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.

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 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 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 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 New Travelling Wave Solution-Based New Riccati Equation for Solving KdV and Modified KdV Equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Hadi Rezazadeh ◽  
Alper Korkmaz ◽  
Abdelfattah EL Achab ◽  
Waleed Adel ◽  
Ahmet Bekir
Keyword(s):  
Large Family ◽  
Algebraic Method ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Hyperbolic Functions ◽  
Kdv Equations ◽  
De Vries ◽  
Wave Forms ◽  
Modified Kdv Equations ◽  
Different Characteristics

AbstractA large family of explicit exact solutions to both Korteweg- de Vries and modified Korteweg- de Vries equations are determined by the implementation of the new extended direct algebraic method. The procedure starts by reducing both equations to related ODEs by compatible travelling wave transforms. The balance between the highest degree nonlinear and highest order derivative terms gives the degree of the finite series. Substitution of the assumed solution and some algebra results in a system of equations are found. The relation between the parameters is determined by solving this system. The solutions of travelling wave forms determined by the application of the approach are represented in explicit functions of some generalized trigonometric and hyperbolic functions and exponential function. Some more solutions with different characteristics are also found.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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