Brian J. McCartin

Learn More
Lamé's 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. 1. Introduction. The eigenstructure(More)
A hybrid numerical method is presented for a linear, first order, hyperbolic partial differential equation (PDE): the nonhomogeneous one-way advection-reaction equation. This PDE is first reduced to an ordinary differential equation (ODE) along a characteristic emanating backward in time from each mesh point. These ODEs are then integrated numerically using(More)
A comprehensive treatment of Rayleigh-Schrödinger perturbation theory for the symmetric matrix eigenvalue problem is furnished with emphasis on the degenerate problem. The treatment is simply based upon theMoore-Penrose pseudoinverse thus distinguishing it from alternative approaches in the literature. In addition to providing a concise matrixtheoretic(More)
We examine the eigenstructure of generalized isosceles triangles and explore the possibilities of analytic solutions to the general eigenvalue problem in other triangles. Starting with work based off of Brian McCartin’s paper on equilateral triangles, we first explore the existence of analytic solutions within the space of all isosceles triangles. We find(More)
Numerical methods based upon angled derivative approximation are presented for a linear first-order system of hyperbolic partial differential equations (PDEs): the hyperbolic heat conduction equations. These equations model the flow of heat in circumstances where the speed of thermal propagation is finite as opposed to the infinite wave speed inherent in(More)