Comparative Analysis of Exact Solutions for the Phi-four Equation

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.

Anahtar Kelimeler:

Ansatz method, He's variational method, Modified simple equation method, Phi-four equation, Sn-ns method

Phi-four Denkleminin Tam Çözümlerinin Karşılaştırmalı Analizi

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,

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