The Numerical Solution Of A Class Of Singular Perturbation Problems

In application fields such as physics and engineering, many practical problemsusually contain . It is often involved in many practical problems, seeking the differ-ential equation with small parameter solution, is often involved in many practicalproblems. The small changes of the equation parameters will lead to significantchanges in production. This kind of problem is called singular perturbation prob-lems. From its source,singular perturbation problems are applied widely, such asoptimal control,optical and other important areas of chemical reaction. It has es-tablished inextricably linked with people's production and livelihood. Therefore,the study of some singular perturbation problem of different nature, and seekingthe solution of singular perturbation problem is particularly important.In this paper, three different ways are used for the solution of singularly arebased on perturbed boundary Value Problems, and using the numerical calculationsoftware Mathematica 5.0 Simulation to verify the feasibility of the method .Firstly, the way of study reproducing kernel method. It is a method of solv-ing problems that apply the reproducing feature of reproducing kernel function inreproducing kernel space. In the process of solving differential equations, after theestablishment of standard orthogonal functions, with the regeneration of the repro-ducing kernel function can find the exact equation expression.Secondly, the way of study Spline interpolation method. "spline" is widelyused in engineering design of slender strips of wood, and it is a convenient andeffective mapping tool, draughtsman fixed the spline with small nails and otherobjects in some scatter points, they formed a connection with smooth features, thiscurve is the spline function. Simple mathematically speaking, spline equivalent topolynomial function, zero corresponds to zero order polynomial spline, and so on,n times spline the equivalent of n order polynomial.Finally, the way of study finite difference method. It is an approximate replacedmethod, the derivation Function is replaced with the difference quotient functionapproximately, by the relations between difference quotient function and the dif-ference differential equation, the solution of difference equation is the approximate solution of the original equation. This article describes the basics of differential anddifference equations tectonic processes, finally, with a numerical example, verify thevalidity of the method.