Numerical Methods for Smooth Solutions of Ordinary Differential Equations
Vu, Thu Van
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https://hdl.handle.net/2142/69518
Description
Title
Numerical Methods for Smooth Solutions of Ordinary Differential Equations
Author(s)
Vu, Thu Van
Issue Date
1983
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Computer Science
Abstract
In parameter estimation, the coefficient parameters of a system of differential equations are often determined by error norm minimization. Minimization codes may require that partial derivatives with respect to parameters be evaluated. If the derivatives are obtained by differencing the numerical solution of the differential equations, the smoothness of that solution with respect to parameter changes is crucial to the performance of minimization codes.
This thesis deals with the smoothness of the numerical solution of ordinary differential equations with respect to parameter variations. Numerical methods and techniques are developed to attain smooth solutions along mesh trajectories and at a fixed point in time. The methods are based on modifications of the Runge-Kutta methods and the Taylor's series methods. Error bounds for smooth one-step methods are also obtained. These bounds assure the same order of convergence in the derivatives as in the solution. A smooth, automatic integrator named RKTSOL is written in FORTRAN 77 to implement the methods discussed. Limited numerical results indicate that this code outperforms nonsmooth codes such as RKF45 and LSODE in computing numerical solutions for approximations to partial derivatives with respect to parameters.
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