Oksana Bihun

Oksana Bihun, Austin Bren, Michael Dyrud et al.

To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the differential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear differential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems.

Oksana Bihun, Mykola Prytula

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.