A generalized new auxiliary equation method and its applications to nonlinear partial differential equations

2007 ◽  
Vol 363 (5-6) ◽  
pp. 356-360 ◽  
Author(s):  
Sheng Zhang ◽  
Tiecheng Xia
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Partial Differential ◽  
Auxiliary Equation Method

scholarly journals Exact Solutions with Variable Coefficient Function Forms for Conformable Fractional Partial Differential Equations by an Auxiliary Equation Method

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Fanwei Meng ◽  
Qinghua Feng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Coefficient Function ◽  
Fractional Partial Differential Equations ◽  
Auxiliary Equation Method

In this paper, an auxiliary equation method is introduced for seeking exact solutions expressed in variable coefficient function forms for fractional partial differential equations, where the concerned fractional derivative is defined by the conformable fractional derivative. By the use of certain fractional transformation, the fractional derivative in the equations can be converted into integer order case with respect to a new variable. As for applications, we apply this method to the time fractional two-dimensional Boussinesq equation and the space-time fractional (2+1)-dimensional breaking soliton equation. As a result, some exact solutions including variable coefficient function solutions as well as solitary wave solutions for the two equations are found.

scholarly journals New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations

2020 ◽  
Vol 5 (1) ◽  
pp. 447-454 ◽  
Author(s):  
Hülya Durur ◽  
Orkun Tasbozan ◽  
Ali Kurt
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Water Waves ◽  
Boussinesq Equations ◽  
Chain Rule ◽  
Auxiliary Equation ◽  
Fractional Partial Differential Equations ◽  
Shallow Water Waves ◽  
Partial Differential ◽  
Auxiliary Equation Method

AbstractThe main purpose of this article is to obtain the new solutions of fractional bad and good modified Boussinesq equations with the aid of auxiliary equation method, which can be considered as a model of shallow water waves. By using the conformable wave transform and chain rule, nonlinear fractional partial differential equations are converted into nonlinear ordinary differential equations. This is an important impact because both Caputo definition and Riemann–Liouville definition do not satisfy the chain rule. By using conformable fractional derivatives, reliable solutions can be achieved for conformable fractional partial differential equations.

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.

A Generalized Auxiliary Equation Method for the Quadratic Nonlinear KG Equation

2013 ◽  
Vol 394 ◽  
pp. 571-576
Author(s):  
Sheng Zhang ◽  
Bo Xu ◽  
Ao Xue Peng
Keyword(s):  
Partial Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Mathematical Tool ◽  
Auxiliary Equation Method ◽  
Klein Gordon ◽  
Generalized Auxiliary Equation Method ◽  
Powerful Mathematical Tool

A generalized auxiliary equation method with symbolic computation is used to construct more general exact solutions of the quadratic nonlinear Klein-Gordon (KG) equation. As a result, new and more general solutions are obtained. It is shown that the generalized auxiliary equation method provides a more powerful mathematical tool for solving nonlinear partial differential equations arising in the fields of nonlinear sciences.

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