An expansion based on Sine-Gordon equation to Solve KdV and modified KdV equations in conformable fractional forms
DOI:
https://doi.org/10.5269/bspm.44592Abstract
An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equations in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions. Some exact solutions in traveling wave forms are explicitly expressed by the proposed method for both the Korteweg-de Vries and modified Korteweg-de Vries equations.
References
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30. Hosseini, K., Mayeli, P., & Ansari, R., Modified Kudryashov method for solving the conformable time-fractional KleinGordon equations with quadratic and cubic nonlinearities, Optik-International Journal for Light and Electron Optics, 130, 737-742, 2017. https://doi.org/10.1016/j.ijleo.2016.10.136
31. Korkmaz, A., Exact solutions of space-time fractional EW and modified EW equations, Chaos, Solitons & Fractals, 96, 132-138, 2017. https://doi.org/10.1016/j.chaos.2017.01.015
32. Zafar, A., Rational exponential solutions of conformable space-time fractional equal-width equations, Nonlinear Engineering, 2018. (in press) https://doi.org/10.1515/nleng-2018-0076
33. Atangana, A., Baleanu, D., & Alsaedi, A., New properties of conformable derivative, Open Mathematics, 13(1), 1-10, 2015. https://doi.org/10.1515/math-2015-0081
34. Abdeljawad, T., On conformable fractional calculus, Journal of computational and Applied Mathematics, 279, 57-66, 2015. https://doi.org/10.1016/j.cam.2014.10.016
35. Yan, C., A simple transformation for nonlinear waves, Physics Letters A, 224, 77-84, 1996. https://doi.org/10.1016/S0375-9601(96)00770-0
2. Korteweg, D. J., de Vries, G. , On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves, Philosophical Magazine, 39 (240): 422-443, 1895. https://doi.org/10.1080/14786449508620739
3. Miura, Robert M., Gardner, Clifford S., Kruskal, Martin D., Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Mathematical Phys., 9 (8): 1204-1209, 1968. https://doi.org/10.1063/1.1664701
4. Wadati, M., & Toda, M., The exact N-soliton solution of the Korteweg-de Vries equation, Journal of the Physical Society of Japan, 32(5), 1403-1411, 1972. https://doi.org/10.1143/JPSJ.32.1403
5. Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Physical Review Letters, 27(18), 1192, 1971. https://doi.org/10.1103/PhysRevLett.27.1192
6. Wazzan, L., A modified tanh-coth method for solving the KdV and the KdV-Burgers' equations, Communications in Nonlinear Science and Numerical Simulation, 14(2), 443-450, 2009. https://doi.org/10.1016/j.cnsns.2007.06.011
7. Wang, M., Li, X., & Zhang, J., The (G'/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372(4), 417-423, 2008. https://doi.org/10.1016/j.physleta.2007.07.051
8. Zheng-De, D., Zhen-Jiang, L., & Dong-Long, L., Exact periodic solitary-wave solution for KdV equation, Chinese Physics Letters, 25(5), 1531, 2008. https://doi.org/10.1088/0256-307X/25/5/003
9. Ma, W. X., & Zhou, Y., Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. Journal of Differential Equations, 264(4), 2633-2659, 2018. https://doi.org/10.1016/j.jde.2017.10.033
10. Korkmaz, A., Explicit exact solutions to some one-dimensional conformable time fractional equations, Waves in Random and Complex Media, 29(1), 124-137, 2019. https://doi.org/10.1080/17455030.2017.1416702
11. Chen, S. T., & Ma, W. X., Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation, Frontiers of Mathematics in China, 1-10, 2018. https://doi.org/10.1007/s11464-018-0694-z
12. Korkmaz, A., Complex wave solutions to mathematical biology models I: Newell-Whitehead-Segel and Zeldovich equations, Journal of Computational and Nonlinear Dynamics, 13(8), 081004, 2018. https://doi.org/10.1115/1.4040411
13. Chen, S. T., & Ma, W. X., Lump solutions of a generalized Calogero-Bogoyavlenskii-Schiff equation, Computers & Mathematics with Applications, 76(7), 1680-1685, 2018. https://doi.org/10.1016/j.camwa.2018.07.019
14. Rezazadeh, H., Korkmaz, A., Eslami, M., Vahidi, J., & Asghari, R., Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method, Optical and Quantum Electronics, 50(3), 150, 2018. https://doi.org/10.1007/s11082-018-1416-1
