Numerical solutions of some nonlinear evolution equations by Chebyshev spectral collocation methods

Abstract

Chebyshev spectral collocation methods (known as El-Gendi methods—as described by El-Gendi in 1969 and Mihaila and Mihaila 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.

Keywords:

• Nonlinear evolution equations
• Numerical solutions
• Chebyshev spectral collocation methods
• Fluid mechanics
• Plasma physics