Advances in Applied Mathematics
Vol.07 No.01(2018), Article ID:23560,5 pages
10.12677/AAM.2018.71013

Singularly Perturbed Problems with a Power-Law Attenuation Boundary Layer

Yujiao Sun

School of Mathematical Science and Engineering, Anhui University of Technology, Maanshan Anhui

Received: Dec. 22nd, 2017; accepted: Jan. 18th, 2018; published: Jan. 26th, 2018

ABSTRACT

Singularly Perturbed Problems with a Power-Law Layer is studied in this paper. The method of boundary layer function is used to construct the formal asymptotic expansion of the solution, and get the attenuated boundary layer functions of power-law. The existence and uniformly valid approximation of solutions are obtained by upper and lower solutions method.

Keywords:Singular Perturbation, Triple Root, Power-Law Boundary Layer, Asymptotic Solution

具有幂率衰减边界层的奇摄动问题

孙玉娇

安徽工业大学,数理科学与工程学院,安徽 马鞍山

收稿日期:2017年12月22日;录用日期:2018年1月18日;发布日期:2018年1月26日

摘 要

本文主要研究退化方程具有三重根的二阶奇摄动方程边值问题。运用边界层函数法构造出解的形式渐近展开式,得到以幂率形式衰减的边界层函数。最后用上下解方法得到形式解存在性和一致有效估计。

关键词 :奇摄动,三重根,幂率衰减边界层,渐近解

Copyright © 2018 by author and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

对于非线性奇摄动二阶方程的边值问题:

ε 2 d 2 u d x 2 = f ( u , x ) , 0 < x < 1 , (1)

u ( 0 , ε ) = u 0 , u ( 1 , ε ) = u 1 . (2)

其中, 0 < ε 1 ,为小参数。 f , φ 是连续可微的函数。

V.F. Butuzov等在文献 [1] 中研究了具有指数衰减的边界层问题。在文献 [2] [3] 中,倪明康和Huai-Ping Zhu对具有幂率衰减边界层的问题进行了讨论。在文献 [4] [5] [6] 中,A.B. Vasil’eva也分别对退化方程具有单根和二重根时的问题进行了研究,得到了渐近解的表达式:

u ( x , ε ) = u ¯ ( x ) + Π ( τ ) + R ( ξ ) , τ = x ε , ξ = x 1 ε .

其中

正则项 u ¯ ( x ) = u 0 ¯ ( x ) + ε u 1 ¯ ( x ) +

左边界层项 Π ( τ , ε ) = Π 0 ( τ ) + ε Π 1 ( τ ) +

右边界层项 Π ( ξ , ε ) = Π 0 ( ξ ) + ε Π 1 ( ξ ) +

本文将继续对退化方程具有三重根进行讨论,设 f ( u , x , ε ) 有如下形式

f ( u , x , ε ) = ( u φ ( x ) ) 3 . (3)

并提出假设:

[ H 1 ] u 0 > φ ( 0 ) , u 1 > φ ( 1 ) , 0 x 1. (4)

[ H 2 ] φ ( x ) > 0 , 0 x 1. (5)

2. 渐近解的构造

设问题(1),(2)具有下述的形式渐近解:

u ( x , ε ) = u ¯ ( x , ε ) + Π ( τ , ε ) , τ = x ε , ξ = x 1 ε . (6)

将(6)式代入(1)式,

ε 2 d 2 u d x 2 = ε 2 d 2 u ¯ d x 2 + d 2 Π d τ 2 . (7)

f ( u , x , ε ) = f ( u ¯ ( x ) , x , ε ) + f ( u ¯ ( ε τ ) + Π ( τ ) , ε τ , ε ) f ( u ¯ ( ε τ ) , ε τ , ε ) . (8)

故有表达式:

ε 2 d 2 u ¯ d x 2 = f ( u ¯ ( x ) , x , ε ) . (9)

d 2 Π d τ 2 = f ( u ¯ ( ε τ ) + Π ( τ ) , ε τ , ε ) f ( u ¯ ( ε τ ) , ε τ , ε ) . (10)

