Paul B. Bailey

Automatic Solution of the Sturm-Liouville Problem

A general purpose algorithm is described for the automatic computation of eigenvalues and eigenfunctions of Sturm-Llouvllle problems, both singular and nonsmgular The method is based on the solution of initial value problems, using an integrator with a built-in global error estimating capabihty Problems arising in the construction of a satisfactory algorithm for this task are described, as are some of the software engineering aspects The resulting algorithm has performed satisfactorily on a sizable list of test problems

Computing Eigenvalues of Singular Sturm-Liouville Problems

We describe a new algorithm to compute the eigenvalues of singular Sturm-Liouville problems with separated self-adjoint boundary conditions for both the limit-circle nonoscillatory and oscillatory cases. Also described is a numerical code implementing this algorithm and how it compares with SLEIGN. The latter is the only effective general purpose software available for the computation of the eigenvalues of singular Sturm-Liouville problems.

Regular approximations of singular Sturm-Liouville problems

Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {Sr{ of regular S-L problems with the properties(i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {Sr{(ii) in the case when 5 is regular or limit-circle at each endpoint, a convergent sequence of eigenvalues from the individual members of {Sr{ has to converge to an eigenvalue of S(iii) in the general case when S is bounded below, property (ii) holds for all eigenvalues below the essential spectrum of S.

On the local and global behavior of acceleration waves

where t denotes the time. Formula (1.1) is the well-known Bernoulli equation. In general, the coefficients/~ and fl are functions of t. A careful examination of these coefficients reveals that there are two common features. Briefly, (1) the coefficient # depends on the material under consideration and the consequences of the physical assumptions introduced regarding the conditions of the material ahead of the waves, (2) the coefficient fl depends on the elastic response of the material alone. In other words, the coefficient # depends on whether we are considering acceleration waves propagating in, e.g., materials with memory, elastic nonconductors of heat, or inhomogeneous elastic materials, and whether we assume that the material regions ahead of the waves are at rest in homogeneous configurations, in mechanical equilibrium, or in dynamic equilibrium. On the other hand, the coefficient fl depends on the instantaneous elastic response of materials with memory, the isentropic elastic response of elastic non-conductors of heat, or the local elastic response of inhomogeneous elastic materials. In this paper, we shall show that knowing the sign of fl(t) not only allows us to determine the local behavior of amplitudes f rom (1.1) but it also tells us a great deal about their global behavior. Some of our conclusions on the global behavior would not be expected f rom the local behavior alone.

Eigenvalue and eigenfunction computations for Sturm-Liouville problems

SLEIGN is a software package for the computation of eigenvalues and eigenfunction:s of regular and singular Sturm Liouville boundary value problems, The package is a modification and extension of a code with the same name developed by Bailey, Gordon, and Shampinej which is described in ACM Z’OMS 4 (1978), 193-208. The modifications and extensions include (1) a restructuring of the FORTRAN program, (2) the coverage of problems with semidefi nite weight functions, and (3) the coverage of problems with indefinite weight functions.

A rigorous derivation of some invariant imbedding equations of transport theory

The invariant imbedding equations are obtained rigorously from the Boltzmann formulations for cases of timedependent transport in a rod, steady- state transport in a slab, and steady-state transport in a hollow sphere, a solid sphere, and a spherical shell enclosing a perfect absorber. (M.J.T.)

A correction to some invariant imbedding equations of transport theory obtained by “particle counting”☆

Comparison of a rigorously derived invariant imbedding equation for transport in a sphere with one obtained by particle-counting methods shows that a term is missing from the latter. This error, due to a geometrical effect, is corrected by a more careful application of the particlecounting technique. (M.J.T.)

A slightly modified Prüfer transformation useful for calculating Sturm-Liouville eigenvalues☆

Examples are given of calculations made using the Pruefer transformation into which a positive constant has been inserted. (AIP)

Algorithm 700: A Fortran software package for Sturm–Liouville problems

The package comprises two user-visible subprograms, sleign and zcount, and a set of special purpose subprograms invoked from them; the integration of the differential equations is done by the GERK software of Shampine and Watts [31 included with the package. zcount is used in the initial step for the semidefinite problem [1] to determine the translated index of the eigenvalue to be calculated by sleign.

Existence and uniqueness of solutions of the second order boundary value problem

Thus il b — a satisfies this inequality, then the iteration procedure converges to a function y(t) which is a solution to the boundary value problem, and there is no other solution. Although this convergence question has been investigated by a number of people over many years (see [ l ] [8 ] , for example), the maximum interval for which Picards iteration procedure converges is still not known. And even if it were, that fact alone would not necessarily tell us anything about the maximum interval for which the boundary value problem has a unique solution, other than that it provides a lower bound. Thus Picards method, though extremely useful for a wide class of problems, does have the one serious limitation of being applicable to only those problems for which the iteration procedure happens to converge.