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Journal of Function Spaces
Volume 2018, Article ID 7035194, 6 pages
https://doi.org/10.1155/2018/7035194
Research Article

A Note on the Fractional Generalized Higher Order KdV Equation

School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China

Correspondence should be addressed to Yongyi Gu; moc.361@iygnoyugdg

Received 30 May 2018; Accepted 19 July 2018; Published 23 August 2018

Academic Editor: Xinguang Zhang

Copyright © 2018 Yongyi Gu. 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 obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics.

1. Introduction

Nonlinear fractional differential equations (NFDEs) are universally applied in signal processing, electrical networks, acoustics, fluid dynamics, biology, chemistry, physics, etc. For example, the singular behaviours [19] and impulsive phenomena [1019] often exhibit some blow-up properties [2025] which occur in a lot of complex physical processes. NFDEs have been attracted extensive attention and have been widely investigated [2638]. Exact solutions of NFDEs play an important role in the study of mathematical physics phenomena. Therefore, seeking exact solutions of NFDEs is an interesting and significant subject.

The fractional generalized higher order KdV equation is a useful model. Applying the generalized exp(-)-expansion method, Lu et al. [39] obtained exact solutions of this equation. In this article, we would like to utilize the complex method [4043] to seek exact solutions to the fractional generalized higher order KdV equation.

2. Preliminaries

Let ,   be a continuous function and denote a constant discrete span. Define the operator as follows:then the fractional difference of of order can be expressed aswhere , and its fractional derivative of order can be expressed asThe above is expressed asFurther, Jumarie’s modified Riemann-Liouville derivative [44, 45] is given bythen its related NFDE is given byLet , , , , andthen the degree of is denoted by . We define the differential polynomial asin which is a finite index set and are constants. The degree of can be denoted by .

The ordinary differential equation (ODE) is given bywhere are constants, .

Suppose that the meromorphic solutions of (9) have at least one pole. Let and insert the Laurent seriesinto (9); if it is determined different Laurent singular parts:then (9) is said to satisfy the weak condition.

Give two complex numbers such that , let be the discrete subset , and L is isomorphic to . Let the discriminant and

A meromorphic function with double periods , which satisfies the equationin which , , and , is called the Weierstrass elliptic function.

If a meromorphic function is a rational function of , an elliptic function, or a rational function of , then we say that belongs to the class .

In 2009, Eremenko et al. [46] considered the following -order Briot-Bouquet equation (BBEq):where and are constant coefficient polynomials. For the -order BBEq, we have the lemma as follows.

Lemma 1 (see [41, 47, 48]). Let , and , and the -order BBEqsatisfies weak condition, then the meromorphic solutions belong to the class . Suppose that for some values of parameters such solutions exist, then other meromorphic solutions should form one-parametric family , . Then each elliptic solution with a pole at can be expressed asin which are determined by (10), , and .
Each rational function solution can be expressed asand it has distinct poles of multiplicity .
Each simply periodic solution is a rational function of and can be expressed asand it has distinct poles of multiplicity .

Lemma 2 (see [48, 49]). Weierstrass elliptic functions have the following addition formula:When , it can be degenerated to rational functions according toWhen , it can also be degenerated to simple periodic functions according to

3. Main Results

The fractional generalized higher order KdV equation [39] is given as

Substituting traveling wave transforminto (22), we getIntegrating (24) yieldswhere and are constants and is the integral constant.

Theorem 3. If , then the meromorphic solutions of (25) have the following forms.
The rational function solutionswhere , , and .
The simply periodic solutionswhere , , and .
The elliptic function solutionswhere , , and .

Proof. Substituting (10) into (25) we have ,  ,  ,  ,  ,  ,  , and is an arbitrary constant.
Therefore, (25) satisfies the weak condition. In the following, we will show the meromorphic solutions of (25).
By (17), we infer that the indeterminant rational solutions of (25) arewith pole at .
Inserting into (25), we havethen we get ,  , and  .
So we can determine thatwhere and .
Therefore the rational solutions of (25) arewhere , , and .
Let . To derive simply periodic solutions, we substitute into (25) to yieldSubstitutinginto (33), we obtain thatwhere and .
Substituting into (35), we achieve simply periodic solutions to (25) with pole at where and .
Thurs, simply periodic solutions of (25) arewhere , , and .
From (16) of Lemma 1, the elliptic solutions of (25) are expressed aswith pole at .
Putting into (25), we getwhere and .
So, the elliptic solutions of (25) arewhere .
We can apply the addition formula to rewrite it aswhere , , and .
Substitute traveling wave transform into the meromorphic solutions of (25) to get traveling wave exact solutions to the fractional generalized higher order KdV equation. So we obtain Theorem 4 as follows.

