Numerical techniques for singularly perturbed boundary value problems

Abstract

Perturbation theory was first proposed for the solution of problems in celestial mechanics in the context of the motions of planets in the solar system Since the planets are very remote from each other and since their mass is small as compared to the mass of the Sun the gravitational forces between the planets can be neglected and the planetary motion is considered to a first approximation as taking place along Keplers orbits which are defined by the equations of the two body problem the two bodies being the planet and the Sun newlineSince astronomic data came to be known with much greater accuracy it became necessary to consider how the motion of a planet around the Sun is affected by other planets This was the origin of the three body problem thus in studying the system Moon Earth Sun the mass ratio between the Moon and the Earth was chosen as the small parameter Lagrange and Laplace were the first to advance the view that the constants which describe the motion of a planet around the Sun are perturbed as it were by the motion of other planets and vary as a function of time hence the name perturbation theory During the last few years much progress has been made in the theory and computer implementation of the numerical treatment of singular perturbation problems A singular perturbation problem is well defined as one in which no single asymptotic expansion is uniformly valid throughout the interval, as the perturbation parameter The singular perturbation problems find place in many area of engineering and applied mathematics for instance fluid mechanics fluid dynamics elasticity aerodynamics plasma dynamics magneto hydrodynamics rarefied gas dynamics Oceanography and other domains of fluid motion A few notable examples are boundary layer problems WKB problems the modeling of steady and unsteady viscous flow problems with large Reynolds numbers and convective heat transport problems with large Peclet numbers Singular perturbation is a field of increasing interest to applied mathematicians newline