Rank of Projection-Algebraic Representations of Some Differential Operators
This work addresses a fundamental limitation of Lie-algebraic methods for approximating differential operators, which is relevant to researchers in numerical analysis and applied mathematics.
The paper proves that the rank of n-dimensional representations of certain differential operators is n-k, limiting the Lie-algebraic method to approximate only some solutions. It proposes a change of variables to overcome this for boundary value problems and generalizes to multiple dimensions with numerical tests.
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.