Spline techniques for numerical solution of singular perturbation boundary value problems

Abstract

Many fields of applications in real life for example fluid mechanics chemical reactions control theory aerodynamics hydrodynamics lubrication theory and electrical networks give rise to differential equation which exhibit a phenomena called singular perturbation Singularly perturbed problems are those in which the highest derivative term is multiplied by small positive parameter and the problems depend on the parameter in such a way that the solutions behave non uniformly as the parameter tend to zero As the parameter is set to zero the reduced differential equation is of lower order than the original differential equation In cases where the problem has a limiting solutions as parameter tend to zero which solves the reduced equation regions of non uniform convergence boundary layers region near the boundary where the solution change rapidly can be expected in general near the boundary due to the loss of boundary conditions in the limiting solution Then a regular perturbation procedure fails and we say that the problem at hand is a singular perturbation problem newline Numerical analysis and asymptotic analysis are two principal approaches for solving singular perturbation problems Since the goals and the problem classes are rather different there has not been much interaction between these approaches Numerical analysis tries to provide quantitative information about a particular problem whereas asymptotic analysis tries to gain insight into the qualitative behaviour of a family of problems and only semi quantitative information about any particular member of the family Numerical methods are intended for broad classes of problems and are intended to minimize demands upon the problem solver Asymptotic methods treat comparatively restricted classes of problems and require the problem solver to have some understanding of the behaviour of the solution newline newline