Linear Two-point Boundary-value Problem with Polynomial Coefficients

Greg Fee

We consider a defect control algorithm for computing an
approximate solution to a linear, second-order,
ordinary differential equation with polynomial coefficients.
As the boundary conditions are given at two points,
 we have a two-point boundary-value problem.
We express the second derivative as a translated Legendre
polynomial with
unknown coefficients and integrate twice to find an approximate solution.
We substitute the polynomial into the differential equation to compute the
 defect and
 express this defect in an orthogonal polynomial basis,
using translated Legendre polynomials. Then we equate the low-order terms
to zero to generate a banded system of linear equations that we solve for
 the unknown
 coefficients. In some cases we are able to approximate the solution over
the entire interval with a relative error less than $10^{-100}$.