fractional modified Riemann-Liouville derivative properties which are used further in this paper. Some of the useful formulas are given as

Some new trial equation methods were defined in the literature [1426]. In this paper, the extended trial equation method will be given. In order to apply this method to fractional nonlinear partial differential equations, we consider the following steps.

Step 1. Consider time fractional partial differential equation in two variables and a dependent variable and take the wave transformation where . Substituting (6) into (5) yields a nonlinear ordinary differential equation

Step 2. Take trial equation as follows: where Using (8) and (9), we can write where and are polynomials. Substituting the above relations into (7) yields an equation of polynomial of : According to the balance principle, we can find a relation of , , and . We can compute some values of , , and .

Step 3. Let the coefficients of all be zero will yield an algebraic equations system: Solving this system, we will determine the values of , and .

Step 4. Reduce (9) to the elementary integral form Using a complete discrimination system for polynomial to classify the roots of , we solve (13) with the help of Mathematica and classify the exact solutions to (7). In addition, we can write the exact traveling wave solutions to (5), respectively.

3. Applications to the Extended Trial Equation Method

In this section, we apply the method given in Section 2 to the generalized third order fractional KdV and the fractional equation, respectively.

3.1. Application to Generalized Fractional KdV Equation

In the case of , (1) reduces to the classical nonlinear generalized third order KdV equation. Many researchers have tried to get the exact solutions of this equation by using a variety of methods. But, we solve the generalized fractional KdV equation for . Compactons, solitary patterns, and periodic and solitary traveling plane waves solutions of this equation are found.

Let us consider the travelling wave solutions of (1), and we perform the transformation , , where is constant to be determined later. Then, integrating this equation with respect to and setting the integration constant to zero, we get By use of the following transformation: (14) turns into the equation From (8)–(10), we can write the following highest order nonlinear terms in order to determine the balance procedure: Balancing the highest order nonlinear terms of and in (16), we have hence This resolution procedure is applied, and we obtain results as follows.

Case 1. If we take , , and , then where , . Thus, we have a system of algebraic equations from the coefficients of polynomial of . Solving the algebraic equation system (12), we get Substituting these results into (13), we have Integrating (22), we procure the solutions to (1) as follows: where Also , , and are the roots of the polynomial equation Substituting the solutions (23) into (8) and (15), we have

If we take and , then the solutions (26)–(28) can reduce to rational function solution 1-soliton wave solution singular soliton solution

where , , , ,  , and . Here, and are the amplitude of the soliton, while and are the inverse width of the solitons and is the velocity.

On the other hand, if we take and , then the Jacobi elliptic function solution (29) can be written in the form where and , . Thus, we can say that the solitons exist for .

Remark 1. We provide a proof by verifying the exact solution (31) by direct substitution. If we take and , then (1) becomes
where , , , and . Here, .
Equation (31) is If , then Substituting the above relations into the partial differential equation, we obtain the following equation: Otherwise, we can write where and . From this equation, we find . Substituting this relation into the above equation yields . Thus, it is seen that the obtained solution (31) is an exact solution of the fractional KdV equation.

Remark 2. The solutions (30)–(33) obtained by using the extended trial equation method for (1) have been checked by Mathematica. To our knowledge, rational function solution, 1-soliton solution, the singular soliton solution, and the Jacobi elliptic function that we find in this paper are not shown in the previous literature. These results are new traveling wave solutions of (1).

Case 2. If we take , , and , then where , . Respectively, solving the algebraic equation system (12) yields as follows: Substituting these results into (13), we get Integrating (41), we obtain the solutions to (1) as follows: where Also , , , and are the roots of the polynomial equation Substituting the solutions (42) into (8) and (15), we have

For simplicity, if we take , then we can write the solutions (45) as follows: where , , , , and . Here, is the amplitude of the soliton, while is the velocity and and are the inverse width of the solitons.

Remark 3. To our knowledge, the solutions (46) that we find in this paper have not been found in the literature, and these are new exact solutions of (1).

