Spatiotemporal Orthogonal Polynomial Approximation for Partial   Differential Equations
  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. 
Samir Kumar Bhowmik and Sharanjeet Dhawan
math.NA