(2+1)-boyutlu Broer-Kaup-Kupershmidt denklemi ve Kolmogorov-Petrovskii-Piskunov denklemine modifiye edilmiş deneme denklem metodu

Yıl 2021, Cilt: 23 Sayı: 2, 673 - 684, 04.07.2021

Şeyma Tülüce Demiray Serife Duman

https://doi.org/10.25092/baunfbed.853756

Cited By: 1

Öz

Lineer olmayan problemlerin çözümünü bulmak için bilim insanları tarafından birçok yöntem geliştirilmiştir. Bu yazıda, modifiye edilmiş deneme denklem metodunun (MEDDM) genel yapısı tanıtılmış ve (2+1)-boyutlu Broer-Kaup-Kupershmidt (BKK), Kolmogorov-Petrovskii-Piskunov (KPP) denklemlerinin bazı tam çözümlerini bulmak için MEDDM kullanılmıştır. İlk olarak, hareketli dalga dönüşümü altında lineer olmayan kısmi diferansiyel denklemin (NLPDE) adi diferansiyel denkleme indirgenmesiyle bir cebirsel denklem sistemi elde edilmiştir. Elde edilen cebirsel denklem sistemleri çözülerek hareketli dalga çözümleri bulunur. Mathematica 9 programı kullanılarak, dalga çözümlerinin fiziksel davranışını analiz etmek için uygun parametreler için üç ve iki boyutlu grafikler çizilmiştir. MEDDM, bazı kısmi diferansiyel denklemlerin tam çözümlerini bulmada büyük önem taşımaktadır.

Anahtar Kelimeler

(2+1)-boyutlu BKK denklemi, KPP denklemi, hareketli dalga çözümü.

The modified trial equation method to the (2+1)-dimensional Broer-Kaup-Kupershmidt equation and Kolmogorov-Petrovskii-Piskunov equation

Öz

Many methods have been developed by scientists to find solutions for nonlinear problems. In this paper, the general structure of the modified trial equation method (MTEM) is introduced, and MTEM is used to find some exact solutions of (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK), Kolmogorov-Petrovskii-Piskunov (KPP) equations. Firstly, an algebraic equation system is obtained by reducing the nonlinear partial differential equation (NLPDE) to the ordinary differential equation under the travelling wave transformation. Travelling wave solutions are found by solving the obtained algebraic equation systems. By using Mathematica 9 program, three and two dimensional graphs for suitable parameters were plotted to analyze the physical behavior of wave solutions. MTEM is of great importance in finding exact solutions of some partial differential equations.

Anahtar Kelimeler

(2+1)-dimensional BKK equation, KPP equation, travelling wave solution.

