Interval analysis is an essential tool in the construction of validated numerical solutions
of Initial Value Problems (IVP) for Ordinary (ODE) and Partial (PDE) Differential
Equations. A validated solution typically consists of guaranteed lower and upper bounds
for the exact solution or set of exact solutions in the case of uncertain data, i.e. it is an
interval function (enclosure) containing all solutions of the problem.
IVP for ODE: The central point of discussion is the wrapping effect. A new concept of
wrapping function is introduced and applied in studying this effect. It is proved that the
wrapping function is the limit of the enclosures produced by any method of certain type
(propagate and wrap type). Then, the wrapping effect can be quantified as the difference
between the wrapping function and the optimal interval enclosure of the solution set (or
some norm of it). The problems with no wrapping effect are characterized as problems for
which the wrapping function equals the optimal interval enclosure. A sufficient condition
for no wrapping effect is that there exist a linear transformation, preserving the intervals,
which reduces the right-hand side of the system of ODE to a quasi-isotone function. This
condition is also necessary for linear problems and "near" necessary in the general case.
Hyperbolic PDE: The Initial Value Problem with periodic boundary conditions for
the wave equation is considered. It is proved that under certain conditions the problem
is an operator equation with an operator of monotone type. Using the established monotone
properties, an interval (validated) method for numerical solution of the problem is
proposed. The solution is obtained step by step in the time dimension as a Fourier series
of the space variable and a polynomial of the time variable. The numerical implementation
involves computations in Fourier and Taylor functoids. Propagation of discontinuo~swaves
is a serious problem when a Fourier series is used (Gibbs phenomenon, etc.). We
propose the combined use of periodic splines and Fourier series for representing discontinuous
functions and a method for propagating discontinuous waves. The numerical implementation involves computations in a Fourier hyper functoid. / Mathematical Sciences / D. Phil. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:uir.unisa.ac.za:10500/17723 |
| Date | 09 1900 |
| Creators | Anguelov, Roumen Anguelov |
| Contributors | Bishop, N. T., Wright, C. J. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 1 online resource (iii, 153 leaves) |
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