设正则项的渐近表达式为:

u ¯ ( x , ε ) = u 0 ¯ ( x ) + ε 2 3 u 1 ¯ ( x ) + ε 4 3 u 2 ¯ ( x ) + . (11)

将(11)式代入(9)得:

ε 2 d 2 d x 2 ( u 0 ¯ ( x ) + ε 2 3 u 1 ¯ ( x ) + ε 4 3 u 2 ¯ ( x ) + ) = ( u 0 ¯ ( x ) + ε 2 3 u 1 ¯ ( x ) + ε 4 3 u 2 ¯ ( x ) + ) 3 . (12)

比较上式两边的同次幂系数:

ε 0 : 0 = ( u 0 ¯ ( x ) φ ( x ) ) 3 . (13)

ε 2 : d 2 u 0 ¯ ( x ) d x 2 = ( u 1 ¯ ( x ) ) 3 . (14)

ε 8 3 : d 2 u 1 ¯ ( x ) d x 2 = 3 u 1 ¯ ( x ) u 2 ¯ ( x ) . (15)

由上式可解出

u 0 ¯ ( x ) = φ ( x ) . (16)

u 1 ¯ ( x ) = ( φ ( x ) ) 1 3 . (17)

u 2 ¯ ( x ) = 1 9 ( φ ( x ) ) 1 . (18)

i 3 时可用类似方法得出。

再设边界层的渐近表达式为

Π ( x , ε ) = Π 0 ( x ) + ε 2 3 Π 1 ( x ) + ε 4 3 Π 2 ( x ) + . (19)

代入(10)得

d 2 Π d τ 2 = ( u 0 ¯ ( x ) + ε 2 3 u 1 ¯ ( x ) + + Π 0 ( x ) + ε 2 3 Π 1 ( x ) + ) 3 ( u 0 ¯ ( x ) + ε 2 3 u 1 ¯ ( x ) + ) 3 . (20)

由(2),(6),(11),(19)得

u ( 0 , ε ) = u 0 ¯ ( 0 ) + ε 2 3 u 1 ¯ ( 0 ) + + Π 0 ( 0 ) + ε 2 3 Π 1 ( 0 ) + = u 0 . (21)

得出如下形式的 Π i ( τ )

Π 0 ( 0 ) = u 0 φ ( 0 ) ; (22)

Π i ( 0 , τ ) = u i ( 0 ) , i = 1 , 2 , 3 , (23)

考虑到 V i ( τ ) 为边界层函数,所以要求

Π i ( ) = 0. (24)

比较(20)式两端同次幂系数,可得 Π 0 函数满足的方程

ε 0 : d 2 Π 0 d τ 2 = Π 0 3 , Π 0 ( 0 ) = u 0 φ ( 0 ) , Π 0 ( ) = 0. (25)

Π 0 ( τ ) = 2 ( u 0 φ ( 0 ) ) τ ( u 0 φ ( 0 ) ) + 2 . (26)

τ 时, Π 0 ( 0 ) 0 。且具有幂率衰减特性。

继续使用上述方法,可得到其余边界层函数,且用同样的方法可确定 R ( ξ )

3. 形式解的一致有效性

定理:若存在函数 α ( x ) , β ( x ) 使得问题

L u = ε 2 u x x f ( u , x )

u ( 0 , ε ) = u 0 , u ( 1 , ε ) = u 1 .

满足下列条件:

1) α ( x ) β ( x )

2) L α ( x ) 0 , L β ( x ) 0

3) α ( 0 ) u 0 β ( 0 ) , α ( 1 ) u 1 β ( 1 )

则问题存在解 u ( x ) 满足 [7]

α ( x ) u ( x ) β ( x ) .

U 0 = u 0 ¯ + Π 0 + R 0 = φ ( x ) + Π 0 + R 0 .

