  Starting with some fundamental concepts, in this article we present the
essential aspects of spectral methods and their applications to the numerical
solution of Partial Differential Equations (PDEs). We start by using Lagrange
and Techbychef orthogonal polynomials for spatiotemporal approximation of PDEs
as a weighted sum of polynomials. We use collocation at some clustered grid
points to generate a system of equations to approximate the weights for the
polynomials. We finish the study by demonstrating approximate solutions of some
PDEs in one space dimension.
