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.

Level 1, systematic review and meta-analysis.

No abstract

Nonlinear Alfvén waves, propagating along a homogeneous magnetic field, are studied using relativistic isotropic hydrodynamics. Alfvén solitons of the moving-wave and wave packet types are considered for modified Korteweg–de Vries (mKdV) equation and the nonlinear Schrödinger (NLS) equation, respectively. The method of characteristics is used and the Bäcklund transformations (BTs) are employed to generate new solutions from the old ones. Thus, families of new solutions for the mKdV and the NLS equations are obtained. The question arises which solitons exist in the pulsar atmosphere.