Some Methods For Singular Perturbation Problems

Singular perturbation problems come from celestial mechanics and appear in the late 19 century.In the 20th century,especially in the 1950s around.When WKB,asymptotic matching expending method,Lighthill skill,the methods of multiple scales,averaging method come one by one.Particularly,since Van Dyke have given the matching theory.It is worth mentioning that Mechanical scientist and mathematicians of China have an outstanding contribution on singular perturbation theory.As a kind of approximate solution methods,singular perturbation method has a very wide range of applications.It is widely used in astronomy,mechanics,mathematics,theoretical physics,chemistry,bioloy and other fields,even it is also applied in the field of social sciences.It can be said that the utilization of the singular perturbation method is it's prominent quality. In this paper,we introduce some typical methods for singular perturbation theory. We also pointed out that the distinction,relation,as well as the application in physics of these methods.Chapterâ… is Introduction.It points out that the perturbation method is a class method of solving asymptotic analysis solution.Perturbation method includes regular perturbation and singular perturbation.We first introduce the importance of singular perturbation method.then use a lot of space to describe the application of singular perturbation method.Chapterâ…¡is the main part of the paper.Firstly.we give some important concepts in briefly,such as asymptotic series,asymptotic expansion,uni- form valid and non-uniform valid.It pointed out that the necessary conditions for uniform valid.The article have 6 parts to introduce stretched parameter method(Lindstedt-Poincare skills),Lighthilk skills,regular method,the methods of multiple scales,averaging method and asymptotic matching expansion method. Each method also includes introduction of history,basic idea,applicability or limitations, compared with other techniques,and practical application.We briefly review the main content.In 19th century astronomers Lindstedt and other people developed stretched parameter method through the process of solving perturbation solution of the equation for controlling the secular term.Makingωand the dependent variable u given by expansion of power seriesε.Sinceωisn't obvious included in the differential equation,we let and obtain the equality is on the derivative of s.Let and let the same power coefficientεare equal.We can get a series of equations which can solveÏ…_n one by one,and chooseω_n to remove the secular terms. This technique has been widely used in physics and mathematics.For example. Keller used this technique for boundary value problems of ordinary differential equations.Usingεparameter power expansion the functionÏ…(x₁,x₂,…x_n,ε)and a independent variable is the essence of Lighthill skill.For example x₁.Lighthill introduce a new dependent variable,and expandÏ…and x₁ into power form ofε, coefficient dependent on sLighthill want to make the two expansion equality above all uniform valid through selecting stretched functionξ_m,which only needs u_m/u_m-1andξ_m/ξ_m-1 is bounded.In stretched parameter method we substitute s=ωt into the differential equation,makeωandÏ…expand in the same time.In regularization method we substitute the transform and the expansion ofωinto the direct expansion,and make the result,expanding onε,eliminate secular term by choosingω_n,i.e..we direct expand the solution of the equation in the following form Let For fixed s,we expand the consequence on small quantityε.By choosingω_n we can eliminate secular term.The advantages of this technique is a small amount of computation.Stretched parameters,skill of Lighthill and regularization method all belong to stretched coordinates method.The kind of the methods has effectively solved many physical problems of uniformly valid expansion,especially,it is successful to solve the problems of hyperbolic differential equation.Stretched coordinates is failure for multiplication of perturbation problem."The method of asymptotic matching expansion" is named by the idea of Bretherton in 1962,it is promotion for boundary layers theory of Prandtl.When the small parameters multiplied by the highest derivative produce non-uniform problems,the method has the obvious advantage.In the application it is necessary to grasp two points,firstly,selecting appropriate internal variable is the key of asymptotic matching expansion method;Secondly, accurately using asymptotic matching principles (the external expansion of n term's)the internal expansion of m term =(the internal expansion of m term's)the external expansion of n term.The above principle is enough for problems which can be solved by method of asymptotic matching expansion.Chapterâ…¢is conclusion.We pointed out that the produce of singular perturbation method is the historical inevitability.We should make great efforts to develop this theory.