The rank of projection-algebraic representations of some differential operators

The Lie-algebraic method approximates differential operators that are formal polynomials of ${1,x,frac{d}{dx}}$ 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 $$K=a_k frac{d^k}{dx^k}+a_{k+1}frac{d^{k+1}}{dx^{k+1}}+ldots +a_{k+p}frac{d^{k+p}}{dx^{k+p}}$$ is $n-k$ and conclude that the Lie-algebraic reductions of differential equations allow to approximate only {it 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.