Steven Pruess

Mathematical software for Sturm-Liouville problems

Software is described for the Sturm-Liouville eigenproblem. Eigenvalues, eigenfunctions, and spectral density functions can be estimated with global error control. The method of approximating the coefficients forms the mathematical basis. The underlying algorithms are briefly described, and several examples are presented.

Properties of splines in tension

Abstract Rigorous arguments are given establishing convergence rates and asymptotic behavior of interpolatory tension splines with variable tension. It is shown that for sufficiently smooth data, convergence is O ( h 4 ) for uniformly bounded tension parameters. For large tension parameters the tension spline is essentially locally linear and applications of this fact are given which allow one to construct convex or monotone approximants.

Estimating the Eigenvalues of Sturm–Liouville Problems by Approximating the Differential Equation

This paper is concerned with computing accurate approximations to the eigenvalues and eigenfunctions of regular Sturm–Liouville differential equations. The method consists of replacing the coefficient functions of the given problem by piecewise polynomial functions and then solving the resulting simplified problem. Error estimates in terms of the approximate solutions are established and numerical results are displayed. Since the asymptotic properties for Sturm–Liouville systems are preserved by the approximation, the relative error in the higher eigenvalues is much more uniform than is the case for finite difference or Rayleigh–Ritz methods.

A rate-based model for the design of gas absorbers for the removal of CO2 and H2S using aqueous solutions of MEA and DEA

A rate-based model was developed for the design of acid gas absorbers using aqueous alkanolamine solutions. The model adopts the film theory and assumes that thermodynamic equilibrium among the reacting species exists in the bulk liquid. The diffusion-reaction equations for the reacting species in the liquid film are solved using collocation techniques. Heat effects accompanying diffusion and reaction are accounted for using appropriate heat balances on each tray. The algorithm adopts a plate-by-plate calculation starting at the bottom of the tower. Tray hydraulics was added to the algorithm to ensure proper operation of the tower. The program was developed to handle either monoethanolamine (MEA) or diethanolamine (DEA) as chemical solvents.

Alternatives to the exponential spline in tension

A general setting is given for smooth interpolating splines depending on a parameter such that as this parameter approaches infinity the spline converges to the piecewise linear interpolant. The theory includes the standard exponential spline in tension, a rational spline, and several cubic splines. An algorithm is given for one of the cubics; the parameter for this example controls the spacing of new knots which are introduced.

Solving linear boundary value problems by approximating the coefficients

A method for solving linear boundary value problems is described which consists of approximating the coefficients of the differential operator. Error estimates for the ap- proximate solutions are established and improved results are given for the case of ap- proximation by piecewise polynomial functions. For the latter approximations, the resulting problem can be solved by Taylor series techniques and several examples of this are given.

Some Remarks on the Numerical Estimation of Fractal Dimension

Fractal geometry is currently of major interest in many fields. A commonly occurring problem involves determining the fractal dimension. Hausdorff (1919) rigorously defined the concept of fractional dimension, and further theoretical work was done by Besicovitch and others (see Falconer, 1985 or 1990, for an extensive list of references). Few applications were made using the concept of fractals until the early 1970s when Mandelbrot began his work. In the past ten years, many researchers (e.g., Grassberger, 1983; Barton and Larson, 1985; Sreenivasan and Meneveau, 1986; and Hunt and Sullivan, 1989) from a wide variety of disciplines have devised algorithms for estimating fractal dimensions.

Interpolation schemes for collocation solutions of two point boundary value problems

Several interpolatory approximations for collocation solutions to systems of two point boundary value problems are studied. These interpolate superconvergent values are produced by the usual Gauss–Legendre collocation algorithm. Comparisons are made to the collocation solution in terms of accuracy, storage requirements and number of operations; numerical examples are given which illustrate typical behavior.

Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions

Abstract In this paper we consider the cooling of a cylindrical rod which is dropped lengthwise into a rectangular container containing a finite amount of liquid. The separation of variables gives Bessels equation in the radial variable with the separation parameter also appearing in the boundary condition at the regular end-point because of the interface condition requiring conservation of heat. We consider here also the case of a rod with nonuniform mass density and show that the methods of Pruess [SIAM J. Numer. Anal., 10 (1973), 55–68; Numer. Math., 24 (1975), 241–247] can be adapted to yield a numerical algorithm for the eigenvalues of the associated Bessel-like equation. The basic expansion theory for the eigenvalue problem with eigenparameter in the boundary conditions has been given in the regular case by Fulton [Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), 293–308] and Walter [Math. Z., 133 (1973), 301–312], and for singular cases which include the present problem by Fulton. The present problem, however, is sufficiently special that the self-adjoint operator arising in the separated problem can be realized as the infinitesimal generator of a contraction semigroup in terms of which the Cauchy initial value problem associated with the heat conduction problem can be solved.

The computation of spectral density functions for singular Sturm-Liouville problems involving simple continuous spectra

The software package SLEDGE has as one of its options the estimation of spectral density functions p(t) for a wide class of singular Strurm-Liouville problems. In this article the underlaying theory and implementation issues are discussed. Several examples exhibiting quite varied asymptotic behavior in p are presented.