Internal 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. With regard to IVP for ODE, the central point of discussion is the wrapping effect. Anew concept of wrapping function is introduced ad 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 effects are characterised as problems for which the wrapping function equals the optimal interval enclosure. A sufficient condition for no 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. With regard to 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 montone 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 discontinuous waves 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.
|Subject||Mathematics, Mathematical statistics and Statistics|
|Subject 2||Mathematics, Mathematical statistics and Statistics|
|Degree Type||Doctoral degree|