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

2018 ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Traveling Wave ◽  
Gordon Equation ◽  
Reaction Diffusion ◽  
Expansion Method ◽  
Reaction Diffusion Equations ◽  
Trigonometric Functions ◽  
Sine Gordon Equation ◽  
Hyperbolic Functions ◽  
Complex Wave ◽  
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 are determined by using some relations between the hyperbolic functions and trigonometric functions based on Sine-Gordon equation and traveling wave transform. The Newel-Whitehead-Segel and Zeldovich equations are solved explicitly. Some real valued solutions are depicted for some particular choice of parameters.

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 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.

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 On The Exact Solutions to Conformable Time Fractional Equations in EW Family Using Sine-Gordon Equation Approach

2017 ◽  
Author(s):  
Alper Korkmaz ◽  
Ozlem Ersoy Hepson ◽  
Kamyar Hosseini ◽  
Hadi Rezazadeh ◽  
Mostafa Eslami
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Gordon Equation ◽  
Solution Procedure ◽  
Equation Approach ◽  
Sine Gordon Equation ◽  
Hyperbolic Functions ◽  
Fractional Equations ◽  
New Exact Solutions ◽  
Homogeneous Balance

New exact solutions to conformable time fractional EW and modified EW equations are constructed by using Sine-Gordon expansion approach. The fractional traveling wave transform and homogeneous balance have significant roles in the solution procedure. The predicted solution is of the form of some finite series of multiplication of powers of cos and sin functions. The relation among trigonometric and hyperbolic functions in sense of Sine-Gordon expansion gives opportunity to construct the solutions in terms of hyperbolic functions.

scholarly journals Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Wei Li ◽  
Huizhang Yang ◽  
Bin He
Keyword(s):  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Burgers Equation ◽  
Expansion Method ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Hilliard Equation ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Cahn Hilliard Equation

Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

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.

The (G'/G,1/G)-Expansion Method for Solving the (2+1)-Dimensional Breaking Soliton Equations

2013 ◽  
Vol 787 ◽  
pp. 1006-1010
Author(s):  
Yun Jie Yang ◽  
Yun Mei Zhao ◽  
Yan He
Keyword(s):  
Rational Functions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Soliton Equations ◽  
Wave Solutions

In this paper, the-expansion method is applied to construct more general exact travelling solutions of the (2+1)-dimensional breaking soliton equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions.

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