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

A 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.

Semi-Analytical Solutions of Time-Fractional KdV and Modified KdV Equations

In this paper, semi-analytical solutions of time-fractional Korteweg-de Vries (KdV) equations are obtained by using a novel variational technique. The method is based on the coupling of Laplace Transform Method (LTM) with Variational Iteration Method (VIM) and it was implemented on regular and modified KdV equations of fractional order in Caputo sense. Correction functionals were used in general Lagrange multipliers with optimality conditions via variational theory. The implementation of this method to non-linear fractional differential equations is quite easy in comparison with other existing methods.

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

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.