6 - NUMERICAL TREATMENT OF SINGULAR/ DISCONTINUOUS INITIAL VALUE PROBLEMS

Abstract

This chapter discusses the numerical treatment of singular/discontinuous initial value problems. The mathematical formulation of physical phenomena in simulation, electrical engineering, control theory, and economics often leads to an initial value problem in which there is a pole in the solution or a discontinuous low order derivative. The switching on and off of electrical circuits and the state of the economy of a nation disrupted by an unforeseen disaster are practical examples of such problems. Alternative strategies are based on non-polynomial interpolating functions either by perturbed polynomials or rational functions. The resultant algorithms often behave nicely in the neighborhood of a singularity provided the mesh size is chosen as to sandwich the point of singularity. The snags with such schemes are the limited analysis and theoretical backing. The theory of ordinary nonlinear differential equations offers no clue as to the location and the nature of singularities in the solution of an equation. Hence, singularities have to be detected heuristically. Although the integration formulas based on rational approximations are quite effective in the neighborhood of singularities, the derivation of these formulas, and even the formulas, are rather too involved and complicated. These formulas do not subject themselves to easy analysis and computer applications.