Legendre expansions of products of functions with applications to nonlinear partial differential equations

Abstract

Given the Fourier–Legendre expansions of f and g, and mild conditions on f and g, we derive the Fourier–Legendre expansion of their product in terms of their corresponding Fourier–Legendre coefficients. We establish upper bounds on rates of convergence. We then employ these expansions to solve semi-analytically a class of nonlinear PDEs with a polynomial nonlinearity of degree 2. The obtained numerical results illustrate the efficiency and performance accuracy of this Fourier–Legendre-based solution methodology for solving a class of nonlinear PDEs.