In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge-Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers', KdV-Burgers', coupled Burgers', 2D Burgers' and system of 2D Burgers' equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds' number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.

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The periodic wave solutions of two nonlinear models for ion-acoustic plasma waves, including the mixed Korteweg-de Vries (KdV) and modified KdV equation and the (3 þ 1)-dimensional modified Zakharov-Kuznetsov equation, are obtained by using the F-expansion method and introducing appropriate transformations. In the limiting cases, the solitary wave solutions of the equations are also obtained.

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Chebyshev spectral collocation methods (known as El-Gendi methods-as described by El-Gendi in 1969 andMihaila andMihaila in 2002) are presented to deal with some nonlinear evolution equations including the Korteweg-de Vries (KdV), the modified KdV, the mixed KdV and modified KdV, and the generalized fifth-order KdV equations, which include as special cases some well-known equations. The problem is reduced to a system of ordinary differential equations that are solved by combinations of backward differential formulas and appropriate explicit schemes (implicit-explicit BDF methods-as described by Akrivis and Smyrlis in 2004). Good numerical results have been obtained and compared with the exact solutions.

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