Abstract
The Phi-4 equation plays an important role in mathematical physics and
it is particular form of the Klein-Gordon equation that models the phenomenon
in particle physics. This significant equation has been studied by many
researchers and many solutions to this equation have been obtained by using
different methods. In this study, the solutions obtained by three important
methods have been focused on: The modified simple equation method, the ansatz
method and He's variational method. Reconsidering the phi-4 equation, the same
solutions and new trigonometric, hyperbolic and elliptic function solutions
have been obtained by using the sn-ns method. The similarities and differences
of the obtained solutions have been compared with each other. In addition to
its easy applicability, the sn-ns method was shown to be highly effective and
reliable method.
Keywords
Ansatz method He's variational method Modified simple equation method Phi-four equation Sn-ns method
References
- Abramowitz, M. and Stegun, I. A., 1972. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover, 1046p.
- Akter, J. and Akbar, M. A., 2015. Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method. Results in Physics, 5, 125-130.
- Calogero, F. and Degasperis, A., 1982. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations. New York: North-Holland, 532p.
- Cao, J., Song, M. and Biswas, A., 2014. Topological Solitons and Bifurcation Analysis of the PHI-Four Equation. Bull. Malays. Math. Sci. Soc., 37, 1209-1219.
- Najafi, M., 2012. Using He's Variational Method to Seek the Traveling Wave Solution of PHI-Four Equation. International Journal of Applied Mathematical Research, 1, 659-665.
- Salas, H.A. and Castillo, J.E., 2011. New exact solutions to sinh-cosh-Gordon equation by using techniques based on projective Riccati equations. Computers and Mathematics with Applications, 61, 470-481.
- Salas H. A., 2011. Exact Solutions for the Ito Equation by the sn-ns Method. Applied Mathematical Sciences, 5, 2283-2287.
Abstract
Matematiksel fizikte önemli bir rol oynayan Phi-4
denklemi, bu olguyu parçacık fiziğinde modelleyen Klein-Gordon denkleminin özel
bir halidir. Bu önemli denklem birçok araştırmacı tarafından çalışılmış ve bu denklemin
birçok çözümü farklı yöntemler kullanılarak elde edilmiştir. Bu çalışmada, üç
önemli yöntemle elde edilen çözümlere odaklanıldı: Modifiye edilmiş basit
denklem yöntemi, ansatz yöntemi ve He'nin varyasyonel yöntemi. Phi-dört denklemi yeniden göz önüne alınarak,
daha önce elde edilmiş çözümlerin yanında yeni trigonometrik, hiperbolik ve
eliptik fonksiyon çözümleri sn-ns yöntemi kullanılarak elde edildi. Elde edilen
çözümlerin benzerlikleri ve farklılıkları birbirleriyle karşılaştırıldı. Kolay
uygulanabilirliğinin yanında, sn-ns metodun oldukça etkin ve güvenilir bir
yöntem olduğu gösterildi.
Keywords
Ansatz metodu He'nin varyasyonel metodu Modifiye edilmiş basit denklem metodu Phi-four denklemi Sn-ns metodu
References
- Abramowitz, M. and Stegun, I. A., 1972. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover, 1046p.
- Akter, J. and Akbar, M. A., 2015. Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method. Results in Physics, 5, 125-130.
- Calogero, F. and Degasperis, A., 1982. Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations. New York: North-Holland, 532p.
- Cao, J., Song, M. and Biswas, A., 2014. Topological Solitons and Bifurcation Analysis of the PHI-Four Equation. Bull. Malays. Math. Sci. Soc., 37, 1209-1219.
- Najafi, M., 2012. Using He's Variational Method to Seek the Traveling Wave Solution of PHI-Four Equation. International Journal of Applied Mathematical Research, 1, 659-665.
- Salas, H.A. and Castillo, J.E., 2011. New exact solutions to sinh-cosh-Gordon equation by using techniques based on projective Riccati equations. Computers and Mathematics with Applications, 61, 470-481.
- Salas H. A., 2011. Exact Solutions for the Ito Equation by the sn-ns Method. Applied Mathematical Sciences, 5, 2283-2287.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | July 15, 2019 |
| Submission Date | January 7, 2019 |
| Acceptance Date | March 20, 2019 |
| Published in Issue | Year 2019 Volume: 9 Issue: 3 |
