An ?- uniform method for a boundary value problem on equidistant meshes

In this thesis we propose a fully discrete ? ? uniform ? nite difference method onan equidistant mesh for a singularly perturbed two-point boundary value problem.We start with a ? tted operator method re ? ecting the singular perturbation natureof the problem through a local boundary value problem. However, to solve thelocal boundary value problem we employ an upwind method on a Shishkin meshin local domain, instead of solving it exactly. Thus we show that it is possibleto develop ? ? uniform method, totally in the context of ? nite differences withoutknowing location of the layer a priori and without solving any differential equationexactly.Imagine a river-a river ? owing strongly and smoothly. Liquid pollution pours intothe water at a certain point or Imagine a drop of ink dropped into a glass of water.What shape does the pollution stain form on the surface of the river or ink in theglass of water? In real life, we need diffusion to explain this problem near by theconvection. We further study the convergence properties of the numerical methodproposed and prove that it nodally convergence to the true solution for any ? .