Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations

TL;DR

This paper develops a method to expand products of functions using Fourier-Legendre series, enabling semi-analytical solutions to certain nonlinear PDEs with demonstrated efficiency and accuracy.

Contribution

It derives explicit Fourier-Legendre expansions for products of functions and applies these to solve nonlinear PDEs semi-analytically, with proven convergence bounds.

Findings

Effective Fourier-Legendre expansion for function products.

Upper bounds on convergence rates established.

Numerical results show high accuracy and efficiency.

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. In this way, expansions of whole number powers of $f$ may be obtained. 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 an important class of nonlinear PDEs.