AN EFFICIENT SPECTRAL METHOD FOR ORDINARY DIFFERENTIAL-EQUATIONS WITHRATIONAL-FUNCTION COEFFICIENTS

We present some relations that allow the efficient approximate inversi on of linear differential operators with rational function coefficient s. We employ expansions in terms of a large class of orthogonal polyno mial families, including all the classical orthogonal polynomials. The se families obey a simple 3-term recurrence relation for differentiati on, which implies that on an appropriately restricted domain the diffe rentiation operator has a unique banded inverse. The inverse is an int egration operator for the family, and it is simply the tridiagonal coe fficient matrix for tile recurrence. Since in these families convoluti on operators (i.e., matrix representations of multiplication by a func tion) are banded for polynomials, we are able to obtain a banded repre sentation for linear differential operators with rational coefficients . This leads to a method of solution of initial or boundary value prob lems that, besides having an operation count that scales linearly with tile order of truncation N, is computationally well conditioned. Amon g the applications considered is the use of rational maps for the reso lution of sharp interior layers.