A transformed rational function method for (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation

Pramana ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. 561-571 ◽  
Author(s):  
SHENG ZHANG ◽  
HONG-QING ZHANG
Keyword(s):  
Rational Function ◽  
Function Method ◽  
Transformed Rational Function Method

Exact solutions of the Rosenau–Hyman equation, coupled KdV system and Burgers–Huxley equation using modified transformed rational function method

2018 ◽  
Vol 32 (24) ◽  
pp. 1850282 ◽  
Author(s):  
Yong-Li Sun ◽  
Wen-Xiu Ma ◽  
Jian-Ping Yu ◽  
Chaudry Masood Khalique
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Function Method ◽  
Bernoulli Equation ◽  
Huxley Equation ◽  
Transformed Rational Function Method ◽  
The Given

In this research, we study the exact solutions of the Rosenau–Hyman equation, the coupled KdV system and the Burgers–Huxley equation using modified transformed rational function method. In this paper, the simplest equation is the Bernoulli equation. We are not only obtain the exact solutions of the aforementioned equations and system but also give some geometric descriptions of obtained solutions. All can be illustrated vividly by the given graphs.

Extended Transformed Rational Function Method to Nonlinear Evolution Equations

2019 ◽  
Vol 20 (6) ◽  
pp. 691-701 ◽  
Author(s):  
Emrullah Yaşar ◽  
Yakup Yıldırım ◽  
Abdullahi Rashid Adem
Keyword(s):  
Rational Function ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Trigonometric Function ◽  
Nonlinear Evolution Equations ◽  
Bilinear Forms ◽  
Complexiton Solutions ◽  
Transformed Rational Function Method ◽  
Hirota Bilinear

AbstractIn this work, we study complexiton solutions to a (2+1)-dimensional (SK) equation and a (3+1)-dimensional nonlinear evolution equation. The complexiton solutions are combinations of trigonometric function waves and exponential function waves. For this goal, the extended transformed rational function method is carried out which is based on the Hirota bilinear forms of the considered equations and provides a systematical and convenient tool for constructing the exact solutions of nonlinear evolution equations.

Application of Extended Transformed Rational Function Method to Some (3+1) Dimensional Nonlinear Evolution Equations

2018 ◽  
pp. 433-437
Keyword(s):  
Rational Function ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Kdv Equation ◽  
Mapping Method ◽  
Nonlinear Evolution Equations ◽  
Petviashvili Equation ◽  
Complexiton Solutions ◽  
Transformed Rational Function Method

The transformed rational function method can be considered as unification of the tanh type methods, the homogeneous balance method, the mapping method, the exp-function method and the F-expansion type methods. In this paper, we present complexiton solutions of (3+1) dimensional Korteweg-de Vries (KdV) equation and a new (3+1) dimensional generalized Kadomtsev-Petviashvili equation by using extended transformed rational function method which provides very useful and effective way to obtain complexiton solutions of nonlinear evolution equations.

Abundant exact solutions to the strain wave equation in micro-structured solids

2021 ◽  
pp. 2150439
Author(s):  
Karmina K. Ali ◽  
R. Yilmazer ◽  
H. Bulut ◽  
Tolga Aktürk ◽  
M. S. Osman
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Exact Solutions ◽  
Rational Function ◽  
Physical Interpretation ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Singular Solutions ◽  
Strain Wave ◽  
Important Place

In this study, the strain wave equation in micro-structured solids which take an important place in solid physics is presented for consideration. The generalized exponential rational function method is used for this purpose which is one of the most powerful methods of constructing abundantly distinct, exact solutions of nonlinear partial differential equations. In micro-structured solids, wave propagation is based on the structure of the strain wave equation. As a consequence, we successfully received many different exact solutions, including non-topological solutions, periodic singular solutions, topological solutions, singular solutions, like periodic lump solutions. Furthermore, in order to better understand their physical interpretation, 2D, 3D, and counter plots are graphed for each of the solutions acquired.

Application of the Exponential Rational Function Method to Some Fractional Soliton Equations

2020 ◽  
pp. 13-32
Author(s):  
Mustafa Ekici ◽  
Metin Ünal
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Function Method ◽  
Space Time ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Kdv Equations ◽  
Fractional Differential ◽  
Exponential Rational Function Method ◽  
Fifth Order

In this chapter, the authors study the exponential rational function method to find new exact solutions for the time-fractional fifth-order Sawada-Kotera equation, the space-time fractional Whitham-Broer-Kaup equations, and the space-time fractional generalized Hirota-Satsuma coupled KdV equations. These fractional differential equations are converted into ordinary differential equations by using the fractional complex transform. The fractional derivatives are defined in the sense of Jumarie's modified Riemann-Liouville. The proposed method is direct and effective for solving different kind of nonlinear fractional equations in mathematical physics.

Optical soliton solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation

2019 ◽  
Vol 33 (32) ◽  
pp. 1950402 ◽  
Author(s):  
Behzad Ghanbari ◽  
J. F. Gómez-Aguilar
Keyword(s):  
Rational Function ◽  
Imaginary Part ◽  
Traveling Waves ◽  
Analytical Solutions ◽  
Efficient Method ◽  
Function Method ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Complex Structures ◽  
Exponential Rational Function Method

In this paper, the generalized exponential rational function method is applied to obtain analytical solutions for the nonlinear Radhakrishnan–Kundu–Lakshmanan equation. We obtain novel soliton, traveling waves and kink-type solutions with complex structures. We also present the two- and three-dimensional shapes for the real and imaginary part of the solutions obtained. It is illustrated that generalized exponential rational function method (GERFM) is simple and efficient method to reach the various type of the soliton solutions.

Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method

2019 ◽  
Vol 33 (09) ◽  
pp. 1950106 ◽  
Author(s):  
Behzad Ghanbari
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Graphical Interpretation ◽  
Wave Solutions ◽  
Complex Models ◽  
Exponential Rational Function Method

In this paper, some new traveling wave solutions to the Hirota–Maccari equation are constructed with the help of the newly introduced method called generalized exponential rational function method. Several families of exact solutions are found corresponding to the equation. To the best of our knowledge, these solutions are new, and have never been addressed in the literature. The graphical interpretation of the solutions is also depicted. Moreover, it is contemplated that the proposed technique can also be employed to another sort of complex models.

THE RATIONAL SOLUTIONS TO A GENERALIZED RICCATI EQUATION AND THEIR APPLICATION

2013 ◽  
Vol 27 (05) ◽  
pp. 1350013 ◽  
Author(s):  
QING LIU ◽  
SHI-YAO SHEN ◽  
ZI-HUA WANG
Keyword(s):  
Riccati Equation ◽  
Rational Function ◽  
Function Method ◽  
New Method ◽  
Equation Method ◽  
Rational Solutions ◽  
Generalized Riccati Equation

Based on the rational solutions to a generalized Riccati equation, a new method which is called as rational function method is proposed. We apply this method to solve the coupled mKdV equations and derive a set of rational solutions. This method is also available for seeking their solutions to other NLPDEs. It shows that the rational function method is more universal and powerful than the auxiliary Riccati equation method.

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