Brian J. McCartin

Theory of exponential splines

Abstract Pruess [12, 14] has shown that exponential splines can produce co-convex and co-monotone interpolants. These results justify the further study of the mathematical properties of exponential splines as they pertain to their utility as numerical approximations. They also warrant the generalization of the exponential spline in fruitful directions. Herein, we present convergence rates and extremal properties for exponential spline approximation, cardinal spline and B-spline bases for the space of exponential splines, and generalizations to higher order tension splines and Hermite tension interpolants.

Eigenstructure of the Equilateral Triangle, Part I: The Dirichlet Problem

Lames formulas for the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.

FDFD analysis of electromagnetic scattering in anisotropic media using unconstrained triangular meshes

A method using finite differences in the frequency domain (FDFD) for modeling field interaction and propagation in anisotropic media with generalized tensor permittivity and permeability, and with complex geometry, is presented. The method uses an unconstrained mesh with triangular cells, which provides an efficient discrete approximation of curved surfaces. The two-dimensional problem with transverse field excitation is studied. Computed results comparing this algorithm to published data for the former case show excellent agreement. The analysis provides information for all view angles simultaneously at a single frequency. The method can be adapted to a time-domain formulation to analyze the effects of wave pulses and multiple frequencies at a single observation angle. >

Eigenstructure of the equilateral triangle, Part II: The Neumann problem

Lames formulas for the eigenvalues and eigenfunctions of the Laplacian with Neumann boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.

Exponential fitting of the delayed recruitment/renewal equation

Abstract From respiratory physiology to laser-based optical devices, the so-called delayed recruitment/renewal equation (1) e d c(t) d t =−c(t)+f(c(t−1)), provides the mathematical model in a diverse spectrum of practical applications. Here, e is inversely proportional to the product of the time-delay inherent in the physical system and its rate of decay. When this time-lag is large relative to the reciprocal of the decay rate, e is small and this delay differential equation (DDE) is singularly perturbed. When this situation obtains, c ( t ) can exhibit initial layers and chaotic oscillations. In order to accurately capture such solution features numerically, one must use an approximation technique tailored to singular perturbation problems. In this work, we develop such a family of exponentially fitted schemes for the numerical approximation of this fundamental DDE. Application of this new technique is then made to a variety of interesting and important problems, not the least of which is the subject of dynamical diseases.

Computation of exponential splines

Pruesss investigations [J. Approx. Theory, 17 (1976), pp. 86–96], [Math. Comp., 33 (1979), pp. 1273–128 1] revealed the shape preservation properties of exponential splines and provided the impetus for further theoretical study of exponential splines [J. Approx. Theory, to appear]. Together, these theoretical results form the backdrop for the detailed analysis of issues in the computation of exponential splines contained herewith. Specifically, first and foremost the construction of tension parameter selection algorithms is considered. The conditioning and iterative solution of the spline equations, as well as the derivation and accuracy of end conditions, are discussed. This inquiry concludes with a potpourri of numerical considerations and the presentation of a variety of numerical examples.

Absorbing Boundary Conditions on Circular and Elliptical Boundaries

Finite difference methods are becoming increasingly popular in the computational electromagnetics community. A major issue in applying these methods to electromagnetic wave scattering is to limit the computational domain to a finite size, which is accomplished by selecting an outer boundary and imposing absorbing boundary conditions to simulate free space. In this paper, the pseudo-differential operator approach is employed to derive absorbing boundary conditions for both circular and elliptical outer boundaries. The pseudo-differential operator approach employed by Engquist and Majda is modified to derive improved absorbing boundary conditions. In the case of the circular outer boundary, the modified pseudo-differential operator approach leads to a condition equivalent to that of Bayliss and Turkels second-order condition. The modified pseudo-differential operator is then used to derive the second-order absorbing boundary condition for the elliptical outer boundary. The effectiveness of the second-order...

A triangular-grid finite-difference time-domain method for electromagnetic scattering problems

A two-dimensional (2-D) finite-difference time-domain (FDTD) method using a triangular grid is introduced for solving electromagnetic scattering problems. The 2-D FDTD method is based on a control region approximation, which is defined by the Dirichlet tessellation of the triangular grid. In general, this discretization scheme is accurate to second-order in time, to first-order in space for non-uniform grids, and to second-order in space for uniform grids. Using triangular grids, arbitrary geometries can be represented by piecewise linear models . In addition, an absorbing boundary condition on a smooth outer boundary, such as a circular boundary, can be implemented. This method is illustrated and verified by calculating scattering from perfectly conducting and coated objects. It is shown that geometrical modeling using a triangular grid is more accurate for electromagnetic scattering problems than those using a rectangular grid, especially when the surface wave is significant.

EIGENSTRUCTURE OF THE EQUILATERAL TRIANGLE. PART III. THE ROBIN PROBLEM

Lames formulas for the eigenvalues and eigenfunctions of the Laplacian on an equilateral triangle under Dirichlet and Neumann boundary conditions are herein extended to the Robin boundary condition. They are shown to form a complete orthonormal system. Various properties of the spectrum and modal functions are explored.

A model-trust region algorithm utilizing a quadratic interpolant

Abstract A new model-trust region algorithm for problems in unconstrained optimization and nonlinear equations utilizing a quadratic interpolant for step selection is presented and analyzed. This is offered as an alternative to the piecewise-linear interpolant employed in the widely used “double dogleg” step selection strategy. After the new step selection algorithm has been presented, we offer a summary, with proofs, of its desirable mathematical properties. Numerical results illustrating the efficacy of this new approach are presented.