15. Yong, X., Ma, W. X., Huang, Y., & Liu, Y., Lump solutions to the Kadomtsev-Petviashvili I equation with a selfconsistent source, Computers & Mathematics with Applications, 75(9), 3414-3419, 2018. https://doi.org/10.1016/j.camwa.2018.02.007
16. Osman, M. S., Korkmaz, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., & Zhou, Q., The unified method for conformable time fractional Schr¨odinger equation with perturbation terms, Chinese Journal of Physics, 56(5), 2500-2506, 2018. https://doi.org/10.1016/j.cjph.2018.06.009
17. Ma, W. X., Yong, X., & Zhang, H. Q., Diversity of interaction solutions to the (2+ 1)-dimensional Ito equation, Computers & Mathematics with Applications, 75(1), 289-295, 2018. https://doi.org/10.1016/j.camwa.2017.09.013
18. Yang, J. Y., Ma, W. X., & Qin, Z. , Lump and lump-soliton solutions to the (2+1)-dimensional Ito equation, Analysis and Mathematical Physics, 8(3), 427-436, 2018. https://doi.org/10.1007/s13324-017-0181-9
19. Yang, J. Y., Ma, W. X., & Qin, Z. Y., Abundant mixed lump-soliton solutions to the BKP equation, East Asian J. Appl. Math, 8(2), 224-232, 2018. https://doi.org/10.4208/eajam.210917.051217a
20. Ma, W. X., Abundant lumps and their interaction solutions of (3+ 1)-dimensional linear PDEs, Journal of Geometry and Physics, 133, 10-16, 2018. https://doi.org/10.1016/j.geomphys.2018.07.003
21. L¨u, D., Cui, Y., L¨u, C., & Wei, C., Novel composite function solutions of the modified KdV equation, Applied Mathematics and Computation, 217(1), 283-288, 2010. https://doi.org/10.1016/j.amc.2010.05.059
22. Wazwaz, A. M., A sine-cosine method for handling nonlinear wave equations, Mathematical and Computer modeling, 40(5-6), 499-508, 2004. https://doi.org/10.1016/j.mcm.2003.12.010
23. He, J. H., & Wu, X. H., Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30(3), 700-708, 2006. https://doi.org/10.1016/j.chaos.2006.03.020
24. Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, 2014. https://doi.org/10.1016/j.cam.2014.01.002
25. Korkmaz, A., & Hosseini, K., Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods, Optical and Quantum Electronics, 49(8), 278, 2017. https://doi.org/10.1007/s11082-017-1116-2
26. Hosseini, K., & Ansari, R., New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Waves in Random and Complex Media, 1-9, 2017. https://doi.org/10.1080/17455030.2017.1296983
27. Korkmaz A., On The Wave Solutions of Conformable Fractional Evolution Equations, Commun. Fac. Sci. Univ. Ank. Series A1, 67(1) 68-79, 2018. https://doi.org/10.1501/Commua1_0000000831
28. Korkmaz A., Exact Solutions to (3 + 1) Conformable Time Fractional Jimbo-Miwa,Zakharov-Kuznetsov and Modified Zakharov-Kuznetsov Equations, Communications in Theoretical Physics, 67(5), 479-482, 2017. https://doi.org/10.1088/0253-6102/67/5/479
29. Hosseini, K., Mayeli, P., & Ansari, R., Bright and singular soliton solutions of the conformable time-fractional KleinGordon equations with different nonlinearities, Waves in Random and Complex Media, 1-9, 2017. https://doi.org/10.1080/17455030.2017.1362133
30. Hosseini, K., Mayeli, P., & Ansari, R., Modified Kudryashov method for solving the conformable time-fractional KleinGordon equations with quadratic and cubic nonlinearities, Optik-International Journal for Light and Electron Optics, 130, 737-742, 2017. https://doi.org/10.1016/j.ijleo.2016.10.136
31. Korkmaz, A., Exact solutions of space-time fractional EW and modified EW equations, Chaos, Solitons & Fractals, 96, 132-138, 2017. https://doi.org/10.1016/j.chaos.2017.01.015
32. Zafar, A., Rational exponential solutions of conformable space-time fractional equal-width equations, Nonlinear Engineering, 2018. (in press) https://doi.org/10.1515/nleng-2018-0076
33. Atangana, A., Baleanu, D., & Alsaedi, A., New properties of conformable derivative, Open Mathematics, 13(1), 1-10, 2015. https://doi.org/10.1515/math-2015-0081
34. Abdeljawad, T., On conformable fractional calculus, Journal of computational and Applied Mathematics, 279, 57-66, 2015. https://doi.org/10.1016/j.cam.2014.10.016
35. Yan, C., A simple transformation for nonlinear waves, Physics Letters A, 224, 77-84, 1996. https://doi.org/10.1016/S0375-9601(96)00770-0
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