Auxiliary Equation Method for Fractional Differential Equations with Modified Riemann–Liouville Derivative

2016 ◽  
Vol 17 (7-8) ◽  
pp. 413-420 ◽  
Author(s):  
Arzu Akbulut ◽  
Melike Kaplan ◽  
Ahmet Bekir
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Auxiliary Equation Method ◽  
Liouville Derivative ◽  
Regularized Long Wave Equation ◽  
Klein Gordon ◽  
Fractional Differential

Abstract:In this work, the auxiliary equation method is applied to derive exact solutions of nonlinear fractional Klein–Gordon equation and space-time fractional Symmetric Regularized Long Wave equation. Consequently, some exact solutions of these equations are successfully obtained. These solutions are formed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions founded by the suggested method indicate that the approach is easy to implement and powerful.

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.

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.

scholarly journals Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Emad A.-B. Abdel-Salam ◽  
Eltayeb A. Yousif
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Expansion Method ◽  
Space Time ◽  
Regularized Long Wave Equation ◽  
Klein Gordon ◽  
Fractional Differential ◽  
Nonlinear Physical Phenomena ◽  
Physical Phenomena ◽  
First Time

The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case.

scholarly journals Exact Solutions of the Space-Time Fractional Bidirectional Wave Equations Using the(G′/G)-Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Wei Li ◽  
Huizhang Yang ◽  
Bin He
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Wave Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Space Time ◽  
Promising Tool ◽  
Liouville Derivative ◽  
Fractional Differential

Based on Jumarie’s modified Riemann-Liouville derivative, the fractional complex transformation is used to transform fractional differential equations to ordinary differential equations. Exact solutions including the hyperbolic functions, the trigonometric functions, and the rational functions for the space-time fractional bidirectional wave equations are obtained using the(G′/G)-expansion method. The method provides a promising tool for solving nonlinear fractional differential equations.

scholarly journals Exact solutions of the space-time fractional equal width equation

2019 ◽  
Vol 23 (4) ◽  
pp. 2307-2313 ◽  
Author(s):  
Hongcai Ma ◽  
Xiangmin Meng ◽  
Hanfang Wu ◽  
Aiping Deng
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Soliton Solutions ◽  
Space Time ◽  
Bright Soliton ◽  
Liouville Derivative ◽  
Fractional Differential

A class of fractional differential equations is investigated in this paper. By the use of modified Remann-Liouville derivative and the tanh-sech method, the exact bright soliton solutions for the space-time fractional equal width are obtained.

scholarly journals A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Özkan Güner ◽  
Adem C. Cevikel
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Burgers Equation ◽  
Fractional Partial Differential Equations ◽  
New Exact Solutions ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

scholarly journals Variable Coefficient Exact Solutions for Some Nonlinear Conformable Partial Differential Equations Using an Auxiliary Equation Method

Computation ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 31
Author(s):  
Sekson Sirisubtawee ◽  
Nuntapon Thamareerat ◽  
Thitthita Iatkliang
Keyword(s):  
Partial Differential Equations ◽  
Solitary Wave ◽  
Differential Equations ◽  
Exact Solutions ◽  
Wave Solution ◽  
Variable Coefficient ◽  
Solitary Wave Solution ◽  
Equation Method ◽  
Auxiliary Equation ◽  
Auxiliary Equation Method

The objective of this present paper is to utilize an auxiliary equation method for constructing exact solutions associated with variable coefficient function forms for certain nonlinear partial differential equations (NPDEs) in the sense of the conformable derivative. Utilizing the specific fractional transformations, the conformable derivatives appearing in the original equation can be converted into integer order derivatives with respect to new variables. As for applications of the method, we particularly obtain variable coefficient exact solutions for the conformable time (2 + 1)-dimensional Kadomtsev–Petviashvili equation and the conformable space-time (2 + 1)-dimensional Boussinesq equation. As a result, the obtained exact solutions for the equations are solitary wave solutions including a soliton solitary wave solution and a bell-shaped solitary wave solution. The advantage of the used method beyond other existing methods is that it provides variable coefficient exact solutions covering constant coefficient ones. In consequence, the auxiliary equation method based on setting all coefficients of an exact solution as variable function forms can be more extensively used, straightforward and trustworthy for solving the conformable NPDEs.

Sign in / Sign up

Export Citation Format

Share Document