一类弱非线性临界奇摄动积分边界问题

Abstract

Abstract:

Based on the boundary layer function method, a class of singularly perturbed problems with integral boundary conditions in weakly nonlinear critical cases are studied. In the framework of this paper, we not only construct the asymptotic expansion of the solution of the original equation, but also prove the uniformly effective asymptotic expansion. At the same time, we give an example to illustrate our results, The comparison images of approximate solution and exact solution under different small parameters are drawn.

Key words: Critical situation, Singular perturbation, Boundary layer function method, Approximate solution

CLC Number:

O175.14

Hao Zhang,Na Wang. A Class of Weakly Nonlinear Critical Singularly Perturbed Integral Boundary Problems[J].Acta mathematica scientia,Series A, 2022, 42(4): 1060-1073.

Trendmd

Figures/Tables 2

References 12

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 for Singular Perturbed Problem. Beijing: Higher Education Press, 2009
2	Vasil'eva A B, Butuzov V F. Asymptotic Expansions of Solutions of Singularly Perturbed Equations[in Russian]. Moscow: Nauka, 1973
3	林宗池, 周明儒. 应用数学中的摄动方法. 江苏: 江苏教育报刊总社, 1995
	Lin Z C , Zhou M R . Perturbation method in Applied Mathematics. Jiangsu: Jiangsu Education Newspapers and Periodicals Headquarters, 1995
4	Il'in A M. Matching of asymptotic expansions of solutions of boundary value problems. Translations of Mathematical Monographs, Vol 102. Providence: American Mathematical Society, 1992
5	Vasil'eva A B , Butuzov V F . Singularly Perturbed Equations in the Critical Case. Madison: University of Wisconsin, 1980
6	Vasil'eva A B , Panteleeva O I . On a system of singularly perturbed second-order quasilinear ordinary differential equations in the critical cases. Computational Mathematics and Mathematical Physics, 2006, 46 (4): 563- 574 doi: 10.1134/S0965542506040051
7	Vasil'eva A B . On systems of two singularly perturbed quasilinear second-order equations. Journal of Mathematical Sciences, 2008, 150 (6): 2467- 2472 doi: 10.1007/s10958-008-0145-6
8	Wang N . A class of singularly perturbed delayed boundary value problem in the critical case. Advances in Difference Equations, 2015, 2015 (1): 1- 21 doi: 10.1186/s13662-014-0331-4
9	Amiraliyev G M , Akir M C . 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 (2): 539- 549 doi: 10.1016/j.amc.2003.11.035
11	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(自然科学版), 2010, 1: 1- 5
	Xie F , Zhang L . Nonlinear second-order singularly perturbed problems with integral boundary conditions. Journal of East China Normal University (Natural Science Edition), 2010, 1: 1- 5
12	武利猛, 倪明康, 李素红, 等. 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, 40A (5): 1192- 1203 doi: 10.3969/j.issn.1003-3998.2020.05.008
	Wu L M , Ni M K , Li S H , et al. Asymptotic solutions of singularly perturbed boundary value problems with integral boundary conditions. Acta Mathematica Scientia, 2020, 40A (5): 1192- 1203 doi: 10.3969/j.issn.1003-3998.2020.05.008

[1]	Shengxin Hua,Guolin Yu,Wenyan Han,Xiangyu Kong. Characterization of Optimality to Constrained Vector Equilibrium Problems via Approximate Subdifferential [J]. Acta mathematica scientia,Series A, 2022, 42(2): 365-378.
[2]	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.
[3]	Meng Li,Guolin Yu. Optimality of Robust Approximation Solutions for Uncertain Convex Optimization Problems [J]. Acta mathematica scientia,Series A, 2020, 40(4): 1007-1017.
[4]	Cheng Ouyang,Weigang Wang,Jiaqi Mo. The Fractional Generalized Disturbed Thermal Wave Equation [J]. Acta mathematica scientia,Series A, 2020, 40(2): 452-459.
[5]	Huaxiong Chen,Yanyan Wang,Mingkang Ni. The Contrast Structure for the Singularly Perturbed Problem with Slow-Fast Layers and Discontinuous Righthand Side [J]. Acta mathematica scientia,Series A, 2019, 39(4): 865-874.
[6]	Xianglin Han,Weigang Wang,Jiaqi Mo. Generalized Solution to the Singular Perturbation Problem for a Class of Nonlinear Differential-Integral Time Delay Reaction Diffusion System [J]. Acta mathematica scientia,Series A, 2019, 39(2): 297-306.
[7]	Liao Shupeng, Shen Jianhe. One-Pulse Travelling Front Solutions of a sine-Gordon Equation with Slowly Varying Parameters [J]. Acta mathematica scientia,Series A, 2018, 38(4): 810-822.
[8]	LIN Su-Rong. The Singular Perturbation of Boundary Value Problem on Infinite Interval for a Class Nonlinear Vector Differential Equation [J]. Acta mathematica scientia,Series A, 2014, 34(3): 727-737.
[9]	LONG Xian-Jun. Optimality Conditions for Approximate Solutions on Nonconvex Vector Equilibrium Problems in Asplund Spaces [J]. Acta mathematica scientia,Series A, 2014, 34(3): 593-602.
[10]	WU Li-Meng, NI Ming-Kang, LIU Hai-Bo. Internal Transition Layer Solution of Singularly Perturbed Optimal Control Problem [J]. Acta mathematica scientia,Series A, 2013, 33(2): 206-215.
[11]	LIN Su-Rong. Singular Perturbation of a Class of Third Order \|Partial Differential Equations under Limit Equation to be Mixed Type [J]. Acta mathematica scientia,Series A, 2011, 31(6): 1579-1585.
[12]	LIANG Fei, GAO Hong-Jun. Existence of Travelling Wave Solutions for a Reaction-diffusion Equation with Nonlocal Delay [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1273-1281.
[13]	YANG Yin, ZHOU Qin. Crank-Nicolson Difference Scheme for Convection-Diffusion Problem by Using \|Moving Mesh Method [J]. Acta mathematica scientia,Series A, 2011, 31(4): 1063-1070.
[14]	NI Ming-Kang, LIN Wu-Zhong. The Asymptotic Solutions of Delay Singularly Perturbed Differential Difference Equations [J]. Acta mathematica scientia,Series A, 2010, 30(6): 1413-1423.
[15]	Lin Surong. Application of the Diagonalization Method in the Robin Boundary Value Problem for the Vector Nonlinear Integro-differential Equations [J]. Acta mathematica scientia,Series A, 2009, 29(2): 406-415.

A Class of Weakly Nonlinear Critical Singularly Perturbed Integral Boundary Problems

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

Figures/Tables 2

References 12

Related Articles 15

Metrics

Comments

Recommended 10