(1/G')-Expansion Method for Exact Solutions of (3+1)-Dimensional Jimbo-Miwa Equation

Abstract

The purpose of this article is obtaining the exact solutions for (3+1)-dimensional Jimbo-Miwa Equation (3+1DJME). The (1/G')-expansion method which is an effective method in solving nonlinear evolution equations (NLEEs) is used. Then, 3D, contour and 2D graphics are presented by giving special values to the constants in the solutions obtained. These graphics are a special solution of the (3+1DJME) and represent a stationary wave of the equation. Ready computer package program is used to obtain the solutions and graphics presented in this study.

Keywords

• (1/G')-expansion method
• (3+1)-dimensional Jimbo-Miwa equation
• exact solution
• traveling wave solution

(3 + 1) Boyutlu Jimbo-Miwa Denkleminin Tam Çözümleri için (1/G')-Açılım Yöntemi

Abstract

Bu makalenin amacı (3+1) boyutlu Jimbo-Miwa denklemi için tam çözümler elde etmektir. Lineer olmayan evrim denklemlerinin çözümünde etkili bir yöntem olan (1/G')-açılım yöntemi kullanılmıştır. Daha sonra elde edilen çözümlerdeki sabitlere özel degerler verilerek 3 boyutlu, kontur ve 2 boyutlu grafikler sunulmuştur. Bu grafikler (3 + 1) boyutlu Jimbo-Miwa denkleminin özel bir çözümü olup ve denklemin duragan bir dalgasını temsil etmektedir. Bu çalışmada sunulan çözümler ve grafiklerin elde edilişinde hazır bilgisayar paket programı kullanılmaktadır.

Keywords

• (1/G')-açılım yöntemi
• (3+1)-boyutlu Jimbo-Miwa denklemi
• tam çözüm
• yürüyen dalga çözümü

Supporting Institution

Yok

References

1. 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.
2. 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.
3. Dusunceli, F., Celik, E., Askin, M., & Bulut, H. (2020). New exact solutions for the doubly dispersive equation using the improved Bernoulli sub-equation function method. Indian Journal of Physics, 1-6.
4. 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.
5. 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.
6. Durur H, 2020. Different types analytic solutions of the (1+ 1)-dimensional resonant nonlinear Schrödinger’s equation using (G′/G)-expansion method. Modern Physics Letters B, 34(03), 2050036.
7. 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.
8. Durur H, Taşbozan O, Kurt A, Şenol M, 2019a. 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.

9. Durur H, Şenol M, Kurt A, Taşbozan O, 2019b. Zaman-Kesirli Kadomtsev-Petviashvili Denkleminin Conformable Türev ile Yaklaşık Çözümleri. Erzincan University Journal of the Institute of Science and Technology, 12(2), 796-806.
10. Dusunceli F, 2019. New Exact Solutions for Generalized (3+1) Shallow Water-Like (SWL) Equation. Applied Mathematics and Nonlinear Sciences, 4(2), 365-370.
11. Faraj B, Modanli M, 2017. Using Difference Scheme Method for the Numerical Solution of Telegraph Partial Differential Equation.
12. Jimbo M, Miwa T, 1983. Solitons and infinite dimensional Lie algebras. Publications of the Research Institute for Mathematical Sciences, 19(3), 943-1001.
13. 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.
14. 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.
15. 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.
16. 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.
17. Liu X Q, Jiang S, 2004. New solutions of the 3+ 1 dimensional Jimbo–Miwa equation. Applied mathematics and computation, 158(1), 177-184.
18. Ma W X, 2016. Lump-type solutions to the (3+ 1)-dimensional Jimbo-Miwa equation. International Journal of Nonlinear Sciences and Numerical Simulation, 17(7-8), 355-359.
19. Manafian, J, 2018. Novel solitary wave solutions for the (3+1)-dimensional extended Jimbo–Miwa equations. Computers & Mathematics with Applications, 76(5), 1246-1260.
20. Öziş T, Aslan İ, 2008. Exact and explicit solutions to the (3+ 1)-dimensional Jimbo–Miwa equation via the Exp-function method. Physics Letters A, 372(47), 7011-7015.
21. Rezazadeh H, Tariq H, Eslami M, Mirzazadeh M, Zhou Q, 2018. New exact solutions of nonlinear conformable time-fractional Phi-4 equation. Chinese Journal of Physics, 56(6), 2805-2816.
22. Siddique I, Rizvi S T R, Batool F, 2010. New exact travelling wave solutions of nonlinear evolution equations. International Journal of Nonlinear Science, 9(1), 12-18.
23. Su-Ping Q, Li-Xin T, 2007. Modification of the Clarkson–Kruskal Direct Method for a Coupled System. Chinese Physics Letters, 24(10), 2720.
24. Tang X Y, Liang Z F, 2006. Variable separation solutions for the (3+ 1)-dimensional Jimbo–Miwa equation. Physics Letters A, 351(6), 398-402.
25. 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.
26. Wazwaz A M, 2017. Multiple-soliton solutions for extended (3+1)-dimensional Jimbo–Miwa equations. Applied Mathematics Letters, 64, 21-26.
27. Yang J Y, Ma W X, 2017. Abundant lump-type solutions of the Jimbo–Miwa equation in (3+ 1)-dimensions. Computers & Mathematics with Applications, 73(2), 220-225.
28. Yavuz M, Özdemır N, 2018. An Integral Transform Solution for Fractional Advection-Diffusion Problem. Mathematical Studies and Applications 2018 4-6 October 2018, 442.
29. Yavuz M, Özdemir N, 2018. A quantitative approach to fractional option pricing problems with decomposition series. Konuralp Journal of Mathematics, 6(1), 102-109.
30. Yokuş A, Kaya D, 2015. Traveling wave solutions of some nonlinear partial differential equations by using extended-expansion method.
31. Yokus A, Tuz M, 2017. An application of a new version of (G′/G)-expansion method. In AIP Conference Proceedings 1798(1), 020165.
32. 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.
33. 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.
34. Durur, H., & Yokuş, A. (2020). Analytical solutions of Kolmogorov–Petrovskii–Piskunov equation. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(2), 628-636.
35. Yokus, A., Durur, H., & Ahmad, H. (2020a). Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system. Facta Universitatis, Series: Mathematics and Informatics, 35(2), 523-531.
36. Yokus, A., Durur, H., Ahmad, H., & Yao, S. W. (2020b). Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation. Mathematics, 8(6), 908.
37. Ahmad, H., Khan, T. A., Durur, H., Ismail, G. M., & Yokus, A. (2020). Analytic approximate solutions of diffusion equations arising in oil pollution. Journal of Ocean Engineering and Science.
38. Yavuz, M. (2017). Novel solution methods for initial boundary value problems of fractional order with conformable differentiation. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1), 1-7.
39. Modanli, M. (2019). On the numerical solution for third order fractional partial differential equation by difference scheme method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3), 1-5.