Singular Perturbation Problems For First Order Ordinary Differential Equations

Asymptotic and perturbative analysis has played a significant role in appliedmathematics and theoretical physics. In many cases, regular perturbation methods are not applicable, and various singular perturbation techniques must be used.But,solutions of singular perturbations to some parameters always show a certain restraint non-uniformity behaviour. How to obtain such precision solutionsuniformly valid asymptotic expansions Show Type is a central issue of singular perturbations theory.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 effectivemethods, 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.Recently, a perturbative renormalization group method was developed by Chen, Goldenfeld, and Oono as a unified tool for asymptotic analysis.This paper consist of three parts.In the first part, we consider the following system of differential equations:dy/dt+ Ay =εF(y), (1)y(0) = y0, (2)whereε> 0 is a small parameter, A is a complex matrix,assumed for simplicity to be diagonalizable, F is a polynomial vector function with respect to y.Then, we use the regular perturbation method to obtain a naive perturbationexpansion up to orderε, we have:y₁^ε(t) = e^-tA[v(t₀) +ε∫_t0^te^sAF(e^-sAv(t₀))ds].In order to eliminate the terms which may contain secular terms, we'll separate the probable secular terms. As following lemma:引理3.2 Assume that F is given by (2.3). Thene^sAF(e^-sAv(t₀)) = R(v(t₀)) + Q(s, v(t₀)), (3)wherewithN_rⁱ = {α∈Nⁿ : |α|≤m; (Λ,α) =sum from k=1 to nα_kλ_k =λ_i}, (6)e₁,…, e_n is the natural basis of Rⁿ.In the second part, we firstly introduce the CGO-RG method to eliminate the secular terms and obtain the uniformly valid approximate solution. Above all, we combine the Multipe scales method and Stretched coordinates method, by which we can obtain the uniformly valid expansion without secular terms. As follows定理3.1 y₁^ε(t) = (y₁1^ε(t),y₁2^ε(t),…)y_ln^ε(t)) is the uniformly valid expansionof the initial-condition problem (2.1), (2.2), wherewithÏ„_i= (1 -∑_{α(?)NÏ„ⁱ}C_αⁱv^α/λ_iv_iε)t, N_Ï„ⁱ is given by (2.13), v = (v₁, v₂,…, v_n) is determinedby the initial condition (2.2).In the third part, we consider the Duffing Equation, and compare the results obtained by our method with the results of others.