3.2. Application to the Fractional Equation

In the case of , (2) reduces to the classical nonlinear generalized equation. Many authors have studied to find the exact solutions of this equation by using a variety of methods [27, 28]. In this paper, we consider the generalized fractional equation for the case of . Compactons, solitary patterns, and periodic and solitary traveling plane waves solutions of this equation are obtained.

Let us consider the travelling wave solutions of (2), and we perform the transformation , , where is constant to be determined later. Then, integrating this equation with respect to and setting the integration constant to zero, we get By use of the following transformation: (47) turns into the equation From (8)–(10), we can write the following highest order nonlinear terms in order to determine the balance procedure: Balancing the highest order nonlinear terms of and in (49), we have hence This resolution procedure is applied and we obtain results as follows.

Case 1. If we take ,  , and , then where , . Thus, we have a system of algebraic equations from the coefficients of polynomial of . Solving the algebraic equation system (12), we get Substituting these results into (13), we have Integrating (55), we procure the solutions to (2) as follows: where . Also , are the roots of the polynomial equation Substituting the solutions (56) into (8) and (48), we have

If we take and , then the solutions (58) can reduce to single kink solution and the hyperbolic function solution, respectively,

where , . Here, is the inverse width of the solitons.

Remark 4. The solutions (59)-(60) obtained by using the extended trial equation method for (2) have been checked by Mathematica. To our knowledge, the hyperbolic function solution and single kink solution, that we find in this paper, are not shown in the previous literature. These results are new traveling wave solutions of (2).

Case 2. If we take , , and , then where , . Thus, we have a system of algebraic equations from the coefficients of polynomial of . Solving the algebraic equation system (12), we get Substituting these results into (13), we have Integrating (63), we procure the solutions to (2) as follows: where . Also , are the roots of the polynomial equation Substituting the solutions (64) into (8) and (48), we have

For simplicity, and , then we can write the solutions (66) as follows:

where , . Here, is the inverse width of the solitons.

Case 3. If we take , , and , then where , . Respectively, solving the algebraic equation system (12) yields Substituting these results into (13), we have Integrating (70), we obtain the solutions to (2) as follows: where Also , , and are the roots of the polynomial equation

4. Conclusions

In this study, the extended trial equation method is applied successfully for solving the nonlinear fractional differential equations. We used it to obtain some soliton and elliptic function or elliptic integral function solutions, which are one-wave solutions, to the generalized third order fractional KdV equation and the fractional equation. The performance of this method is reliable and effective, and also this method gives more general solutions. We think that the proposed method can also be applied to other fractional nonlinear differential equations with nonlinear evolution. In our future studies, we will define the multiple trial equation method for the fractional or nonfractional nonlinear partial differential equations.

Appendix

In [29], the first kind of the Legendre elliptic integral is given by

where is known as the modulus. Otherwise, the inverse function is called as the Jacobi elliptic sine function. This function can be represented by .

When , the Jacobi elliptic function is transformed into . If we take , is degenerated as .

Acknowledgments

The research has been supported by Yozgat University Foundation.

