Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
H. Jafari ◽  
H. Tajadodi ◽  
D. Baleanu
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Telegraph Equation ◽  
Fisher Equation ◽  
Fractional Evolution Equations ◽  
Liouville Derivative ◽  
Nonlinear Telegraph Equation ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

The fractional Fan subequation method of the fractional Riccati equation is applied to construct the exact solutions of some nonlinear fractional evolution equations. In this paper, a powerful algorithm is developed for the exact solutions of the modified equal width equation, the Fisher equation, the nonlinear Telegraph equation, and the Cahn–Allen equation of fractional order. Fractional derivatives are described in the sense of the modified Riemann–Liouville derivative. Some relevant examples are investigated.

scholarly journals A Novel Analytical Technique to Obtain Kink Solutions for Higher Order Nonlinear Fractional Evolution Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Qazi Mahmood Ul Hassan ◽  
Jamshad Ahmad ◽  
Muhammad Shakeel
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Fractional Evolution Equations ◽  
Analytical Technique ◽  
Fractional Complex Transform ◽  
Fifth Order

We use the fractional derivatives in Caputo’s sense to construct exact solutions to fractional fifth order nonlinear evolution equations. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted in a number of exact solutions.

AN EFFICIENT METHOD FOR FINDING THE EXACT SOLUTION OF NONLINEAR EVOLUTION EQUATIONS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1703-1706 ◽  
Author(s):  
XIQIANG ZHAO ◽  
DENGBIN TANG ◽  
CHANG SHU
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

In this paper, based on the idea of the homogeneous balance method, the special truncated expansion method is improved. The Burgers-KdV equation is discussed and its many exact solutions are obtained with the computerized symbolic computation system Mathematica. Our method can be applied to finding exact solutions for other nonlinear partial differential equations too.

Abundant New Multiple Soliton-like Solutions and Rational Solutions of the (2+1)-Dimensional Broer-Kaup Equation

2001 ◽  
Vol 56 (12) ◽  
pp. 816-824 ◽  
Author(s):  
Zhenya Yan
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Heat Equations ◽  
Backlund Transformation ◽  
Rational Solutions ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

Abstract In this paper we firstly improve the homogeneous balance method due to Wang, which was only used to obtain single soliton solutions of nonlinear evolution equations, and apply it to (2 + 1)-dimensional Broer-Kaup (BK) equations such that a Backlund transformation is found again. Considering further the obtained Backlund transformation, the relations are deduced among BK equations, well-known Burgers equations and linear heat equations. Finally, abundant multiple soliton-like solutions and infinite rational solutions are obtained from the relations.

scholarly journals Exact solutions of the Kudryashov–Sinelshchikov equation and nonlinear telegraph equation via the first integral method

2012 ◽  
Vol 17 (4) ◽  
pp. 481-488 ◽  
Author(s):  
Mohammad Mirzazadeh ◽  
Mostafa Eslami
Keyword(s):  
Exact Solutions ◽  
Commutative Algebra ◽  
Traveling Wave ◽  
Integral Method ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
Telegraph Equation ◽  
First Integral Method ◽  
Wave Solutions ◽  
Nonlinear Telegraph Equation

In this article we find the exact traveling wave solutions of the Kudryashov–Sinelshchikov equation and nonlinear telegraph equation by using the first integral method. This method is based on the theory of commutative algebra. This method can be applied to nonintegrable equations as well as to integrable ones.

Auto-B¨Acklund Transformations And Exact Solutions For The Generalized Two-Dimensional Korteweg-De Vries-Burgers-Type Equations And Burgers-Type Equations

2003 ◽  
Vol 58 (7-8) ◽  
pp. 464-472
Author(s):  
Biao Li ◽  
Yong Chen ◽  
Hongqing Zhang
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformation ◽  
Type Equation ◽  
Balance Method ◽  
Two Dimensional ◽  
Backlund Transformation ◽  
De Vries ◽  
Homogeneous Balance Method ◽  
Burgers Type Equations ◽  
Homogeneous Balance

In this paper, based on the idea of the homogeneous balance method and with the help of Mathematica, we obtain a new auto-Bäcklund transformation for the generalized two-dimensional Kortewegde Vries-Burgers-type equation and a new auto-Bäcklund transformation for the generalized twodimensional Burgers-type equation by introducing two appropriate transformations. Then, based on these two auto-Bäcklund transformation, some exact solutions for these equations are derived. Some figures are given to show the properties of the solutions.

scholarly journals Analytical Wave Solutions for Foam and KdV-Burgers Equations Using Extended Homogeneous Balance Method

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 729 ◽  
Author(s):  
U.M. Abdelsalam ◽  
M. G. M. Ghazal
Keyword(s):  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Industrial Applications ◽  
Nonlinear Evolution Equations ◽  
Balance Method ◽  
Wave Solutions ◽  
Engineering Sciences ◽  
Foam Drainage Equation ◽  
Homogeneous Balance Method ◽  
Homogeneous Balance

In this paper, extended homogeneous balance method is presented with the aid of computer algebraic system Mathematica for deriving new exact traveling wave solutions for the foam drainage equation and the Kowerteg-de Vries–Burgers equation which have many applications in industrial applications and plasma physics. The method is effective to construct a series of analytical solutions including many types like periodical, rational, singular, shock, and soliton wave solutions for a wide class of nonlinear evolution equations in mathematical physics and engineering sciences.

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