The Double (G^( ^' )/G,1/G)-Expansion Method and Its Applications for Some Nonlinear Partial Differential Equations

Abstract

The double (′ / , /)-expansion method is used to find exact travelling wave solutions to the fractional differantial equations in the sense of Jumarie's modified Riemann-Liouville derivative. We exploit this method for the combined KdV-negative-order KdV equation (KdV-nKdV) and the Calogero-Bogoyavlinskii-Schiff equation (CBS) of fractional order. We see that these solutions are concise and easy to understand the physical phenomena of the nonlinear partial differential equations. These solutions can be shown in terms of trigonometric, hyperbolic and rational functions.

Güç İli̇şki̇leri̇ Bağlaminda Ataerki̇l Şi̇ddeti̇ Anlamak

“…Bu noktada psikolojik şiddet psikolojik istismar, mobbing, duygusal şiddet ya da duygusal istismar gibi alternatif isimlerle tanımlanan, fail ve mağdur arasında gerçekleşen bir saldırganlık biçimi olarak kabul edilebilir. Bu bağlamda failin mağduru duygusal olarak sindirmek, aşağılamak ya da cezalandırmak için mağdura bağırması, küfür etmesi, hakaret etmesi; mağduru korkutması, tehdit etmesi , küçük düşürmesi, devamlı eleştirmesi; mağdurun hareketlerini, giyimini, başkalarıyla ilişkilerini denetleyerek toplumdan uzaklaştırması ve mağduru sınırlaması, mağdurda değersizlik hissi uyandırması gibi yollarla düzenli olarak mağdura duygusal baskı uygulaması psikolojik şiddet kapsamında değerlendirilebilir (Boyacıoğlu vd., 2019;Akay, 2014;Patricia 2012;Ünal, 2005). Örneğin bir psikolojik manipülasyon ve taciz yöntemi olarak adlandırılan ve bir psikolojik şiddet türü olarak ele alınan "gaslighting", mağduru kendi hafızasını, algılarını ve akıl sağlığını sorgulayıp irdelemeye yöneltir, mağdur üzerinde duygusal baskı yaratarak mağdurun kendisini değersiz hissetmesine sebep olur.…”

A study on the solutions of (1+1)-dimensional Mikhailov-Novikov-Wang equation

“…NLEEs have very important applications in areas such as mathematical physics, optical fibers, mathematical chemistry, hydrodynamics, fluid dynamics, geochemistry, control theory, meteorology, optics, mechanics, chemical kinematics, biophysics, biogenetics, and so on. A number of methods have been developed by various researchers in order to obtain solutions for NLEEs, which have such important areas of use in the scientific world: Modified direct algebraic, modified Kudryashov and trigonometric-quantic B-spline methods [1], improved Bernoulli sub-equation function method [2], modified extended tanh-function method [3], new extended generalized Kudryashov and generalized new Kudryashov methods [4], new function method [5], exp (−ϕ(ξ)) method [6], double (G ′ /G, 1/G)-expansion method [7], modified extended tanh-expansion based method [8], modified simple equation method [9], Jacobi elliptic function expansion method [10], modified (1/G ′ )-expansion method [11].…”

Application of the rational (G' /G)-expansion method for solving some coupled and combined wave equations

“…Many researchers have been proposed various different methods to find solutions for nonlinear partial differential equations and nonlinear fractional differential equations [36][37][38][39][40]. Such as the inverse scattering transform method [1], the Hirota's bilinear method [2], truncated Painlevé expansion method [3], the tanh-function expansion method [4], the Jacobi elliptic function expansion method [5], the homogeneous balance method [6][7][8], the trial function method [9], the exp-function method [10,34], differential transform method [33], the Bäcklund transform method [11], the generalized Riccati equation method [12][13][14][15], the sub-ODE method [17][18][19][20], the original (G ′ /G)-expansion method [16,29], the double (G ′ /G,1/G)-expansion method [35] etc.. Since there is not a common method that can be used to solve all types of nonlinear evolution equations.…”