### Abstract:

This thesis consists of four distinct but complementary chapters. The first chapter contains, on the one hand, some to reminders on Sobolev spaces and polynomial spaces with their associated norms, which constitute the framework of the numerical analysis of spectral methods and, on the other hand, it devotes to the reminders on orthogonal polynomials without and with weights and their main properties, and introduces some associated transformations where orthogonal polynomials are widely used in approximation theory. The second chapter exposes in a clear and elegant the forms of Gaussian quadrature, Gauss-Lobatto quadrature formula and weighted formula and the study of some estimates. These quadrature formulas are considered to constriction of numerical quadrature formulas of high precision. In the third chapter is the study of the approximation of the differential problem by the spectral method, wich is a technique for approximating solutions of partial differential equations. Their main characteristics are that the discrete solutions are sought in high degree polynomial spaces. The discretization is then performed by replacing the space of test functions by a space of polynomials and by calculating the integrals by means of appropriate quadrature formulas. The quadrature formula used is the Gauss-Lobatto-Legendre formula applied in the x direction. This method is equivalent to a collocation method. We see that the error decreases exponentially when the discretization parameter N becomes large. The purpose of the fourth chapter is to develop a spectral method to solve the given problem using the matrixs; to convert the differential problem in a system of ordinary differential equations easier to solve or a linear system; to explain how the discrete problem identified in the Chapter Three can be implemented on computer; to study the behaviour of condition numbers; to solve the linear system and give a graphic illustration in the case of explicit and implicit solutions. Programming is done by Maple.