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arXiv · 1011.3782

Rank of Projection-Algebraic Representations of Some Differential Operators

Oksana Bihun·Mykola Prytula
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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.

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Oksana Bihun, Mykola Prytula. 2010-11-16. Rank of Projection-Algebraic Representations of Some Differential Operators. https://arxiv.org/abs/1011.3782

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