Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Rabia Djellouli, David Klein, Matthew Levy

TL;DR

The paper develops a rigorous Fourier–Legendre product formula that expresses the Legendre coefficients of the product $f\cdot g$ in terms of the coefficients of $f$ and $g$ and a explicit combinatorial factor. It establishes convergence rates that tie the decay of coefficients to the smoothness of $f$ and $g$, and provides uniform convergence bounds for the product series. The theory is then applied to transform nonlinear PDEs with quadratic nonlinearities into nonlinear ODE systems for Legendre coefficients, enabling semi-analytical solutions with high accuracy from a small number of modes. Numerical experiments illustrate the method’s effectiveness, including accurate recovery of coefficients and rapid convergence of truncated Legendre expansions, with potential applications to diffusion-dominated PDEs and climate-model type problems.

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.

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

TL;DR

The paper develops a rigorous Fourier–Legendre product formula that expresses the Legendre coefficients of the product in terms of the coefficients of and and a explicit combinatorial factor. It establishes convergence rates that tie the decay of coefficients to the smoothness of and , and provides uniform convergence bounds for the product series. The theory is then applied to transform nonlinear PDEs with quadratic nonlinearities into nonlinear ODE systems for Legendre coefficients, enabling semi-analytical solutions with high accuracy from a small number of modes. Numerical experiments illustrate the method’s effectiveness, including accurate recovery of coefficients and rapid convergence of truncated Legendre expansions, with potential applications to diffusion-dominated PDEs and climate-model type problems.

Abstract

Given the Fourier-Legendre expansions of and , and mild conditions on and , 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 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.

Paper Structure

This paper contains 10 sections, 13 theorems, 85 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Product Theorem
  3. Rates of Convergence
  4. Application
  5. Model Prototype
  6. Solution Methodology
  7. Illustrative Numerical Results
  8. Concluding Remarks
  9. Exact and Computed Legendre Coefficients Numerical Values
  10. Conditions in Theorem \ref{['L2']} are Sufficient Conditions: An illustrative Example

Key Result

Lemma 1

For any integers $j,k,\ell$ satisfying $0\leq j\leq\min(k-\ell,\ell)\leq\ell\leq k$, where $A_{jk\ell}$ is given by Eq.ajkl.

Figures (8)

  • Figure 1: Error bound according to Corollary \ref{['coeffcor']} for $\mu_2$ for values of $M$ on the horizontal axes. The vertical axis for the graph on the left is the right side of Inequality \ref{['j=1,k=2']}, and the vertical axis on the right is the logarithm of the right side of Inequality \ref{['j=1,k=2']}.
  • Figure 2: Error bound according to Corollary \ref{['coeffcorj=2']} for $\mu_2$ for values of $M$ on the horizontal axes. The vertical axis for the graph on the left is the right side of Inequality \ref{['j=2,k=2']}, and the vertical axis on the right is the logarithm of the right side of Inequality \ref{['j=2,k=2']}.
  • Figure 3: Legendre coefficient values as functions of time $a_n(t)$: exact vs computed
  • Figure 4: Exact solution of IBVP given by Eq.\ref{['solutionT']} vs. Truncated computed series given by Eq.\ref{['TN_eq']}. Sensitivity to the sum truncation $N'$ at various times
  • Figure 5: Sensitivity of the relative error given by Eq.\ref{['relerror']} to the truncation number $N'$ at various times.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • ...and 16 more