Renormalization Group Theory For A Kind Of Three Order Singular Perturbation Problems

Singular perturbations theory has been studying a variety of physical phenomena and processes irregular one important tool. From the mathematical point of view,solutions of singular perturbations to some parameters always show a certain restraint non-uniformity behaviour. How to obtain such precision solutions uniformly valid asymptotic expansions Show Type singular perturbations theory is a central issue.Solutions was adopted unanimously as uniformly valid asymptotic expansions show , on the one hand, can help us understand the solution asymptotic behaviour, on the other hand, it was for the adoption of the numerical solutions provide theoretical basis.Construction of such approaches solutions, people have developed many effective methods, such as asymptotic matching, averaging, multiple scales, stretched coordinate and WKB methods. But in the use of these methods, in order to ensure expansion of solutions uniformly valid.To irregular part, such as the Boundary Layer, or by the location and thickness of surface may appear small fraction of the market parameters, such as the need to have some understanding. There is also a need for closer asymptotic matching necessary. This makes them much more restricted applications.In recent years, Chen, Goldenfeld and Oono[4, 5] and Ziane[21] successful restructuring of the use of various types of research methods group singular perturbations. Their results show that : a group approach to the restructuring of singular perturbations, simpler than using traditional methods more effective. First, in the construction asymptotic expansions, it does not need to perturbations of the structure so special series of assumptions, or the need for a asymptotic matching, but generated its own approaches applicable to the asymptotic series. Secondly, through the gradual construction of the exhibition, sometimes we can more accurately than with traditional methods more effective solutions generic information. But so far, the results have only experience and form, and there is no theoretical proof.We first introduce the application of renormalization group method to the Rayleigh equation[l]sdt^y-"\dt 3ydtJ yBy using the renormalization group method, we can obtain the following uniformly valid expansionof the solutions of Rayleigh equation, where R(t) satisfies the so-called renormalization group equationR(t) = R(0)/[e-? + \R(0)2(l - e-*\W + O(e2t),here i?(0),9(0) can be determined by the initial conditions.Then we consider the over-damped linear oscillator:wheree is a small parameter. The uniformly valid expansion obtained for the solution of this equation can be obtained by using the renormalization group method as/+\ __ /-< -(1+eH I s~< — l/t+(l+f)t i r}(c1\where C\, C% is the constants which can be determined by the initial conditions. In chapter 3, we consider the following three order singular perturbation problemJ3-. J2.. a..-- o, (i)(2)where e > 0 is a small parameter, a, /?, 7 are given real numbers.The following is our main result.Theorem Assume that P(t),R(t),Q(t) are C1 functions, and let p0 = P(0),r0 = i?(O))9o = Q(O),Pi = P'(0),ri = #(0). If A = pg - 4r0 # 0, then the solution of initial value problem (l)-(2) has the following expansion which is uniformly validy(t) = Cie-^ -wherei = |(-Po + a/A), A2 = i(-p0 -can be determined by the initial condition (2).