scholarly journals A New Type of Generalized F-Expansion Method and its Application to Sine-Gordon Equation

2017 ◽  
Author(s):  
Yusuf Pandır
Keyword(s):  
Gordon Equation ◽  
Expansion Method ◽  
Sine Gordon Equation ◽  
New Type

Complex Wave Solutions to Mathematical Biology Models I: Newell–Whitehead–Segel and Zeldovich Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Gordon Equation ◽  
Reaction Diffusion ◽  
Expansion Method ◽  
Reaction Diffusion Equations ◽  
Trigonometric Functions ◽  
Sine Gordon Equation ◽  
Hyperbolic Functions ◽  
Complex Wave ◽  
Complex Valued ◽  
Homogeneous Balance

Complex and real valued exact solutions to some reaction-diffusion equations are suggested by using homogeneous balance and Sine-Gordon equation expansion method. The predicted solution of finite series of some hyperbolic functions is determined by using some relations between the hyperbolic functions and the trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel–Whitehead–Segel (NWSE) and Zeldovich equations (ZE) are solved explicitly. Some complex valued solutions are depicted in real and imaginary components for some particular choice of parameters.

scholarly journals Abundant Interaction Solutions of Sine-Gordon Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
DaZhao Lü ◽  
YanYing Cui ◽  
ChangHe Liu ◽  
ShangWen Wu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Symbolic Computation ◽  
Gordon Equation ◽  
Expansion Method ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Interaction Solutions ◽  
Linear Partial Differential Equations

With the help of computer symbolic computation software (e.g.,Maple), abundant interaction solutions of sine-Gordon equation are obtained by means of a constructed Wronskian form expansion method. The method is based upon the forms and structures of Wronskian solutions of sine-Gordon equation, and the functions used in the Wronskian determinants do not satisfy linear partial differential equations. Such interaction solutions are difficultly obtained via other methods. And the method can be automatically carried out in computer.

scholarly journals Solutions to Benjamin-Bona-Mahony Equation in Two Space Dimensions by Various Methods

2018 ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Gordon Equation ◽  
Expansion Method ◽  
Algebraic Method ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Hyperbolic Tangent ◽  
Real Solutions ◽  
Two Space Dimensions ◽  
Bbm Equation

Four methods in two different families have been constructed to derive the exact solutions to Benjamin-Bona-Mahony equation in two space dimensions. Simply defined hyperbolic tangent, hyperbolic secant and hyperbolic cosecant ansatzes and the expansion method based on the Sine-Gordon equation in two dimensions are directly substituted into the governing ODE reduced from the two dimensional BBM equation. Classical algebraic method is used to find the relations among the target parameters representing the nonzero coefficients in the predicted solutions and the wave transform parameters. Some complex and real solutions have been constructed in explicit forms.

scholarly journals THE NONLINEAR AND NONLOCAL INTEGRABLE SINE‐GORDON EQUATION

2005 ◽  
Vol 10 (4) ◽  
pp. 367-376 ◽  
Author(s):  
P. Miškinis
Keyword(s):  
Analytical Solutions ◽  
Gordon Equation ◽  
Classical Solutions ◽  
Sine Gordon Equation ◽  
Exact Analytical Solutions ◽  
New Type ◽  
Interaction Term ◽  
Nonlocal Deformation

A new type of the nonlocal sine‐Gordon equation with the generalized interaction term is suggested. Its limit cases, symmetries and exact analytical solutions are obtained. This type of the nonlocal sine‐Gordon equation is shown to possess one‐, two‐ and N‐solitonic solutions which are a nonlocal deformation of the corresponding classical solutions of the sine‐Gordon equation. Pasiūlyta nauja nelokali sine‐Gordono evoliucine lygtis su apibendrintu saveikos nariu. Nustatyti šios lygties ribiniai atvejai, Lagranžianas, simetrijos, tikslūs analiziniai sprendiniai. Parodyta, kad šios rūšies nelokali sine‐Gordono lygtis turi vieno, dvieju bei N‐solitoninius sprendinius, kurie yra atitinkamu klasikiniu sine‐Gordono lygties sprendiniu nelokalios deformacijos. Nelokalios sine‐Gordono lygties integruojamumas siejamas su geometrinemis dvimačiu nelokaliai deformuotu paviršiu savybemis.

scholarly journals Some New Exact Solutions of (1+2)-Dimensional Sine-Gordon Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wei-Xiong Chen ◽  
Ji Lin
Keyword(s):  
Gordon Equation ◽  
Direct Method ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Sine Gordon Equation ◽  
Interaction Solutions ◽  
New Exact Solutions ◽  
Propagation Properties ◽  
A Direct Method

We use a generalized tanh function expansion method and a direct method to study the analytical solutions of the (1+2)-dimensional sine Gordon (2DsG) equation. We obtain some new interaction solutions among solitary waves and periodic waves, such as the kink-periodic wave interaction solution, two-periodic solitoff solution, and two-toothed-solitoff solution. We also investigate the propagation properties of these solutions.

scholarly journals Complex Wave Solutions to Mathematical Biology Models II: Two Dimensional Fisher and Nagumo Equations

2018 ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Space Dimension ◽  
Gordon Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Wave Solutions ◽  
Complex Wave ◽  
Nagumo Equations ◽  
Complex Valued

We extended the usage of the expansion method based on Sine-Gordon equation to the two dimensional Fisher equation. The relation between the trigonometric and hyperbolic functions are derived from the Sine-Gordon equation dened in two space dimension. The complex-valued traveling wave solutions to the two dimensional Fisher and Nagumo equations are set in forms a nite series of multiplications of powers of sech(.) and tanh(.) functions.

scholarly journals Sine-Gordon Expansion Method for Exact Solutions to Conformable Time Fractional Equations in RLW-Class

2017 ◽  
Author(s):  
Alper Korkmaz ◽  
Ozlem Ersoy Hepson ◽  
Kamyar Hosseini ◽  
Hadi Rezazadeh ◽  
Mostafa Eslami
Keyword(s):  
Exact Solutions ◽  
Gordon Equation ◽  
Expansion Method ◽  
Algebraic Equations ◽  
Sine Gordon Equation ◽  
Fractional Equations ◽  
Polynomial Type ◽  
Long Wave ◽  
Previous Step ◽  
Homogeneous Balance

The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave(RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters.

scholarly journals An Expansion Based on Sine-Gordon Equation to Solve KdV and modified KdV Equations in Conformable Fractional Forms

2017 ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Alper Korkmaz ◽  
Kamyar Hosseini ◽  
Hadi Rezazadeh ◽  
Mostafa Eslami
Keyword(s):  
Exact Solutions ◽  
Gordon Equation ◽  
Expansion Method ◽  
Sine Gordon Equation ◽  
Hyperbolic Functions ◽  
Kdv Equations ◽  
De Vries ◽  
Balance Technique ◽  
Modified Kdv Equations ◽  
Complex Valued

An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equation in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions.

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