构造

β ( x ) = U 0 + λ 1 ε 2 3 , α ( x ) = U 0 λ 2 ε 2 . (27)

L β ( x ) = d 2 Π 0 d τ 2 + d 2 R 0 d ξ 2 + ε 2 φ ( x ) ( Π 0 + R 0 + λ 1 ε 2 3 ) 3 = ε 2 φ ( x ) λ 1 3 ε 2

λ 1 3 > φ ( x ) 时, L β ( x ) 0

L α ( x ) = d 2 Π 0 d τ 2 + d 2 R 0 d ξ 2 + ε 2 φ ( x ) ( Π 0 + R 0 λ 2 ε 2 ) 3 = ε 2 φ ( x ) λ 2 3 ε 6 + + 3 λ 2 ε 2 ( Π 0 2 + R 0 2 ) .

对足够大的 λ 2 L α ( x ) 0

β ( 0 ) = Π 0 ( 0 ) + R 0 ( 1 ε ) + φ ( 0 ) + λ 1 ε 2 3 = u 0 φ ( 0 ) + R 0 ( 1 ε ) + φ ( 0 ) + λ 1 ε 2 3 > u 0 ;

β ( 1 ) = Π 0 ( 1 ε ) + R 0 ( 0 ) + φ ( 1 ) + λ 1 ε 2 3 = u 1 φ ( 1 ) + Π 0 ( 1 ε ) + φ ( 1 ) + λ 1 ε 2 3 > u 1 ;

α ( 0 ) = Π 0 ( 0 ) + R 0 ( 1 ε ) + φ ( 0 ) λ 2 ε 2 = u 0 φ ( 0 ) + R 0 ( 1 ε ) + φ ( 0 ) λ 2 ε 2 < u 0 ;

α ( 1 ) = Π 0 ( 1 ε ) + R 0 ( 0 ) + φ ( 1 ) λ 1 ε 2 = u 1 φ ( 1 ) + Π 0 ( 1 ε ) + φ ( 1 ) λ 1 ε 2 < u 1 .

则问题(1),(2)的渐近解 U ( x ) 满足

λ 1 ε 2 3 λ 2 ε 2 = α U 0 u ( x , ε ) U 0 β U 0 λ 1 ε 2 3 .

亦即

| u ( x , ε ) U 0 | λ 1 ε 2 3 . (28)

特别地,对于首次近似有

| u ( x , ε ) ( φ ( x ) + Π 0 + R 0 ) | = O ( ε 2 3 ) . (29)

致 谢

感谢审稿老师及编辑老师提出的宝贵意见。

文章引用

孙玉娇. 具有幂率衰减边界层的奇摄动问题
Singularly Perturbed Problems with a Power-Law Attenuation Boundary Layer[J]. 应用数学进展, 2018, 07(01): 104-108. http://dx.doi.org/10.12677/AAM.2018.71013

参考文献 (References)

  1. 1. Butuzov, V.F., Nefedov, N.N., Recke, L. and Schneider, K.R. (2013) On a Singularly Perturbed Initial Value Problem in the Case of a Double Root of the Degenerate Equation. Nonlinear Analysis, 83, 1-11.
    https://doi.org/10.1016/j.na.2013.01.013

  2. 2. 倪明康, 丁海云. 具有代数衰减的边界层问题[J]. 数学杂志, 2001, 31(3): 488-494.

  3. 3. Zhu, H.-P. (1997) The Dirichlet Problem for Singularly Perturbed Quasilinear Second Order Differential System. Journal of Mathematical Analysis and Applications, 210, 308-336.
    https://doi.org/10.1006/jmaa.1997.5405

  4. 4. Vasil’eva, A.B. (2009) Two-Point Boundary Value Problem for a Singularly Perturbed Equation with a Reduced Equation Having Multiple Roots. Computational Mathematics and Mathematical Physics, 49, 1021-1032.

  5. 5. Vasil’eva, A.B. (2011) Boundary Layers in the Solution of Singularly Perturbed Boundary Value Problem with a Degenerate Equation Having Roots of Multiplicity Two. Computational Mathematics and Mathematical Physics, 51, 351-354.

  6. 6. Vasil’eva, A.B. and Nefedov, N.N. (1990) Asymptotic Methods in the Theory of Singular Perturbation. Vysshaya Shkola, Moscow. (In Russian)

  7. 7. 倪明康, 林武忠. 微分不等式理论[M]. 北京: 科学出版社, 2012.

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