The acceleration technique known as Iterated Defect Correction (IDeC) for the numerical solution of singular initial value problems is investigated. IDeC based on the implicit Euler method performs satisfactorily and can thus be used for the eÆcient solution of singular boundary value problems with the shooting method. Higher order one-step methods like the box scheme or the trapezoidal rule ca...

Symmetry analysis is a powerful tool that enables the user to construct exact solutions of a given differential equation in a fairly systematic way. For this reason, the Lie point symmetry groups of most well-known differential equations have been catalogued. It is widely believed that the set of symmetries of an initial-value problem (or boundary-value problem) is a subset of the set of symmet...

Diagonal Padd approximations to the time evolution operator for initial value problems are applied in a novel way to the numerical solution of these problems by explicitly factoring the polynomials of the approximation. A remarkable gain over conventional methods in efficiency and accuracy of solution is obtained.

The (Cauchy) initial value formulation of General Relativity is developed, and the maximal vacuum Cauchy development theorem is reviewed. There is a short discussion of the evolution equations and associated guage choices, and global results are mentioned briefly in the conclusion.

Multivariate linear recurrences appear in such diverse elds of mathematics as combinatorics, probability theory, and numerical resolution of partial diierential equations. Whereas in the univariate case the solution of a constant-coeecient recurrence always has a rational generating function, this is no longer true in the multivariate case where this generating function can be non-rational, non...

Finding the exact solution of initial value problems is the main topic of this thesis. Because differential equations in practice can be hard to solve, we require algorithms to solve them on a computer. Since the solution of the equation needs to be stored on a computer, we first discuss the representation of real functions. We show that approximating a function by polynomials gives a suitable ...

This paper reviews various aspects of stiffness in the numerical solution of initial-value problems for systems of ordinary differential equations. In the literature on numerical methods for solving initial value problems the term "stiff" has been used by various authors with quite different meanings, which often causes confusion. This paper attempts to clear up this confusion by reviewing some...