Aspects of interval analysis applied to initial-value problems for ordinary differential equations and hyperbolic partial differential equations

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 SciencesD. Phil. (Mathematics

Spline-Fourier Approximations of Discontinuous Waves

In the Fourier series approximation of real functions discontinuities of the functions or their derivatives cause problems like Gibbs phenomenon or slow uniform convergence. In the case of a finite number of isolated discontinuities the problems can be to a large extend rectified by using periodic splines in the series. This modified Fourier series (Spline-Fourier series) is applied to the numerical solution of the wave equation (in periodic form) where discontinuities in the data functions or their derivatives appear quite often. The solution is sought in the form of a Spline-Fourier series about the space variable and close bounds are obtained using a certain iterative procedure of Newton type

Spline-Fourier Approximations of Discontinuous Waves

: In the Fourier series approximation of real functions discontinuities of the functions or their derivatives cause problems like Gibbs phenomenon or slow uniform convergence. In the case of a finite number of isolated discontinuities the problems can be to a large extend rectified by using periodic splines in the series. This modified Fourier series (Spline-Fourier series) is applied to the numerical solution of the wave equation (in periodic form) where discontinuities in the data functions or their derivatives appear quite often. The solution is sought in the form of a Spline-Fourier series about the space variable and close bounds are obtained using a certain iterative procedure of Newton type. Key Words: Validated numerics, Fourier hyper functoid, Wave equation 1 Introduction This work is inspired by the ideas for validated approximations of function space problems [3] and essentially it is a further application of the Fourier Hyper Functoid described in [2] where the coefficie..