Kaynakça

• Evirgen, F., Yavuz, M., An Alternative Approach for Nonlinear Optimization Problem with Caputo – Fabrizio Derivative, In ITM Web of Conferences, 22, EDP Sciences, (2018).
• Sarp, U., Evirgen, F., Ikikardes, S., Applications of differential transformation method to solve systems of ordinary and partial differential equations, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20, 2, 135-156, (2018).
• Yavuz, M., Sene, N., Approximate solutions of the model describing fluid flow using generalized ρ-Laplace transform method and heat balance integral method, Axioms, 9, 4, 123, (2020).
• Yavuz, M., European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels, Numerical Methods for Partial Differential Equations, 1-23, (2020).
• Yavuz, M., Abdeljawad, T., Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel, Advances in Difference Equations, 2020, 367, 2-18, (2020).
• Song, M., Li, S., Cao, J., New Exact Solutions for the (2+1)-Dimensional Broer-Kaup-Kupershmidt Equations, Abstract and Applied Analysis, 2010, 9, (2010).
• Gurefe, Y., Sonmezoglu, A., Misirli, E., Application of an Irrational Trial Equation Method to High-Dimensional Nonlinear Evolution Equations, Journal of Advanced Mathematical Studies, 5, 2, 41-47, (2012).
• Rouhparvar, H., Travelling Wave Solution of the Kolmogorov-Petrovskii-Piskunov Equation by the First Integral Method, Malaysian Mathematical Sciences Society, 37, 1, 181-190, (2014).
• Feng, J., Li, W., Wan Q., Using (G'/G)-expansion method to seek the traveling wave solution of Kolmogorov-Petrovskii-Piskunov Equation, Applied Mathematics and Computation, 217, 12, 5860-5865, (2011).
• Ma, W. X., Fuchssteiner, B., Explicit and Exact Solutions to a Kolmogorov-Petrovskii-Piskunov Equation, International Journal of Non-Linear Mechanics, 31, 3, 329-338, (1996).
• Hirota, R., The Direct Method in Soliton Theory, Cambridge University Press, 252-253, Cambridge, (2004).
• W. X. Ma, Tiecheng Xia, Pfaffianized systems for a generalized Kadomtsev Petviashvili Equation, Physica Scripta, 87, 5, 8, (2013).
• Liu, G. T., Fan, T. Y., New applications of developed Jacobi elliptic function expansion methods, Physics Letters A, 345, 1-3, 161–166, (2005).
• Wazwaz, A. M., The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Applied Mathematics and Computation, 167, 2, 1196-1210, (2005).
• Zarea, S. A., The tanh method A tool for solving some mathematical models, Chaos, Solitons & Fractals, 41, 2, 979-988, (2009).
• Malfliet, W., Hereman, W., The tanh method Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54, 6, 563-568, (1996).
• Irshad, A., Mohyud-Din, S.T., Ahmed, N., Khan, U., A New Modification in Simple Equation Method and its applications on nonlinear equations of physical nature, Results in Physics, 7, 2017, 4232-4240, (2017).
• Tuluce Demiray, S., Pandir, Y., Bulut, H., Generalized Kudryashov Method for Time-Fractional Differential Equation, Abstract and Applied Analysis, 6, 1-13, (2014).
• Mahmud, F., Samsuzzoha, Md., Akbar, M.A., The Generalized Kudryashov Method to obtain exact traveling wave solutions of the PHI-four Equation and the Fisher Equation, Results in Physics, 7, 4296-4302, (2017).
• Tuluce Demiray, S., Bulut, H., Generalized Kudryashov method for nonlinear fractional double sinh–Poisson Equation, Journal of Nonlinear Science and Applications, 9, 3, 1349-1355, (2016).
• Habib, M. A., Shahadat Ali, H.M., Miah, M., Akbar, M.A., The generalized Kudryashov method for new closed form traveling wave solutions to some NLEEs, Aims Mathematics, 4, 3, 896–909, (2019).
• Gurefe, Y., Misirli, E., Exp-function method for solving nonlinear evolution equations with higher order nonlinearity, Computers and Mathematics with Applications, 61, 8, 2025-2030, (2011).
• Naher, H., Abdullah , F.A., Akbar, M. A., The Exp-function Method for new exact solutions of the nonlinear partial differantial equations, International Journal of the Physical Sciences, 6, 29, 6706-6716, (2011).
• Fan, E., Hongqing, Z., A note on the homogenous balance method, Physics Letters, 246, 5, 403-406, (1998).
• Abdelsalam, U. M., Ghazal, M. G. M., Analytical Wave Solutions for Foam and KdV-Burgers Equations Using Extended Homogeneous Balance Method, Mathematics, 7, 8, 1-12, (2019).
• Injrou, S., New exact Solutions for Generalized Fitzhug-Nagumo Equation by Homogeneous balance Method, Journal for Research and Scientific Studies, 37, 4, 57-65, (2015).
• Bulut, H., Pandir, Y., Baskonus, H.M., The Modified Trial Equation Method for Fractional Wave Equation and Time Fractional Generalized Burgers Equation, Abstract and Applied Analysis, 2013, 8, (2013).