Theorem 4. If , then traveling wave solutions of (25) have the following forms.
The rational function solutionswhere , , and .
The simply periodic solutionswhere , , and .
The elliptic function solutionswhere , , and .

4. Conclusions

In this note, we have used the complex method to construct exact solutions to the mentioned NFDE. Although we do not show that the meromorphic solutions of the fractional generalized higher order KdV equation belong to the class , we can still obtain the meromorphic solutions to this NFDE and then get its traveling wave exact solutions. The results demonstrate that the applied method is direct and efficient method, which allows us to do tedious and complicated algebraic calculation. We can utilize these ideas to other NFDEs.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the NSF of China (11701111 and 11271090); the NSF of Guangdong Province (2016A030310257); Guangdong Universities (Basic and Applied Research) Major Project (2017KZDXM038); Guangzhou City Social Science Federation “Yangcheng Young Scholars” Project (18QNXR35).

References

  1. X. Hao, L. Liu, and Y. Wu, “On positive solutions of m-point nonhomogeneous singular boundary value problem,” Nonlinear Analysis, vol. 73, no. 8, pp. 2532–2540, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  2. X. Zhang and L. Liu, “A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3501–3508, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. P. Li and G. Ren, “Some classes of equations of discrete type with harmonic singular operator and convolution,” Applied Mathematics and Computation, vol. 284, pp. 185–194, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Y. Guan, Z. Zhao, and X. Lin, “On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques,” Boundary Value Problems, vol. 2016, no. 1, article 141, 2016. View at Publisher · View at Google Scholar · View at Scopus
  5. X. Hao, “Positive solution for singular fractional differential equations involving derivatives,” Advances in Difference Equations, Article ID 139, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  6. Z. Zheng and Q. Kong, “Friedrichs extensions for singular Hamiltonian operators with intermediate deficiency indices,” Journal of Mathematical Analysis and Applications, vol. 461, no. 2, pp. 1672–1685, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  7. F. Yan, M. Zuo, and X. Hao, “Positive solution for a fractional singular boundary value problem with p-Laplacian operator,” Boundary Value Problems, article no. 51, 10 pages, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. Jiang, L. Liu, and Y. Wu, “Positive solutions to singular fractional differential system with coupled boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 11, pp. 3061–3074, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. J. Liu and Z. Zhao, “Existence of positive solutions to a singular boundary-value problem using variational methods,” Electronic Journal of Differential Equations, vol. 2014, no. 135, 9 pages, 2014. View at Google Scholar · View at MathSciNet
  10. X. Hao, L. Liu, and Y. Wu, “Positive solutions for second order impulsive differential equations with integral boundary conditions,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 101–111, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. X. Hao, M. Zuo, and L. Liu, “Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities,” Applied Mathematics Letters, vol. 82, pp. 24–31, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  12. J. Wang, A. G. Ibrahim, and D. O'Regan, “Topological structure of the solution set for fractional non-instantaneous impulsive evolution inclusions,” Journal of Fixed Point Theory and Applications, vol. 20, no. 2, Art. 59, 25 pages, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Y. M. Xu and H. J. Zhang, “Positive solutions of an infnite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5806–5818, 2012. View at Google Scholar
  14. J. Liu and Z. Zhao, “An application of variational methods to second-order impulsive differential equation with derivative dependence,” Electronic Journal of Differential Equations, vol. 2014, no. 62, pp. 1–13, 2014. View at Google Scholar · View at MathSciNet
  15. L. Gao, D. Wang, and G. Wang, “Further results on exponential stability for impulsive switched nonlinear time-delay systems with delayed impulse effects,” Applied Mathematics and Computation, vol. 268, pp. 186–200, 2015. View at Publisher · View at Google Scholar
  16. X. Hao and L. Liu, “Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces,” Mathematical Methods in the Applied Sciences, vol. 40, no. 13, pp. 4832–4841, 2017. View at Google Scholar · View at MathSciNet · View at Scopus
  17. Y. L. Guan, Z. Q. Zhao, and X. L. Lin, “On the Existence of Solutions for Impulsive Fractional Differential Equations,” Advances in Mathematical Physics, vol. 2017, Article ID 1207456, 12 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  18. Y. Wang and Z. Zhao, “Existence and multiplicity of solutions for a second-order impulsive differential equation via variational methods,” Advances in Difference Equations, Paper No. 46, 9 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. Liu and Z. Zhao, “Multiple solutions for impulsive problems with non-autonomous perturbations,” Applied Mathematics Letters, vol. 64, pp. 143–149, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Q. Gao, F. Li, and Y. Wang, “Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation,” Central European Journal of Mathematics, vol. 9, no. 3, pp. 686–698, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. F. S. Li and J. L. Li, “Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 1005–1014, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. F. Li and Q. Gao, “Blow-up of solution for a