李超,王曉云

(1.中北大學理學院,山西太原 030051;2.山西警官高等專科學?；A(chǔ)部,山西太原 030021)

具有多參數(shù)的奇攝動非線性邊值問題的攝動解

李超1,王曉云2

(1.中北大學理學院,山西太原 030051;2.山西警官高等?？茖W?；A(chǔ)部,山西太原 030021)

討論含多個參數(shù)的高階非線性方程的攝動解,在適當?shù)臈l件下,先構(gòu)造出外部解,再根據(jù)不同的邊界層,利用伸展變量和冪級數(shù)展開式理論,構(gòu)造問題的形式漸近解,最后利用微分不等式理論證明漸近解的一致有效性和漸近形態(tài),把奇攝動非線性問題中的參數(shù)推廣到多個參數(shù).

非線性;多參數(shù);奇攝動;邊值問題;邊界層校正

1 引言

對于含小參數(shù)非線性系統(tǒng)(奇攝動非線性系統(tǒng))邊界層研究一直是眾多致力于非線性系統(tǒng)工作者非常關(guān)注的問題[1-6].近幾十年來,處理奇攝動非線性系統(tǒng)邊界層的很多方法已經(jīng)陸續(xù)被提出,并不斷被優(yōu)化.其中一些經(jīng)典的方法有邊界層法、匹配漸近展開法、多重尺度法等.另外,一些學者還利用微分不等式方法來研究奇攝動非線性系統(tǒng),如文獻[7-15]利用微分不定式等方法研究了奇攝動非線性常微分方程邊值問題、反應擴散方程、橢圓型方程邊值問題、奇攝動問題的激波層解和大氣物理問題等.文獻[16]利用微分不等式研究了兩個參數(shù)的非線性方程的奇攝動解.本文在文獻[16]的基礎(chǔ)上,討論了一類具有n個參數(shù)的非線性奇攝動邊值問題,并證明了解的一致有效性.

考慮如下非線性奇攝動邊值問題:

為常數(shù).為了方便,此處把(1)-(3)式記為系統(tǒng)(N).

2 方程的形式漸近解

2.1 外部解

依次可求得

顯然,由(9)式得到的解Y0就是退化問題(4)-(5)的解Y0(x),且由(10)式可以依次地得到Y(jié)i,這樣就得到外部解

但是它并不滿足j=1,2,…,n時的邊界條件,所以需要構(gòu)造在x=a附近的第一,第二,…,第n邊界層校正項.

2.2 第一邊界層校正

為了得到系統(tǒng)(N)形式漸近解的一致有效展開式,下面在不同的邊界層引入伸長不同的變量來構(gòu)造函數(shù).

定義1.1中間方程:方程(1)某一系數(shù)趨于零時得到的方程稱為中間方程.

現(xiàn)考慮(6)式的中間方程

2.3 第二邊界層校正

2.4 第n邊界層校正

3 解的一致有效性證明

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Singu larly perturbed solution for nonlinear equations with some parameters

Li Chao1,Wang Xiaoyun1,2
(1.College of Science,North University of China,Taiyuan 030051,China; 2.Departm ent of Mathem atics,Shanxi Police Academ y,Taiyuan 030021,China)

A class of nonlinear speed system perturbed solution non local prob lem is discussed in this paper.In approp riate conditions,first constructed out external solution,again according to different boundary layer and using stretchy variable and power series launched theory,the asym ptotic expansions of solution of this p roblem is shown and p roved to be uniform ly effective using the theory of differential inequality in the whole interval. This paper extends the perturbed problem of the parameters of the parameters of the p romotion to multip le param eters.

nonlinear,m any param eters,singularly perturbed,boundary value prob lem, boundary layer correction

O178

A

1008-5513(2012)03-0370-08

2012-01-23.

山西省自然科學基金(2011011002-1);中國博士后科學基金(20100471857).

李超(1987-),碩士生,研究方向：應用數(shù)學.

2010 MSC:15A 42