Exact Solutions for the Modified KdV and the Generalized KdV Equations via Exp-Function Method

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 periodic 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

Equivalent HPM with ADM and Convergence of the HPM to a Class of Nonlinear Integral Equations

The purpose of this study is to implement homotopy perturbation method, for solving nonlinear Volterra integral equations. In this work, a reliable approach for convergence of the HPM when applied to a class of nonlinear Volterra integral equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The results obtained by using HPM, are compared to those obtained by using Adomian decomposition method alone. The numerical results, demonstrate that HPM technique, gives the approximate solution with faster convergence rate and higher accuracy than using the standard AD

Variety interaction between k-lump and k-kink solutions for the (3+1)-D Burger system by bilinear analysis

In this paper, we investigate the (3+1)-dimensional Burger system which is employed in soliton theory and generated by considering the Hirota bilinear equation. We conclude some novel analytical solutions, including 2-lump-type, interaction between 2-lump and one kink, two lump and two kink of type I, two lump and two kink of type II, two lump and one periodic, two lump and kink-periodic, and two lump and periodic-periodic wave solutions for the considered system by symbolic estimations. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences. Then we apply explicit numerical methods, most of which are recently introduced by many scholars, to reproduce the analytical solutions. The test results show that the best algorithms, especially the Hirota bilinear, are very efficient and severely outperform the other methods