带有积分边界条件的奇异摄动边值问题的渐近解

数学物理学报 ›› 2020, Vol. 40 ›› Issue (5): 1192-1203.

带有积分边界条件的奇异摄动边值问题的渐近解

武利猛¹(),倪明康²,李素红¹,陆海波³

¹ 河北科技师范学院数学与信息科技学院河北秦皇岛 066004
² 华东师范大学数学科学学院上海 200241
³ 上海应用技术大学经管学院上海 201418

收稿日期:2019-09-27 出版日期:2020-10-26 发布日期:2020-11-04
作者简介:武利猛, E-mail:xiaovikdo@163.com
基金资助:
国家自然科学基金(11871217);国家自然科学基金(11901152);国家重点研发计划(2019YFC1407903)

Asymptotic Solution of Singularly Perturbed Boundary Value Problem with Integral Boundary Condition

Limeng Wu¹(),Mingkang Ni²,Suhong Li¹,Haibo Lu³

¹ School of Mathematics and Information Technology, Hebei Normal University of Science and Technology, Hebei Qinhuangdao 066004
² School of Mathematical Sciences, East China Normal University, Shanghai 200241
³ School of Economics and Management, Shanghai Institute of Technology, Shanghai 201418

Received:2019-09-27 Online:2020-10-26 Published:2020-11-04
Supported by:
the NSFC(11871217);the NSFC(11901152);the NKRDPC(2019YFC1407903)

摘要/Abstract

摘要：

研究了一类带有积分边界条件的奇异摄动边值问题，利用奇异摄动几何理论证明了阶梯状空间对照结构解的存在性.基于解的结构，利用边界层函数法构造了原问题一致有效的形式渐近解.通过一个例子，验证了主要结果.

关键词: 奇异摄动, 渐近解, 边界层函数法

Abstract:

In this paper, we consider a class of singularly perturbed boundary value problem with integral boundary condition. Based on the singularly perturbed geometric theory, the existence of step-like contrast structure solution is proved. By virtue of the structure of the solution, we construct the uniformly valid formal asymptotic solution by the boundary layer function method. Finally, an example is given to show the main result.

Key words: Singular perturbation, Asymptotic solution, Boundary layer function method

中图分类号:

O175.14

武利猛,倪明康,李素红,陆海波. 带有积分边界条件的奇异摄动边值问题的渐近解[J]. 数学物理学报, 2020, 40(5): 1192-1203.

Limeng Wu,Mingkang Ni,Suhong Li,Haibo Lu. Asymptotic Solution of Singularly Perturbed Boundary Value Problem with Integral Boundary Condition[J]. Acta mathematica scientia,Series A, 2020, 40(5): 1192-1203.

图/表 1

参考文献 21

1	倪明康, 林武忠. 奇异摄动问题中的渐近理论. 北京:高等教育出版社, 2009
	Ni M K , Lin W Z . Asymptotic Theory of Singularly Perturbed Problem. Beijing:Higher Education Press, 2009
2	Mo J Q , Wang H , Lin W T . The solvability for a class of singularly perturbed quasi-linear differential system. Acta Mathematica Scientia, 2008, 28B (3): 495- 500
3	Mo J Q , Lin W T , Wang H . A class of homotopic solving method for ENSO model. Acta Mathematica Scientia, 2009, 29B (1): 101- 110
4	Du Z J , Kong L J . Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems. Applied Mathematics Letters, 2010, 23, 980- 983 doi: 10.1016/j.aml.2010.04.021
5	刘树德, 孙建山, 谢元静. 一类奇摄动拟线性边值问题的激波解. 数学物理学报, 2012, 32A (2): 312- 319
	Liu S D , Sun J S , Xie Y J . Shock solutions for some singularly perturbed quasilinear boundary value problems. Acta Mathematica Scientia, 2012, 32A (2): 312- 319
6	刘帅, 沈建和, 周哲彦. 二阶半线性奇摄动边值问题的渐近解. 数学物理学报, 2014, 34A (5): 1104- 1110
	Liu S , Shen J H , Zhou Z Y . Asymptotic solution for second-order semilinear singularly perturbed boundary value problems. Acta Mathematica Scientia, 2014, 34A (5): 1104- 1110
7	Ionkin N I . Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differential Equations, 1977, 13, 294- 304
8	Nicoud F , Schonfeld T . Integral boundary conditions for unsteady biomedical CFD applications. Internat J Numer Methods Fluids, 2002, 40, 457- 465 doi: 10.1002/fld.299
9	Amiraliyev G M , Cakir M . Numerical solution of the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Mechanics, 2002, 23 (7): 755- 764 doi: 10.1007/BF02456971
10	Cakir M , Amiraliyev G M . A finite difference method for the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Computation, 2005, 160, 539- 549 doi: 10.1016/j.amc.2003.11.035
11	Xie F , Jin Z Y , Ni M K . On the step-type contrast structure of second-order semilinear differential equation with integral boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations, 2010, 1, 1- 14
12	谢峰, 张莲. 具有积分边界条件的非线性二阶奇摄动问题. 华东师范大学学报, 2010, 1, 1- 5
	Xie F , Zhang L . Nonlinear second order singularly perturbed problem with integral boundary condition. Journal of East China Normal University, 2010, 1, 1- 5
13	Tin S K , Koppell N , Jones C K R T . Invariant manifolds and singularly perturbed boundary value problems. SIAM J Numer Anal, 1994, 31 (6): 1558- 1576 doi: 10.1137/0731081
14	Wu L M , Ni M K , Lu H B . Internal layer solution of singularly perturbed optimal control problem with integral boundary condition. Qualitative Theory of Dynamical Systems, 2018, 17, 49- 66 doi: 10.1007/s12346-017-0261-0
15	陆海波, 倪明康, 武利猛. 奇异奇摄动系统的几何方法. 华东师范大学学报, 2013, 3, 140- 148
	Lu H B , Ni M K , Wu L M . Geometric singular perturbation approach to singular singularly perturbed systems. Journal of East China Normal University, 2013, 3, 140- 148
16	Lin X B . Shadowing lemma and singularly perturbed boundary value problems. SIAM Journal on Applied Mathematics, 1989, 49, 26- 54 doi: 10.1137/0149002
17	Li X J , Zhang Q . Existence of solution for a p-Laplacian multi-point boundary value problem at resonance. Qualitative Theory of Dynamical Systems, 2018, 17, 143- 154 doi: 10.1007/s12346-017-0259-7
18	Du Z J , Li J , Li X W . The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach. J Funct Anal, 2018, 275, 988- 1007 doi: 10.1016/j.jfa.2018.05.005
19	Lin X J , Liu J , Wang C . The existence and asymptotic estimates of solutions for a third-order nonlinear singularly perturbed boundary value problem. Qualitative Theory of Dynamical Systems, 2019, 18, 687- 710 doi: 10.1007/s12346-018-0307-y
20	Wang C , Zhang X . Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type Ⅲ. J Differential Equations, 2019, 267, 3397- 3441 doi: 10.1016/j.jde.2019.04.008
21	Lin X J , Liu J , Wang C . The existence, uniqueness and asymptotic estimates of solutions for third-order full nonlinear singularly perturbed vector boundary value problems. Boundary Value Problems, 2020, 2020, 1- 17 doi: 10.1186/s13661-019-01311-5

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