Vakhnenko-Parkes Denkleminin Hiperbolik Tipte Yürüyen Dalga Çözümü

Abstract

Bu makalede, (1/𝐺′)-açılım metodunun yapısı uygulanmıştır. İndirgenmiş Ostrovsky denkleminin bir diğer adı olan Vakhnenko-Parkes (V-P) denklemi dikkate alınmış ve (V-P) denkleminin (1/𝐺′)-açılım metodunu kullanılarak tam çözümleri inşa edilmiştir. Bu yöntem lineer olmayan kısmi diferansiyel denklemlerin analitik çözümlerini bulmak için daha kolay ve etkili bir metottur. Metot sembolik hesaplama için daha kolay ve daha hızlı görünüyor.Bu makalede, (1/𝐺′)-açılım metodunun yapısı uygulanmıştır. İndirgenmiş Ostrovsky denkleminin bir diğer adı olan Vakhnenko-Parkes (V-P) denklemi dikkate alınmış ve (V-P) denkleminin (1/𝐺′)-açılım metodunu kullanılarak tam çözümleri inşa edilmiştir. Bu yöntem lineer olmayan kısmi diferansiyel denklemlerin analitik çözümlerini bulmak için daha kolay ve etkili bir metottur. Metot sembolik hesaplama için daha kolay ve daha hızlı görünüyor.

Keywords

• (1/𝐺′)-açılım metodu,
• Vakhnenko-Parkes denklemi,
• tam çözüm, yürüyen dalga çözümü.

Traveling Wave Solution of Vakhnenko-Parkes Equation

Abstract

In this article, a structure of (1/G')-expansion method is proposed. Another name of the reduced Ostrovsky equation, Vakhnenko-Parkes (V-P) equation is taken into consideration and exact solutions have been constructed of the (V-P) equation using (1/G')-expansion method. This method is easier and efficient method for finding analytic solutions of nPDEs. The method appears to be easier and faster for symbolic computation.In this article, a structure of (1/G')-expansion method is proposed. Another name of the reduced Ostrovsky equation, Vakhnenko-Parkes (V-P) equation is taken into consideration and exact solutions have been constructed of the (V-P) equation using (1/G')-expansion method. This method is easier and efficient method for finding analytic solutions of nPDEs. The method appears to be easier and faster for symbolic computation.

Keywords

• (1/G')-expansion method
• the Vakhnenko-Parkes equation
• exact solution
• traveling wave solution.

References

1. Manafian, J. (2018). “Novel solitary wave solutions for the (3+ 1)-dimensional extended Jimbo–Miwa equations”. Computers & Mathematics with Applications, 76(5), 1246-1260.
2. Wazwaz, A. M. (2007). “The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations”. Applied Mathematics and Computation, 188(2), 1467-1475.
3. Yokuş, A., & Kaya, D. (2015). “Traveling wave solutions of some nonlinear partial differential equations by using extended-expansion method”.
4. Yokus, A., & Tuz, M. (2017). “An application of a new version of (G′/G)-expansion method”. In AIP Conference Proceedings (Vol. 1798, No. 1, p. 020165). AIP Publishing.
5. Durur, H. (2019). “Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method”. Modern Physics Letters B, 2050036.
6. Yavuz, M., & Özdemır, N. (2018). “An Integral Transform Solution for Fractional Advection-Diffusion Problem”. Mathematical Studies and Applications, 442.
7. Baskonus, H. M., Sulaiman, T. A., Bulut, H., & Aktürk, T. (2018). “Investigations of dark, bright, combined dark-bright optical and other soliton solutions in the complex cubic nonlinear Schrödinger equation with δ-potential”. Superlattices and Microstructures, 115, 19-29.
8. Cattani, C., Sulaiman, T. A., Baskonus, H. M., & Bulut, H. (2018). “On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems”. Optical and Quantum Electronics, 50(3), 138.

