Solution of Sturm–Liouville Problems Using Modified Neumann Schemes

Abstract

Abstract. The main purpose of this paper is to describe the extension of the successful modified integral series methods for Schrödinger problems to more general Sturm-Liouville eigenvalue problems. We present a robust and reliable modified Neumann method which can handle a wide variety of problems. This modified Neumann method is closely related to the second-order Pruess method, but provides for higher order approximations. We show that the method can be successfully implemented in a competitive automatic general-purpose software package.

Solving Sturm–Liouville problems by piecewise perturbation methods, revisited

“…In [16] we constructed some higher order methods by including more Neumann terms. Each such Neumann term corresponds to a CPM correction term.…”

“…In [16], practical Neumann algorithms were constructed by truncating the Neumann series and applying a Filon-Legendre approach to approximate the oscillating integrals in the modified scheme (70). The Filon method, in fact, fits a polynomial to the nonoscillatory part of the integrand and solves the resulting integral analytically.…”

“…This scheme is theoretically equivalent to the Neumann method with one integral term and first degree approximating polynomials (see [16]), and is thus also fourth order. Note that although the procedure is a bit complicated, surprisingly simple formulae are obtained in the end.…”

“…This means that we write the coefficient functions as P (x) = 1/p(x) =P + ∆ P (x), q(x) =q + ∆ q (x) and w(x) =w + ∆ w (x), and we define Z(δ) =P (q − Ew)δ 2 . Note that, as in [16], we replace the P function by a piecewise polynomial rather than its inverse p. This ensures that the perturbation corrections will have a closed analytic form.…”

“…In [16] we described how a modified Neumann series method can be applied directly to the general SL problem (1). This modified Neumann method forms the first higher-order CA method for SL equations which are not necessarily in the Schrödinger form.…”