Recently , the spectral and pseudo spectral methods play an important role in determining the approximate solutions of partial differential equations(PDEs) for various real problems , such as dynamic , wave movement , weather prediction and turbulence modeling . These methods are of more accuracy than the other existing methods. Spectral methods are a In this thesis , after dealing with elementary concepts and explain the attribute of Chebyshev and Legendre polynomials and a brief history of alternating direction implicit and spectral collocation methods , we discuss the Crank-Nicolson and Laplace modified alternating direction implicit Legandre and Chebyshev spectral collocation methods for linear , variable coefficient , parabolic initial-boundary value problem on a rectangular domain with solution subject to non-zero Dirichlet boundary condition. The discretization of problems by the above methods yields matrices with banded structures . Also the system of equations obtained from these methods can be solved by using the well-known method of preconditioned BICGSTAB. Then the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation is given . The numerical experiments verify the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems. The Burgers' equation is a useful model for many interesting physical problems such as traffic , shock wave , and turbulence problems . It is one of a few well-known nonlinear PDEs , which has been solved analytically for restricted set of arbitrary initial conditions . Hopf-Cole transformation is a poweful analytical tool for Burgers' equation for getting various exact solutions . In the sequel , the system of two-dimensional Burgers' equations are solved by the Chebyshev spectral collocation method . Two new method are used , first one is based on the two-dimensional Hopf-Cole transformation , which transforms the system of two-dimensional Burgers' equations into a linear heat equation , and the second one is based on linearization .