References

  1. J. A. T. Machado, M. F. Silva, R. S. Barbosa et al., “Some applications of fractional calculus in engineering,” Mathematical Problems in Engineering, vol. 2010, Article ID 639801, 34 pages, 2010. View at Publisher · View at Google Scholar
  2. M. Dalir and M. Bashour, “Applications of fractional calculus,” Applied Mathematical Sciences, vol. 4, no. 21–24, pp. 1021–1032, 2010. View at Zentralblatt MATH · View at MathSciNet
  3. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  4. 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 B.V., Amsterdam, The Netherlands, 2006. View at Zentralblatt MATH · View at MathSciNet
  5. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  6. B. Lu, “The first integral method for some time fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, no. 2, pp. 684–693, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. El-Kahlout, T. O. Salim, and S. El-Azab, “Exact solution of time fractional partial differential equation,” Applied Mathematical Sciences, vol. 2, no. 49–52, pp. 2577–2590, 2008. View at Zentralblatt MATH · View at MathSciNet
  8. R. Sahadevan and T. Bakkyaraj, “Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations,” Journal of Mathematical Analysis and Applications, vol. 393, no. 2, pp. 341–347, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Dehghan, J. Manafian, and A. Saadatmandi, “Solving nonlinear fractional partial differential equations using the homotopy analysis method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 448–479, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. G. Jumarie, “Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function,” Journal of Applied Mathematics & Computing, vol. 23, no. 1-2, pp. 215–228, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Z. Z. Ganji, D. D. Ganji, A. D. Ganji, and M. Rostamian, “Analytical solution of time-fractional Navier-Stokes equation in polar coordinate by homotopy perturbation method,” Numerical Methods for Partial Differential Equations, vol. 26, no. 1, pp. 117–124, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  14. C.-S. Liu, “Trial equation method and its applications to nonlinear evolution equations,” Acta Physica Sinica, vol. 54, no. 6, pp. 2505–2509, 2005. View at Zentralblatt MATH · View at MathSciNet
  15. C.-S. Liu, “Trial equation method for nonlinear evolution equations with rank inhomogeneous: mathematical discussions and applications,” Communications in Theoretical Physics, vol. 45, pp. 219–223, 2006. View at Publisher · View at Google Scholar
  16. C.-S. Liu, “A new trial equation method and its applications,” Communications in Theoretical Physics, vol. 45, pp. 395–397, 2006. View at Publisher · View at Google Scholar
  17. C.-S. Liu, “Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations,” Computer Physics Communications, vol. 181, no. 2, pp. 317–324, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y.-J. Cheng, “Classification of traveling wave solutions to the Vakhnenko equations,” Computers & Mathematics with Applications, vol. 62, no. 10, pp. 3987–3996, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Y.-J. Cheng, “Classification of traveling wave solutions to the modified form of the Degasperis-Procesi equation,” Mathematical and Computer Modelling, vol. 56, no. 1-2, pp. 43–48, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics,” Pramana, vol. 77, no. 6, pp. 1023–1029, 2011. View at Publisher · View at Google Scholar
  21. Y. Gurefe, A. Sonmezoglu, and E. Misirli, “Application of an irrational trial equation method to high-dimensional nonlinear evolution equations,” Journal of Advanced Mathematical Studies, vol. 5, no. 1, pp. 41–47, 2012. View at Zentralblatt MATH · View at MathSciNet
  22. Y. Pandir, Y. Gurefe, U. Kadak, and E. Misirli, “Classification of exact solutions for some nonlinear partial differential equations with generalized evolution,” Abstract and Applied Analysis, vol. 2012, Article ID 478531, 16 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  23. Y. Gurefe, E. Misirli, A. Sonmezoglu, and M. Ekici, “Extended trial equation method to generalized nonlinear partial differential equations,” Applied Mathematics and Computation, vol. 219, no. 10, pp. 5253–5260, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. Y. Pandir, Y. Gurefe, and E. Misirli, “Classification of exact solutions to the generalized Kadomtsev-Petviashvili equation,” Physica Scripta, vol. 87, Article ID 025003, 12 pages, 2013.
  25. M. V. Demina and N. A. Kudryashov, “From Laurent series to exact meromorphic solutions: the Kawahara equation,” Physics Letters A, vol. 374, no. 39, pp. 4023–4029, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. N. A. Kudryashov and M. V. Demina, “Traveling wave solutions of the generalized nonlinear evolution equations,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 551–557, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. A.-M. Wazwaz, “The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 18–23, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  28. P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelengths,” Physical Review Letters, vol. 70, pp. 564–567, 1993. View at Publisher · View at Google Scholar
  29. Z. Fu, S. Liu, S. Liu, and Q. Zhao, “New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations,” Physics Letters A, vol. 290, no. 1-2, pp. 72–76, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet