二阶非线性奇摄动方程脉冲状空间对照结构
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摘要
近年来，大量的工作[1]-[4]是研究奇摄动问题中产生强烈反差解的内部层现象，这类解我们习惯上称之为空间对照结构．这种特殊的内部层问题最早在[5]中已有所反映，但当时并没有提出空间对照结构的概念，最早提出空间对照结构这个概念的是[6]。空间对照结构可以分为阶梯状空间对照结构(它在相平面或相空间上对应于异宿轨道)参见文献[7]-[13]和脉冲状空间对照结构(它在相平面或相空间上对应于同宿轨道)，该问题的研究具有很重要的意义，在化学自组织理论中意义尤为深刻。
到目前为止，对脉冲状空间对照结构的研究仅局限于对半线性方程的研究，如果方程的右端包含dy／dx问题会相当复杂。本文就是对一类含有dy／dx的二阶非线性方程
μ~2 d~2y／dx~2=F(μ dy／dx,y,x),-1＜x＜1,
y(-1,μ)=y(1,μ)=0,0＜μ＜1.进行研究，指出该方程在一定条件下可以产生脉冲状空间对照结构，并用边界函数法构造该问题的渐近解，证明解的存在性并得到渐近解的误差估计。
Recently, the phenomena of inner layer which occur strongly contrast solution were studied in much work. Usually, we call it contrast structure. This special inner layer was concerned first in [5], but the conception of contrast structure wasn't put forward until the article [6] was born. The contrast structure is divided into step-type contrast structure(which corresponds to heteroclinic orbit in phase plane or phase space) and spike-type contrast structure (which corresponds to homoclinic orbit in phase plane or phase space).
So far, the study of spike-type contrast structure is only confined in semi-linear equation. The problem will be changed into more and more complex
if the right side of the equation involves the terms of dy/dx . In this paper, we
will study a kind of second order equation (which involves dy/dx):
and point out that this equation will has a contrast structure solution under certain conditions. Using the method of boundary function we construct the formula asymptotic solution. At the same time, the existence of the solution is proved and the error evaluation of the asymptotic solution is obtained.
到目前为止，对脉冲状空间对照结构的研究仅局限于对半线性方程的研究，如果方程的右端包含dy／dx问题会相当复杂。本文就是对一类含有dy／dx的二阶非线性方程
μ~2 d~2y／dx~2=F(μ dy／dx,y,x),-1＜x＜1,
y(-1,μ)=y(1,μ)=0,0＜μ＜1.进行研究，指出该方程在一定条件下可以产生脉冲状空间对照结构，并用边界函数法构造该问题的渐近解，证明解的存在性并得到渐近解的误差估计。
Recently, the phenomena of inner layer which occur strongly contrast solution were studied in much work. Usually, we call it contrast structure. This special inner layer was concerned first in [5], but the conception of contrast structure wasn't put forward until the article [6] was born. The contrast structure is divided into step-type contrast structure(which corresponds to heteroclinic orbit in phase plane or phase space) and spike-type contrast structure (which corresponds to homoclinic orbit in phase plane or phase space).
So far, the study of spike-type contrast structure is only confined in semi-linear equation. The problem will be changed into more and more complex
if the right side of the equation involves the terms of dy/dx . In this paper, we
will study a kind of second order equation (which involves dy/dx):
and point out that this equation will has a contrast structure solution under certain conditions. Using the method of boundary function we construct the formula asymptotic solution. At the same time, the existence of the solution is proved and the error evaluation of the asymptotic solution is obtained.
引文
[1] Niegoli.G,Plirogen.E，在非均匀系统中自组织化过程，莫斯科和平出版社，1973．
[2] Plak. R. C, Mihrelov. A. C. 在非均匀化学物理中的自组织化过程．莫斯科科学出版社，1983．
[3] Romanovshge． U. M, Sjteganov. M. B, Chronovshge. D. C,数学的生物物理，莫斯科科学出版社．
[4] 尼科历斯，普里戈任．非均匀系统中的自组织，[M]．莫斯科和平出版社，1979．
[5] A.B.Vasileva , V.F.Butuzov. Asymptotic expansions of singularly perturbed differential equations, [M]. Nauka, Moscow, 1973,(in Russian)
[6] V. F. Butuzov, A. B. Vasileva. 具有空间对照结构的渐近解．1987，Vol．42，No．6，831-841．
[7] A. B. Vasileva, Davdav. A. C. 由两个两阶的奇摄动方程组组成的方程组的阶梯型对照结构，计算数学和数学物理杂志，1996，Vol．36，No．5，75-89
[8] A. B. Vasileva, M. A. Davydova. 某类二阶非线性奇异摄动方程阶梯型空间对照结构，[J]．计算数学和数学物理杂志．1998．No 6，938-947．
[9] A.B.Vasileva, On Contrast Steplike Structure for a System of Singularly Perturbed Equation, Zh. Vychisl. Mat. Mat. Fiz. 1994, Vol.34, No.10, 1401-1411.
[10] A.B.Vasileva, Contrast structure of step type for a Singularly Perturbed quasilinea second order differential equation Zh. Vychisl. Mat. Mat. Fiz. 1995, Vol.35, No.4, 411-418.
[11] 倪明康，古稀．一类变分问题阶梯解的存在性．[J]．华东师范大学学报(自然科学版)．No．1，2006．[12] A.B.Vasileva, O.E.Omel'chenko, periodic contrast structure of step type for singularly perturbed parabolic equations, Differ, Uravn,2000, Vol.36, No.2, 236-246.
[13] A.B.Vasileva, O.E.Omel'chenko, contrast structures of step type for a singu-larly perturbed elliplic equation in an annulus Zh. Vychisl. Mat. Fiz. 2000, Vol.40, No.1, 118-130.
[14] V.Esipova, Differential Equations, 11, 1457. (1975)
[15] A.B.Vasileva, V.Butuzov, Singularly Perturbed Equations in Criticle Case, Technical Report MRC-TSR 2039, University of Wisconsin, Madison, WI, 1980.
[16] A.Tikhonov,A.B.Vasileva, and Sveshnikov, Differential Equations, Springer-Verlag, Berlin, Heidelberg, New Yoryo, 1985.
[17] 倪明康，林武忠．一类条件稳定的多小参数奇摄动方程组边值问题，[J]．华东师范大学学报(自然科学版)．N．1，1989，16-23．
[18] 倪明康．具有空间对照结构渐近解的构造(Ⅰ)，[J]．华东师范大学学报(自然科学版)．No．1，1993，1-11．
[19] A.B.Vasileva, V.F.Butuzov, and L.Kalachev. The boundary function method for singular perturbation problems, [M]. SIAM, Philadelphia, PA, 1995.
[2] Plak. R. C, Mihrelov. A. C. 在非均匀化学物理中的自组织化过程．莫斯科科学出版社，1983．
[3] Romanovshge． U. M, Sjteganov. M. B, Chronovshge. D. C,数学的生物物理，莫斯科科学出版社．
[4] 尼科历斯，普里戈任．非均匀系统中的自组织，[M]．莫斯科和平出版社，1979．
[5] A.B.Vasileva , V.F.Butuzov. Asymptotic expansions of singularly perturbed differential equations, [M]. Nauka, Moscow, 1973,(in Russian)
[6] V. F. Butuzov, A. B. Vasileva. 具有空间对照结构的渐近解．1987，Vol．42，No．6，831-841．
[7] A. B. Vasileva, Davdav. A. C. 由两个两阶的奇摄动方程组组成的方程组的阶梯型对照结构，计算数学和数学物理杂志，1996，Vol．36，No．5，75-89
[8] A. B. Vasileva, M. A. Davydova. 某类二阶非线性奇异摄动方程阶梯型空间对照结构，[J]．计算数学和数学物理杂志．1998．No 6，938-947．
[9] A.B.Vasileva, On Contrast Steplike Structure for a System of Singularly Perturbed Equation, Zh. Vychisl. Mat. Mat. Fiz. 1994, Vol.34, No.10, 1401-1411.
[10] A.B.Vasileva, Contrast structure of step type for a Singularly Perturbed quasilinea second order differential equation Zh. Vychisl. Mat. Mat. Fiz. 1995, Vol.35, No.4, 411-418.
[11] 倪明康，古稀．一类变分问题阶梯解的存在性．[J]．华东师范大学学报(自然科学版)．No．1，2006．[12] A.B.Vasileva, O.E.Omel'chenko, periodic contrast structure of step type for singularly perturbed parabolic equations, Differ, Uravn,2000, Vol.36, No.2, 236-246.
[13] A.B.Vasileva, O.E.Omel'chenko, contrast structures of step type for a singu-larly perturbed elliplic equation in an annulus Zh. Vychisl. Mat. Fiz. 2000, Vol.40, No.1, 118-130.
[14] V.Esipova, Differential Equations, 11, 1457. (1975)
[15] A.B.Vasileva, V.Butuzov, Singularly Perturbed Equations in Criticle Case, Technical Report MRC-TSR 2039, University of Wisconsin, Madison, WI, 1980.
[16] A.Tikhonov,A.B.Vasileva, and Sveshnikov, Differential Equations, Springer-Verlag, Berlin, Heidelberg, New Yoryo, 1985.
[17] 倪明康，林武忠．一类条件稳定的多小参数奇摄动方程组边值问题，[J]．华东师范大学学报(自然科学版)．N．1，1989，16-23．
[18] 倪明康．具有空间对照结构渐近解的构造(Ⅰ)，[J]．华东师范大学学报(自然科学版)．No．1，1993，1-11．
[19] A.B.Vasileva, V.F.Butuzov, and L.Kalachev. The boundary function method for singular perturbation problems, [M]. SIAM, Philadelphia, PA, 1995.