scholarly journals The Extended Trial Equation Method for Some Time Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yusuf Pandir ◽  
Yusuf Gurefe ◽  
Emine Misirli
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Equation Method ◽  
Wave Transformation ◽  
Jacobi Elliptic Function ◽  
Partial Differential ◽  
Extended Trial Equation Method ◽  
Trial Equation Method ◽  
Liouville Derivative ◽  
Extended Trial

Nonlinear fractional partial differential equations have been solved with the help of the extended trial equation method. Based on the fractional derivative in the sense of modified Riemann-Liouville derivative and traveling wave transformation, the fractional partial differential equation can be turned into the nonlinear nonfractional ordinary differential equation. For illustrating the reliability of this approach, we apply it to the generalized third order fractional KdV equation and the fractionalKn,nequation according to the complete discrimination system for polynomial method. As a result, some new exact solutions to these nonlinear problems are successfully constructed such as elliptic integral function solutions, Jacobi elliptic function solutions, and soliton solutions.

Extended Trial Equation Method for Nonlinear Partial Differential Equations

2015 ◽  
Vol 70 (4) ◽  
pp. 269-279 ◽  
Author(s):  
Khaled A. Gepreel ◽  
Taher A. Nofal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Equation Method ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Partial Differential ◽  
Extended Trial Equation Method ◽  
Trial Equation Method ◽  
Extended Trial

AbstractThe main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber–Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.

scholarly journals The Jacobi Elliptic Equation Method for Solving Fractional Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Elliptic Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Equation Method ◽  
Hyperbolic Function ◽  
Jacobi Elliptic Function ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Liouville Derivative

Based on a nonlinear fractional complex transformation, the Jacobi elliptic equation method is extended to seek exact solutions for fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. For demonstrating the validity of this method, we apply it to solve the space fractional coupled Konopelchenko-Dubrovsky (KD) equations and the space-time fractional Fokas equation. As a result, some exact solutions for them including the hyperbolic function solutions, trigonometric function solutions, rational function solutions, and Jacobi elliptic function solutions are successfully found.

Optical solitons in DWDM system by extended trial equation method

Optik ◽  
2017 ◽  
Vol 141 ◽  
pp. 157-167 ◽  
Author(s):  
Mehmet Ekici ◽  
Abdullah Sonmezoglu ◽  
Qin Zhou ◽  
Anjan Biswas ◽  
Malik Zaka Ullah ◽  
...  
Keyword(s):  
Optical Solitons ◽  
Equation Method ◽  
Extended Trial Equation Method ◽  
Trial Equation Method ◽  
Dwdm System ◽  
Extended Trial

Optical solitons for paraxial wave equation in Kerr media

2019 ◽  
Vol 33 (03) ◽  
pp. 1950020 ◽  
Author(s):  
Kashif Ali ◽  
Syed Tahir Raza Rizvi ◽  
Badar Nawaz ◽  
Muhammad Younis
Keyword(s):  
Wave Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Optical Solitons ◽  
Equation Method ◽  
Kerr Media ◽  
Extended Trial Equation Method ◽  
Trial Equation Method ◽  
Paraxial Wave Equation ◽  
Extended Trial

This paper retrieves Jacobi elliptic, periodic, bright and singular solitons for paraxial nonlinear Schrödinger equation (NLSE) in Kerr media. We use extended trial equation method to obtain these solitons solutions. For the existence of the soliton solutions, constraint conditions are also presented.

Optical soliton perturbation for Gerdjikov–Ivanov equation by extended trial equation method

Optik ◽  
2018 ◽  
Vol 158 ◽  
pp. 747-752 ◽  
Author(s):  
Anjan Biswas ◽  
Mehmet Ekici ◽  
Abdullah Sonmezoglu ◽  
Fayequa B. Majid ◽  
Houria Triki ◽  
...  
Keyword(s):  
Optical Soliton ◽  
Equation Method ◽  
Extended Trial Equation Method ◽  
Trial Equation Method ◽  
Soliton Perturbation ◽  
Extended Trial

scholarly journals On Exact Solutions of Phi-4 Partial Differential Equation Using the Enhanced Modified Simple Equation Method

2018 ◽  
Vol 6 (4) ◽  
Author(s):  
Ziad Salem Rached
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Simple Equation ◽  
Equation Method ◽  
Partial Differential ◽  
New Exact Solutions ◽  
Modified Simple Equation Method

Constructing exact solutions of nonlinear ordinary and partial differential equations is an important topic in various disciplines such as Mathematics, Physics, Engineering, Biology, Astronomy, Chemistry,… since many problems and experiments can be modeled using these equations. Various methods are available in the literature to obtain explicit exact solutions. In this correspondence, the enhanced modified simple equation method (EMSEM) is applied to the Phi-4 partial differential equation. New exact solutions are obtained.

Nematicons in liquid crystals by extended trial equation method

2017 ◽  
Vol 26 (01) ◽  
pp. 1750005 ◽  
Author(s):  
Mehmet Ekici ◽  
Mohammad Mirzazadeh ◽  
Abdullah Sonmezoglu ◽  
Malik Zaka Ullah ◽  
Qin Zhou ◽  
...  
Keyword(s):  
Liquid Crystals ◽  
Shock Waves ◽  
Governing Equation ◽  
Plane Waves ◽  
Equation Method ◽  
Integration Algorithm ◽  
Extended Trial Equation Method ◽  
Trial Equation Method ◽  
First Time ◽  
Extended Trial

This paper employs extended trial equation method to retrieve nematicons in liquid crystals from its governing equation. In addition, several other forms of solution naturally emerged from the integration algorithm. These are shock waves, singular solitons, snoidal waves, periodic singular waves, plane waves and others. These variety of solutions are being reported for the first time in the context of liquid crystals.

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