Fully spectral methods for the solution of high order differential equations

In the recent years spectral methods are used for solving stiff and non-stiff partial differential equations and ordinary differential equations. Various types of spectral methods for steady and unsteady problems are proposed to solve stiff and non-stiff partial differential equations effciently. In this article some schemes for solving stiff partial differential equations are derived. There are twofold: first method is based on Chebyshev polynomials for solving high-order boundary value problems. Second methods are based on Fourier-Galerkin and collocation spectral methods in space and Runge-Kutta, exponential time differencing, Taylor expansion and contour integral in time for solving stiff PDEs. Numerical results show the effciency of proposed schemes.