Research Article
BibTex RIS Cite

Year 2022, Volume: 43 Issue: 4, 703 - 707, 27.12.2022

Yusuf Gürefe Yusuf Pandır Tolga Aktürk

https://doi.org/10.17776/csj.1083033
Cited By: 1

Abstract

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.
  • [11] Gurefe N., Kocer E.G., Gurefe Y., Chebyshev-Tau Method for the Linear Klein-Gordon Equation, Int. J. Phys. Sci., 7 (43) (2012) 5723-5728.
  • [12] Shen G., Sun Y., Xiong Y., New Travelling-Wave Solutions for Dodd-Bullough Equation, J. Appl. Math., 2013 (2013) 1-6.
  • [13] Sun Y., New Travelling Wave Solutions for Sine-Gordon Equation, J. Appl. Math., 2014 (2014) 1-5.
  • [14] Bulut H., Akturk T., Gurefe Y., Travelling Wave Solutions of the (N+1)-dimensional Sine-Cosine-Gordon Equation, AIP Conf. Proc., 1637 (2014) 145-149.
  • [15] Bulut H., Akturk T., Gurefe Y., An Application of the New Function Method to the Generalized Double Sinh-Gordon Equation, AIP Conf. Proc., 1648 (2015) 370014.
  • [16] Akturk T., Gurefe Y., Bulut H., New Function Method to the (n+1)-dimensional Nonlinear Problems, Int. J. Optim. Control: Theor. Appl., 7 (3) (2017) 234-239.
  • [17] Akturk T., Gurefe Y., Pandır Y., An Application of the New Function Method to the Zhiber-Shabat Equation, Int. J. Optim. Control: Theor. Appl., 7 (3) (2017) 271-274.
  • [18] Gurefe Y., Akturk T., Pandir Y., An Application of the New Function Method for a Coupled Sine-Gordon Equation, Adv. Differ. Equ. Control Process., 19 (3) (2018) 287-294.
  • [19] Wazwaz A.M., The tanh and the Sine-Cosine Methods for Compact and Noncompact Solutions of the Nonlinear Klein-Gordon Equation, Appl. Math. Comput., 167(2) (2005) 1179-1195.
  • [20]Duncan D.B., Symplectic Finite Difference Approximations of the Nonlinear Klein-Gordon Equation, SIAM J. Numer. Anal., 34 (5) (1997) 1742-1760.

Analysis of Exact Solutions of a Mathematical Model by New Function Method

Year 2022, Volume: 43 Issue: 4, 703 - 707, 27.12.2022

Yusuf Gürefe Yusuf Pandır Tolga Aktürk

https://doi.org/10.17776/csj.1083033
Cited By: 1

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.
  • [11] Gurefe N., Kocer E.G., Gurefe Y., Chebyshev-Tau Method for the Linear Klein-Gordon Equation, Int. J. Phys. Sci., 7 (43) (2012) 5723-5728.
  • [12] Shen G., Sun Y., Xiong Y., New Travelling-Wave Solutions for Dodd-Bullough Equation, J. Appl. Math., 2013 (2013) 1-6.
  • [13] Sun Y., New Travelling Wave Solutions for Sine-Gordon Equation, J. Appl. Math., 2014 (2014) 1-5.
  • [14] Bulut H., Akturk T., Gurefe Y., Travelling Wave Solutions of the (N+1)-dimensional Sine-Cosine-Gordon Equation, AIP Conf. Proc., 1637 (2014) 145-149.
  • [15] Bulut H., Akturk T., Gurefe Y., An Application of the New Function Method to the Generalized Double Sinh-Gordon Equation, AIP Conf. Proc., 1648 (2015) 370014.
  • [16] Akturk T., Gurefe Y., Bulut H., New Function Method to the (n+1)-dimensional Nonlinear Problems, Int. J. Optim. Control: Theor. Appl., 7 (3) (2017) 234-239.
  • [17] Akturk T., Gurefe Y., Pandır Y., An Application of the New Function Method to the Zhiber-Shabat Equation, Int. J. Optim. Control: Theor. Appl., 7 (3) (2017) 271-274.
  • [18] Gurefe Y., Akturk T., Pandir Y., An Application of the New Function Method for a Coupled Sine-Gordon Equation, Adv. Differ. Equ. Control Process., 19 (3) (2018) 287-294.
  • [19] Wazwaz A.M., The tanh and the Sine-Cosine Methods for Compact and Noncompact Solutions of the Nonlinear Klein-Gordon Equation, Appl. Math. Comput., 167(2) (2005) 1179-1195.
  • [20]Duncan D.B., Symplectic Finite Difference Approximations of the Nonlinear Klein-Gordon Equation, SIAM J. Numer. Anal., 34 (5) (1997) 1742-1760.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Yusuf Gürefe 0000-0002-7210-5683

Yusuf Pandır 0000-0003-0274-7901

Tolga Aktürk 0000-0002-8873-0424

Publication Date December 27, 2022
Submission Date March 4, 2022
Acceptance Date December 9, 2022
Published in Issue Year 2022Volume: 43 Issue: 4

Cite

APA Gürefe, Y., Pandır, Y., & Aktürk, T. (2022). Analysis of Exact Solutions of a Mathematical Model by New Function Method. Cumhuriyet Science Journal, 43(4), 703-707. https://doi.org/10.17776/csj.1083033

Cited By

New solutions of time-space fractional coupled Schrödinger systems
AIMS Mathematics
https://doi.org/10.3934/math.20231383
 Download Cover Image