Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods

Chebyshev spectral collocation methods (known as El-Gendi method [S.E. El-Gendi, Chebyshev solution of differential integral and integro-differential equations, Comput. J. 12 (1969) 282–287; B. Mihaila, I. Mihaila, Numerical approximation using Chebyshev polynomial expansions: El-gendi’s method revisited, J. Phys. A 35 (2002) 731–746]) are extended to deal with the generalized Kuramoto–Sivashinsky equation. The problem is reduced to a system of ordinary differential equations that are solved by combinations of backward differential formula and appropriate explicit schemes (implicit–explicit BDF methods [G. Akrivis, Y.S. Smyrlis, Implicit–explicit BDF methods for the Kuramoto–Sivashinsky equation, Appl. Numer. Math. 51 (2004) 151–169]). Good numerical results have been obtained and compared with the exact solutions.