Browsing by Subject "Differential equations."
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Item Adaptive methods for the Helmholtz equation with discontinuous coefficients at an interface.(2008-03-03T17:35:23Z) Rogers, James W., Jr.; Sheng, Qin.; Mathematics.; Baylor University. Dept. of Mathematics.In this dissertation, we explore highly efficient and accurate finite difference methods for the numerical solution of variable coefficient partial differential equations arising in electromagnetic wave applications. We are particularly interested in the Helmholtz equation due to its importance in laser beam propagation simulations. The single lens environments we consider involve physical domains with subregions of differing indices of refraction. Coefficient values possess jump discontinuities at the interface between subregions. We construct novel numerical solution methods that avoid computational instability and maintain high accuracy near the interface. The first class of difference methods developed transforms the differential equation problem to a new boundary value problem for which a numerical solution can be readily computed on rectangular subregions with constant wavenumbers. The second class of numerical methods implemented combines adaptive domain transformation with coefficient smoothing to yield a boundary value problem well-suited for numerical solution on a uniform grid in the computational space. The resulting finite difference schemes do not have treat the grid points near the interface as a special case. A novel matrix analysis technique is implemented to examine the stability of these new methods. Computational verifications are carried out.Item Comparison of smallest eigenvalues and extremal points for third and fourth order three point boundary value problems.(2011-09-14) Neugebauer, Jeffrey T.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.The theory of u₀-positive operators with respect to a cone in a Banach space is applied to the linear differential equations u⁽⁴⁾ + λ₁p(x)u = 0 and u⁽⁴⁾ + λ₂q(x)u = 0, 0 ≤ x ≤ 1, with each satisfying the boundary conditions u(0) = u′(r) = u″(r) = u‴(1) = 0, 0 < r < 1. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained. These results are then extended to the nth order problem using two different methods. One method involves finding the Green's function for –u⁽ⁿ⁾ = 0 satisfying the higher order boundary conditions, and the other involves making a substitution that allows us to work with a variation of the fourth order problem. Extremal points via Krein-Rutman theory are then found. Analogous results are then obtained for the eigenvalue problems u‴ + λ₁p(x)u = 0 and u‴ + λ₂q(x)u = 0, with each satisfying u(0) = u′(r) = u″(1) = 0, 0 < 1/2 < r < 1.Item Existence of positive solutions to singular right focal boundary value problems.(Orlando, FL : International Publications., 2005-05) Maroun, Mariette.; Henderson, Johnny.; Mathematics.; Baylor University. Dept. of Mathematics.In this dissetation, we seek positive solutions for the n^th order ordinary differential equation, y^(n)=f(x,y), satisfying the right focal boundary conditions, y^(i)(0)=y^(n-2)(p)=y^(n-1)(1)=0, i=0,...,n-3, where p is a fixed value between 1/2 and 1, and where f(x,y) has certain singulrities at x=0, at y=0, and possibly at y=infinity.