Solving a Differential Equation by a Spectral Method

Abstract

Spectral methods are approximation techniques for the computation of the solutions to ordinary and partial differential equations. They are based on a polynomial expansion of the solution. The precision of these methods is limited only by the regularity of the solution, in contrast to the finite difference method and the finite element methods. The approximation is based primarily on the variational formulation of the continuous problem. The test functions are polynomials and the integrals involved in the formulation are computed by suitable quadrature formulas. This project proposes to implement a spectral method to solve the following boundary value problem defined on the interval Ω = (−1, 1):

$$ \left\{ \begin{gathered} - u'' + cu = f, \hfill \\ u\left( { - 1} \right) = 0, \hfill \\ u\left( 1 \right) = 0, \hfill \\ \end{gathered} \right. $$

((5.1))

with f ∈ L ² (Ω) and c a positive real number.