9. Durur, H., Taşbozan, O., Kurt, A., & Şenol, M. (2019). “New Wave Solutions of Time Fractional Kadomtsev-Petviashvili Equation Arising In the Evolution of Nonlinear Long Waves of Small Amplitude”. Erzincan University Journal of the Institute of Science and Technology, 12(2), 807-815.
10. Su-Ping, Q., & Li-Xin, T. (2007). “Modification of the Clarkson–Kruskal Direct Method for a Coupled System”. Chinese Physics Letters, 24(10), 2720.
11. Yokuş, A., & Durur, H. (2019). “Complex hyperbolic traveling wave solutions of Kuramoto-Sivashinsky equation using (1/G') expansion method for nonlinear dynamic theory”. Journal of Balıkesir University Institute of Science and Technology, 21(2), 590-599.
12. Yokuş, A., & Kaya, D. (2015). “Conservation laws and a new expansion method for sixth order Boussinesq equation”. In AIP Conference Proceedings (Vol. 1676, No. 1, p. 020062). AIP Publishing.
13. Durur, H., & Yokuş, A. (2019). “(1/G')-Açılım Metodunu Kullanarak Sawada–Kotera Denkleminin Hiperbolik Yürüyen Dalga Çözümleri”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 19(3), 615-619.
14. Kumar, D., Seadawy, A. R., & Joardar, A. K. (2018). “Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology”. Chinese journal of physics, 56(1), 75-85.
15. Kaya, D., & Yokus, A. (2002). “A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations”. Mathematics and Computers in Simulation, 60(6), 507-512.
16. Kaya, D., & Yokus, A. (2005). “A decomposition method for finding solitary and periodic solutions for a coupled higher-dimensional Burgers equations”. Applied Mathematics and Computation, 164(3), 857-864.
17. Yavuz, M., & Özdemir, N. (2018). “A quantitative approach to fractional option pricing problems with decomposition series”. Konuralp Journal of Mathematics, 6(1), 102-109.
18. Darvishi, M., Arbabi, S., Najafi, M., & Wazwaz, A. (2016). “Traveling wave solutions of a (2+1)-dimensional Zakharov-like equation by the first integral method and the tanh method”. Optik, 127(16), 6312-6321.
19. Aziz, I., & Šarler, B. (2010). “The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets”. Mathematical and Computer Modelling, 52(9-10), 1577-1590.
20. Kurt, A., Tasbozan, O., & Durur, H. (2019). “The Exact Solutions of Conformable Fractional Partial Differential Equations Using New Sub Equation Method”. Fundamental Journal of Mathematics and Applications, 2(2), 173-179.
21. Baskonus, H. M., & Bulut, H. (2016). “Exponential prototype structures for (2+ 1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics”. Waves in Random and Complex Media, 26(2), 189-196.
22. Dusunceli, F. (2019). “New Exact Solutions for Generalized (3+ 1) Shallow Water-Like (SWL) Equation”. Applied Mathematics and Nonlinear Sciences, 4(2), 365-370.
23. Durur, H., Şenol, M., Kurt, A., & Taşbozan, O. (2019). “Zaman-Kesirli Kadomtsev-Petviashvili Denkleminin Conformable Türev ile Yaklaşık Çözümleri”. Erzincan University Journal of the Institute of Science and Technology, 12(2), 796-806.
24. Yokus, A., Baskonus, H. M., Sulaiman, T. A., & Bulut, H. (2018). “Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem”. Numerical Methods for Partial Differential Equations, 34(1), 211-227.
25. Vakhnenko, V. O., & Parkes, E. J. (1998). “The two loop soliton solution of the Vakhnenko equation”. Nonlinearity, 11(6), 1457.
26. Abazari, R. (2010). “Application of G′ G-expansion method to travelling wave solutions of three nonlinear evolution equation”. Computers & Fluids, 39(10), 1957-1963.
27. Vakhnenko, V. O., & Parkes, E. J. (2012). “Solutions associated with discrete and continuous spectrums in the inverse scattering method for the Vakhnenko-Parkes equation”. Progress of Theoretical Physics, 127(4), 593-613.
28. Roshid, H. O., Kabir, M. R., Bhowmik, R. C., & Datta, B. K. (2014). “Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp (− ϕ (ξ))-expansion method”. SpringerPlus, 3(1), 692.
29. Majid, F., Triki, H., Hayat, T., Aldossary, O. M., & Biswas, A. (2012). “Solitary wave solutions of the Vakhnenko-Parkes equation”. Nonlinear Analysis: Modelling and Control, 17(1), 60-66. Volterra V. (1959). “Theory of Functionals and of Integral and Integro-differential Equations”, Dover Publications, New York.
30. Ye, Y., Song, J., Shen, S., & Di, Y. (2012). “New coherent structures of the Vakhnenko–Parkes equation”. Results in Physics, 2, 170-174.
31. Gu, Y., Yuan, W., Aminakbari, N., & Jiang, Q. (2017). “Exact solutions of the Vakhnenko-Parkes equation with complex method”. Journal of Function Spaces.
32. Baskonus, H. M., Bulut, H., & Emir, D. G. (2015). “Regarding New Complex Analytical Solutions for the Nonlinear Partial Vakhnenko-Parkes Differential Equation via Bernoulli Sub-Equation Function Method”. Mathematics Letters, 1, 1-9.
33. Liu, X., & He, C. (2013). “New Traveling Wave Solutions to the Vakhnenko-Parkes Equation”. ISRN Mathematical Physics.
34. Faraj, B., & Modanli, M. (2017). “Using Difference Scheme Method for the Numerical Solution of Telegraph Partial Differential Equation”.