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In this study, with the aid of Wolfram Mathematica 11, the modified exp (− (η))-expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well-known nonlinear evolutionary equation, namely; the two-component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two-component second order KdV evolutionary system with the finite forward difference method by using the Fourier-Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of L 2 and L ∞ norm error. We present the comparison between the exact and numerical solutions of the two-component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp (− (η))-expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models. K E Y W O R D Sanalytical solutions, numerical solutions, the FDM, the MEFM, the twocomponent second order KdV evolutionary system Numer Methods Partial Differential Eq. 2018;34:211-227.wileyonlinelibrary.com/journal/num

In this paper, a powerful sine-Gordon expansion method (SGEM) with aid of a computational program is used in constructing a new hyperbolic function solutions to one of the popular nonlinear evolution equations that arises in the field of mathematical physics, namely; longren-wave equation. We also give the 3D and 2D graphics of all the obtained solutions which are explaining new properties of model considered in this paper. Finally, we submit a comprehensive conclusion at the end of this paper.

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