Rank of Projection-Algebraic Representations of Some Differential Operators

TL;DR

This paper investigates the rank of finite-dimensional representations of differential operators using Lie-algebraic methods, revealing limitations in approximating solutions and proposing variable transformations to improve boundary value problem solutions.

Contribution

It proves the rank of the n-dimensional representation of certain differential operators is n-k and introduces a method to overcome approximation limitations in boundary value problems.

Findings

Rank of the operator representation is n-k.

Lie-algebraic reductions only approximate some solutions.

Variable changes can improve boundary value problem solutions.

Abstract

The Lie-algebraic method approximates differential operators that are formal polynomials of {1,x,d/dx} by linear operators acting on a finite dimensional space of polynomials. In this paper we prove that the rank of the n-dimensional representation of the operator K=a_k d^k/dx^k+a_{k+1} d^{k+1}/dx^{k+1}+... +a_{k+p} d^{k+p}/dx^{k+p} is n-k and conclude that the Lie-algebraic reductions of differential equations allow to approximate only some of solutions of the differential equation K[u]=f. We show how to circumvent this obstacle when solving boundary value problems by making an appropriate change of variables. We generalize our results to the case of several dimensions and illustrate them with numerical tests.