Conformal Maps to Multiply-Slit Domains and Applications

By exploiting conformal maps to vertically slit regions in the complex plane, a recently developed rational spectral method [Tee and Trefethen, 2006] is able to solve PDEs with interior layer-like behaviour using significantly fewer collocation points than traditional spectral methods. The conformal maps are chosen to 'enlarge the region of analyticity' in the solution: an idea which can be extended to other numerical methods based upon global polynomial interpolation. Here we show how such maps can be rapidly computed in both periodic and nonperiodic geometries, and apply them to some challenging differential equations

Existence and Uniqueness of Mild Solutions for the Damped Burgers Equation in Weighted Sobolev Spaces on the Half Line

This paper addresses an initial boundary value problem for the damped Burgers equation in weighted Sobolev spaces on half line. First, it introduces two normed spaces and present relations between them, which in turn enables us to analysis the existence and uniqueness of a local mild solution and of a global strong solution in these weighted spaces. The paper also studies the well-posedness of this equation in a semi-infinite interval

Numerical and Analytical Studies of Electromagnetic Waves: Hermite Methods, Supercontinuum Generation, and Multiple Poles in the SEM

The dissertation consists of three parts: Hermite methods, scattering from a lossless sphere, and analysis of supercontinuum generation. Hermite methods are a new class of arbitrary order algorithms to solve partial differential equations (PDE). In the first chapter, we discuss the fundamentals of Hermite methods in great detail. Hermite interpolation is discussed as well as the different time evolution schemes including Hermite-Taylor and Hermite-Runge-Kutta schemes. Further, an order adaptive Hermite method for initial value problems is described. Analytical studies and numerical simulations in both 1D and 2D are presented. To handle geometry, a hybrid Hermite discontinuous Galerkin method is introduced. A discontinuous Galerkin method is used next to the boundaries to handle the geometry and the boundary conditions, while a Hermite method is used in the interior of the computation domain to enhance the performance. Numerical simulations of 1D wave propagation and the solutions to 2D Maxwell\u27s TM equations are presented along with performance and accuracy data. In the second chapter, we study the scattering problem concerning the scattering poles from a lossless sphere for both acoustic and electromagnetic waves. We show that in certain cases there exist only first order scattering poles, but in some other cases, arbitrary order scattering poles can be found by imposing certain lossless impedance boundary conditions on the spherical scatterer. A method to construct arbitrary order scattering poles is discussed. The impedance loading function is required to satisfy Foster\u27s theorem so that the scattering problem is lossless. In the last chapter, we analyse the generation of supercontinua in photonic crystal fibers. We depart from the commonly used approach where a Taylor series expansion of the propagation constant is used to model the dispersive properties in a generalized nonlinear Schrodinger equation (gNLSE). Instead, we develop a mathematical model starting from numerically calculated group velocity dispersion (GVD) curves. Then, we construct a certain function over a broad frequency window and integrate the gNLSE in a way so that the spectral dependence of the propagation constant is preserved. We found that the generation of broadband supercontinua in air-silica microstructured fibers results from a delicate balance of dispersion and nonlinearity. Numerical simulations show that if the nonlinear self-steepening is strong enough, the model produces a shock that is not arrested by dispersion, whereas for weaker nonlinearity the pulse propagates the full extent of the fiber with the generation of a supercontinuum