Analysis of Exact Solutions of a Mathematical Model by New Function Method
Abstract
In this article, the new function method is used to obtain the wave solutions of the nonlinear Klein-Gordon equation. Since the Klein-Gordon equation is a nonlinear partial differential equation containing exponential functions, it was decided to apply the new function method, which was defined on the assumption of a nonlinear auxiliary differential equation containing exponential functions. Thus, it aims to reach wave solutions not found in the literature. The considered method can be easily applied to this type of nonlinear problem that is difficult to solve and gives us solutions. Here, two new exact solutions are obtained. Then two and three-dimensional density and contour graphs are drawn by selecting the appropriate parameters to analyze the physical behavior of these solutions. The Mathematica package program was effectively used in all calculations and graphic drawings.
Keywords
New function method, Klein-Gordon Equation with an exponential nonlinearity, Mathematical model
References
- [1] Liu C.S., Trial Equation Method and Its Applications to Nonlinear Evolution Equations, Acta. Phys. Sin., 54 (2005) 2505-2509.
- [2] Kudryashov N.A., One Method for Finding Exact Solutions of Nonlinear Differential Equations, Commun. Nonl. Sci. Numer. Simul., 17 (2012) 2248-2253.
- [3] Lu B., The First Integral Method for Some Time Fractional Differential Equations, J. Math. Anal. Appl., 395 (2) (2012) 684-693.
- [4] Gurefe Y., Misirli E., Sonmezoglu A., Ekici M., Extended Trial Equation Method to Generalized Nonlinear Partial Differential Equations, Appl. Math. Comput., 219 (10) (2013) 5253-5260.
- [5] Hussain A., Jabeen F., Abbas N., Optical Soliton Solutions of Multi-Dimensional Boiti-Leon-Manna-Pempinelli Equations, Mod. Phys. Lett. B, 36(10) (2021) 2250035.
- [6] Aktürk, T., Modified Exponential Function Method for Nonlinear Mathematical Models with Atangana Conformable Derivative, Rev. Mex. Fis., 67(4) (2021) 1-18.
- [7] Sherriffe D., Behera D., Analytical Approach for Travelling Wave Solution of Non-Linear Fifth-Order Time-Fractional Korteweg-De Vries Equation, Pramana-J. Phys., 96(64) (2022) 1-8.
- [8] Zaman U.H.M., Arefin M.A., Akbar M.A., Uddin M.H., Analytical Behavior of Soliton Solutions to the Couple Type Fractional-Order Nonlinear Evolution Equations Utilizing a Novel Technique, Alex. Eng. J., 61(12) (2022) 11947-11958.
- [9] He J.H., Some Asymptotic Methods for Strongly Nonlinear Equations, Int. J. Modern Phys. B, 20 (10) (2006) 1141-1199.
- [10] Alkarawi A.H., Al-Saiq I.R., Adomian Decomposition Method Applied to Klein Gordon and Nonlinear Wave Equation, J. Interdiscip. Math., 24 (5) (2021) 1149-1157.