nonlinear Petrovsky type equation with memory,” Applied Mathematics and Computation, vol. 274, pp. 383–392, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. F. Li and J. Li, “Global existence and blow-up phenomena for p-Laplacian heat equation with inhomogeneous Neumann boundary conditions,” Boundary Value Problems, vol. 2014, no. 218, 14 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  24. F. Sun, L. Liu, and Y. Wu, “Finite time blow-up for a class of parabolic or pseudo-parabolic equations,” Computers & Mathematics with Applications, vol. 75, no. 10, pp. 3685–3701, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  25. X. Peng, Y. Shang, and X. Zheng, “Lower bounds for the blow-up time to a nonlinear viscoelastic wave equation with strong damping,” Applied Mathematics Letters, vol. 76, pp. 66–73, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  26. X. Du and A. Mao, “Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations,” Journal of Function Spaces, vol. 2017, Article ID 3793872, 7 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  27. Q. Feng and F. Meng, “Explicit solutions for space-time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method,” Optik - International Journal for Light and Electron Optics, vol. 127, no. 19, pp. 7450–7458, 2016. View at Publisher · View at Google Scholar · View at Scopus
  28. Y. Guo, “Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations,” Bulletin of the Korean Mathematical Society, vol. 47, no. 1, pp. 81–87, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Y. Guo, “Solvability of boundary-value problems for nonlinear fractional differential equations,” Ukrainian Mathematical Journal, vol. 62, no. 9, pp. 1409–1419, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. A. Qian, “Sing-changing solutions for nonlinear problems with strong resonance,” Electronic Journal of Differential Equations, vol. 2012, no. 17, 8 pages, 2012. View at Google Scholar · View at MathSciNet
  31. Q. Feng and F. Meng, “Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method,” Mathematical Methods in the Applied Sciences, vol. 40, no. 10, pp. 3676–3686, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. F. Li, “Limit behavior of the solution to nonlinear viscoelastic Marguerre–von Kármán shallow shell system,” Journal of Differential Equations, vol. 249, no. 6, pp. 1241–1257, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. A. Qian, “Infinitely many sign-changing solutions for a Schrödinger equation,” Advances in Difference Equations, vol. 2011, no. 39, 6 pages, 2011. View at Google Scholar · View at MathSciNet
  34. H. Liu and F. Meng, “Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent,” Advances in Difference Equations, Paper No. 291, 14 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. K. M. Zhang, “On a sign-changing solution for some fractional differential equations,” Boundary Value Problems, vol. 2017, no. 59, 8 pages, 2017. View at Google Scholar · View at MathSciNet
  36. J. Shao, Z. Zheng, and F. Meng, “Oscillation criteria for fractional differential equations with mixed nonlinearities,” Advances in Difference Equations, p. 323, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  37. R. Xu and F. Meng, “Some new weakly singular integral inequalities and their applications to fractional differential equations,” Journal of Inequalities and Applications, vol. 2016, no. 78, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  38. Y. Wang and J. Jiang, “Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian,” Advances in Difference Equations, Paper No. 337, 19 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  39. D. Lu, C. Yue, and M. Arshad, “Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation,” Advances in Mathematical Physics, vol. 2017, Article ID 6743276, 6 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  40. W. Yuan, Y. Li, and J. Lin, “Meromorphic solutions of an auxiliary ordinary differential equation using complex method,” Mathematical Methods in the Applied Sciences, vol. 36, no. 13, pp. 1776–1782, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. W. J. Yuan, Y. D. Shang, Y. Huang, and H. Wang, “The representation of meromorphic solutions to certain ordinary differential equations and its applications,” Scientia Sinica Mathematica, vol. 43, no. 6, pp. 563–575, 2013. View at Publisher · View at Google Scholar
  42. W. Yuan, Q. Chen, J. Qi, and Y. Li, “The general traveling wave solutions of the Fisher equation with degree three,” Advances in Mathematical Physics, Art. ID 657918, 5 pages, 2013. View at Google Scholar · View at MathSciNet
  43. Y. Gu, W. Yuan, N. Aminakbari, and Q. Jiang, “Exact Solutions of the Vakhnenko-Parkes Equation with Complex Method,” Journal of Function Spaces, vol. 2017, Article ID 6521357, 6 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  44. 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 MathSciNet · View at Scopus
  45. 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 MathSciNet · View at Scopus
  46. A. E. Eremenko, L. Liao, and T. W. Ng, “Meromorphic solutions of higher order Briot-Bouquet differential equations,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 146, no. 1, pp. 197–206, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  47. N. A. Kudryashov, “Meromorphic solutions of nonlinear ordinary differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 10, pp. 2778–2790, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  48. S. Lang, Elliptic Functions, Springer, New York, NY, USA, 2nd edition, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  49. R. Conte and M. Musette, “Elliptic general analytic solutions,” Studies in Applied Mathematics, vol. 123, no. 1, pp. 63–81, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus