We propose novel methods to utilize orthogonal polynomial approximation in higher dimension spaces, which enable us to modify classical differential equation solvers to perform high precision, long-term orbit propagation. These methods have immediate application to efficient propagation of catalogs of Resident Space Objects (RSOs) and improved accounting for the uncertainty in the ephemeris of these objects. More fundamentally, the methodology promises to be of broad utility in solving initial and two point boundary value problems from a wide class of mathematical representations of problems arising in engineering, optimal control, physical sciences and applied mathematics. We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10^−9ms^−2, globally) are required near the Earths surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with both speed and storage efficiency op- timized using radial adaptation. The second class of problems addressed includes orbit propagation and solution of associated boundary value problems. The successive Chebyshev-Picard path approximation method is shown well-suited to solving these problems with over an order of magnitude speedup relative to known methods. Furthermore, the approach is parallel-structured so that it is suited for parallel implementation and further speedups. Used in conjunction with orthogonal Finite Element Model (FEM) gravity approximations, the Chebyshev-Picard path approximation enables truly revolutionary speedups in orbit propagation without accuracy loss.

Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 3 uppsatser.

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 90-94).

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 97-104).

Thesis (M.S)--Ohio University, June, 2005. / Includes bibliographical references (p. 55-56)

Thesis (M.S)--Ohio University, June, 2005. / Title from PDF t.p. Includes bibliographical references (p. 55-56)

A recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industry-standard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initial-value technique is used for its solution by replacing the problem with an asymptotically equivalent first order problem, which is, in turn, solved as an initial value problem by using cubic splines. Numerical examples are provided to show that the method presented provides a fine approximation of the exact solution. The first chapter provides some background material to the cubic spline and boundary value problems. The works of several authors and a comparison of different solution methods are also discussed. Finally, some background into the specific singularly perturbed boundary value problems is introduced. The second chapter contains calculations and derivations necessary for the cubic spline and the initial value technique which are used in the solutions to the boundary value problems. The third chapter contains some worked numerical examples and the numerical data obtained along with most of the tables and figures that describe the solutions. The thesis concludes with some reflections on the results obtained and some discussion of the error bounds on the calculated approximations to the exact solutions for the numeric examples discussed

We prove that the solution of the Boussinesq equation with relatively small initial data exists globally and decays exponentially under some boundary conditions. / Ph. D.

El objetivo de esta tesis doctoral es desarrollar nuevos métodos basados en splines para la resolución de sistemas de ecuaciones diferenciales del tipo Y'(x)=f(x,Y(x)) , a<x<b Y(a)=Y_a (1) donde Y_a, Y(x) son matrices rxq, comenzando con splines de tipo cúbico y con un algoritmo similar al propuesto por Loscalzo y Talbot en el caso escalar [20], intentando poder aumentar el orden del spline, lo que con el método dado en [20] no puede hacerse de forma convergente. Trataremos también de aplicar dicho método al problema Y''(x)=f(x,Y(x),Y'(x)) , a<x<b Y(a)=Y_a Y'(a)=Y_b (2) sin aumentar la dimensión del problema para evitar el sobrecoste computacional. Los métodos presentados se compararán con los existentes en la literatura y serán implementados en algoritmos para ponerlos, debidamente documentados, a disposición de la comunidad científica. / Tung, MM. (2013). Spline approximations for systems of ordinary differential equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/31658 / Premios Extraordinarios de tesis doctorales

This study seeks to develop a method that generalises the use of Adams-Bashforth to solve or treat partial differential equations with local and non-local differentiation by deriving a two-step Adams-Bashforth numerical scheme in Laplace space. The resulting solution is then transformed back into the real space by using the inverse Laplace transform. This is a powerful numerical algorithm for fractional order derivative. The error analysis for the method is studied and presented. The numerical simulations of the method as applied to the four-dimensional model, Caputo-Lu-Chen model and the wave equation are presented. In the analysis, the bifurcation dynamics are discussed and the periodic doubling processes that eventually caused chaotic behaviour (butterfly attractor) are shown. The related graphical simulations that show the existence of fractal structure that is characterised by chaos and usually called strange attractors are provided. For the Caputo-Lu-Chen model, graphical simulations have been realised in both integer and fractional derivative orders. / Mathematical Sciences / M. Sc. (Applied Mathematics)