Investigation of traveling wave solutions of nonlinear mathematical models by the modified exponential function method

Öz

In this work, traveling wave solutions of (1+1)-dimensional Landau-Ginzburg-Higgs and Duffing nonlinear partial differential equations, which are examples of mathematical modeling, are obtained and analyzed using the modified exponential function method. In order to facilitate the physical interpretation of the mathematical models represented by these equations, simulations of the behavior of the mathematical model as three-dimensional, contour, density and two-dimensional graphics are given using a package program with the help of appropriate parameters. It has been shown that the modified exponential function method effectively investigates the solutions of (1+1)-dimensional Landau-Ginzburg-Higgs and Duffing equations.

Anahtar Kelimeler

• Traveling wave solutions
• (1+1)-dimensional landau-ginzburg-higgs equation
• duffing equation
• the modified exponential function method

Lineer olmayan matematiksel modellerin hareketli dalga çözümlerinin genişletilmiş üstel fonksiyon metodu kullanılarak incelenmesi

Öz

Bu çalışmada, matematiksel modelleme örnekleri olan (1+1)-boyutlu Landau-Ginzburg-Higgs ve Duffing doğrusal olmayan kısmi diferansiyel denklemlerin yürüyen dalga çözümleri elde edilmiş ve genişletilmiş üstel fonksiyon metodu kullanılarak analiz edilmiştir. Bu denklemlerin temsil ettiği matematiksel modellerin fiziksel yorumunu kolaylaştırmak için uygun parametreler yardımıyla bir paket program kullanılarak matematiksel modelin davranışının üç boyutlu, kontur, yoğunluk ve iki boyutlu grafikleri olarak simülasyonları verilmiştir. Genişletilmiş üstel fonksiyon metodunun (1+1)-boyutlu Landau-Ginzburg-Higgs ve Duffing denklemlerinin çözümlerinin araştırılmasında etkili bir yöntem olduğu gösterilmiştir.

Anahtar Kelimeler

• Yürüyen dalga çözümleri
• (1+1)-ölçülü landau-ginzburg-higgs denklemi
• duffing denklemi
• genişletilmiş üstel fonksiyon metodu

Kaynakça

1. Liu, C. S., Trial equation method and its applications to nonlinear evolution equations, Acta Physica Sinica, 54, 6, 2505-2509, (2005).
2. Hammouch, Z., Yavuz, M. and Özdemir, N., Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1, 1, 11-23, (2021).
3. Hosseini, K. and Ansari, R., New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method, Waves in Random and Complex Media, 27, 4, 628-636, (2017).
4. Baskonus, H. M., Bulut, H. and Sulaiman, T.A., Investigation of various travelling wave solutions to the extended (2+ 1)-dimensional quantum ZK equation, The European Physical Journal Plus, 132, 11, 1-8, (2017).
5. Wazwaz, A.M., The tanh method for traveling wave solutions of nonlinear equations, Applied Mathematics and Computation, 154, 3, 713-723, (2004).
6. Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6, 2, 75-83, (2016).
7. He, J. H. and Wu, X. H., Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30, 3, 700-708, (2006).
8. Barman, H. K., Akbar, M. A., Osman, M. S., Nisar, K. S., Zakarya, M., Abdel-Aty, A. H. and Eleuch, H., Solutions to the Konopelchenko-Dubrovsky equation and the Landau-Ginzburg-Higgs equation via the generalized Kudryashov technique, Results in Physics, 24, 104092, (2021).

9. Eze, E. O., Obasi, U. E. and Agwu, E. U., Stability Analysis of Periodic Solutions of Some Duffing’s Equations, Open Journal of Applied Sciences, 9, 4, 198-214, (2019).
10. Naher, H. and Abdullah, F. A., New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation, AIP Advances, 3, 3, 032116, (2013).
11. Islam, M. E. and Akbar, M. A., Stable wave solutions to the Landau-Ginzburg-Higgs equation and the modified equal width wave equation using the IBSEF method, Arab Journal of Basic and Applied Sciences, 27, 1, 270-278, (2020).
12. Bekir, A. and Unsal, O., Exact solutions for a class of nonlinear wave equations by using first integral method, International Journal of Nonlinear Science, 15, 2, 99-110, (2013).
13. Evans, D. J. and Raslan, K. R., The tanh function method for solving some important non-linear partial differential equations, International Journal of Computer Mathematics, 82, 7, 897-905, (2005a).
14. Hu, W. P., Deng, Z. C., Han, S. M. and Fa, W., Multi-symplectic Runge-Kutta methods for Landau-Ginzburg-Higgs equation, Applied Mathematics and Mechanics, 30, 8, 1027-1034, (2009).
15. Iftikhar, A., Ghafoor, A., Zubair, T., Firdous, S. and Mohyud-Din, S. T., -expansion method for traveling wave solutions of (2+ 1) dimensional generalized KdV, Sin Gordon and Landau-Ginzburg-Higgs Equations, Scientific Research and Essays, 8, 28, 1349-1359, (2013).
16. Cevikel, A. C., Aksoy, E., Günerb, Ö. and Bekir, A., Dark-bright soliton solutions for some evolution equations, International Journal of Nonlinear Science, 16, 3, 195-202, (2013).
17. Salas, A. H., Exact solution to Duffing equation and the pendulum equation, Applied Mathematical Sciences, 8, 176, 8781-8789, (2014).
18. Al-Jawary, M. A. and Abd-Al-Razaq, S. G., Analytic and numerical solution for Duffing equations, International Journal of Basic and Applied Sciences, 5, 2, 115-119, (2016).
19. Akbar, M. A. and Ali, N. H. M., Exp-function method for Duffing equation and new solutions of (2+ 1) dimensional dispersive long wave equations, Progress in Applied Mathematics, 1, 2, 30-42, (2011).
20. Bülbül, B. and Sezer, M., Numerical solution of Duffing equation by using an improved Taylor matrix method, Journal of Applied Mathematics, (2013).
21. Marinca, V. and Herişanu, N., Explicit and exact solutions to cubic Duffing and double-well Duffing equations, Mathematical and Computer Modelling, 53, 5-6, 604-609, (2011).
22. Tabatabaei, K. and Gunerhan, E., Numerical solution of Duffing equation by the differential transform method, Applied Mathematics & Information Sciences Letters, 2, 1, 1-6, (2014).