An application of the Exp-function method (EFM) to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for the modified KDV equation and the generalized KDV equation. The EFM was used to construct peri- odic wave and solitary wave solutions of nonlinear evolution equations (NLEEs). This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the Exp-function method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics and applied mathematics.

In this paper, the homotopy perturbation method (HPM) is employed to obtain the exact solutions of KDV and Sawada-Kotera equations. To illustrate the simplicity and reliability of the method, several examples are provided. The results obtained using HPM are compared to the results of Adomian decomposition method (ADM) and then ability of each method are also discussed.

In this paper, an application of homotopy perturbation method is applied to finding the solutions of a generalized fifth order KDV (gfKDV) equation. Then we obtain the exact solitary-wave solutions and numerical solutions of the gfKDV equation for the initial conditions. The numerical solutions are compared with the known analytical solutions. Their remarkable accuracy are finally demonstrated for the gfKDV equation.

The variational theory is an inextricable part of both continuum mechanics and physics, and plays an important role in mathematics and nonlinear science, however it is difficult to find a variational formulation for a nonlinear system, and it is more difficult for a fractional differential system. This paper is to search for a variational formulation for the Schrödinger-KDV system with M-fractional derivatives. The fractional complex transformation is used to convert the system into a traditional differential system, and the semi-inverse method is further applied to establish a needed variational principle.

In this article, a class of solitary wave solutions of the coupled KDV equations by using the homotopy analysis method (HAM). The results obtained by this method have a good agreement with the exact ones. Numerical simulations indicate that the HAM is still valid and effective for solving the nonlinear coupled differential equations.

In this paper, we use the rational radial basis functions ( RRBFs) method to solve the Korteweg-de Vries (KDV) equation, particularly when the equation has a solution with steep front or sharp gradients. We approximate the spatial derivatives by RRBFs method then we apply an explicit fourth-order Runge-Kutta method to advance the resulting semi-discrete system in time. Numerical examples show that the presented scheme preserves the conservation laws and the results obtained from this method are in good agreement with analytical solutions.

In this paper, an efficient meshfree collocation scheme based on meshfree radial basis function is implemented for the numerical solution of 7th-order Korteweg-de Vires (KDV) equations. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. The proposed numerical scheme is tested on several test problems. The efficiency and accuracy of the suggested scheme is analyzed via ||L|| and ||L||2 error norms.