带有积分边界条件的奇异摄动边值问题的渐近解

Abstract

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

CLC Number:

O175.14

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.

Trendmd

References

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
1	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
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
5	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
6	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
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
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
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
12	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
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
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
15	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
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
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
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
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
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

[1]	Limeng Wu, Suhong Li, Mingkang Ni, Haibo Lu. Contrast Structure of Singularly Perturbed Integral Boundary Value Problem [J]. Acta mathematica scientia,Series A, 2025, 45(5): 1565-1576.
[2]	Zheng Yanping, Shen Jianhe. Traveling Fronts in a Social Tension-Outbursts Multi-Scale Reaction-Diffusion Equation with Allee Effect [J]. Acta mathematica scientia,Series A, 2025, 45(3): 776-789.
[3]	Fu Yuechen, Ni Mingkang. The Inner Layer of a Class of Singularly Perturbed High-Order Equations with Discontinuous Right-Hand Side [J]. Acta mathematica scientia,Series A, 2024, 44(5): 1153-1166.
[4]	Li Min, Pu Xueke. Long-Wavelength Limit for the Two-Fluid Euler-Poisson Equation [J]. Acta mathematica scientia,Series A, 2023, 43(2): 399-420.
[5]	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.
[6]	Cheng Ouyang,Weigang Wang,Jiaqi Mo. The Fractional Generalized Disturbed Thermal Wave Equation [J]. Acta mathematica scientia,Series A, 2020, 40(2): 452-459.
[7]	Cheng Ouyang,Weigang Wang,Jiaqi Mo. The Asymptotic Path Curve for Fermi Gases Optical Lattices Model [J]. Acta mathematica scientia,Series A, 2019, 39(6): 1555-1562.
[8]	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.
[9]	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.
[10]	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.
[11]	Ouyang Cheng, Mo Jiaqi. Study of Nonlinear Duffing Disturbed Oscillator for Resonance Mechanism [J]. Acta mathematica scientia,Series A, 2018, 38(1): 190-196.
[12]	LIU Shuai, CHEN Jian-He, ZHOU Zhe-Yan. Asymptotic Solution for Second-order Semilinear Singularly Perturbed Boundary Value Problems [J]. Acta mathematica scientia,Series A, 2014, 34(5): 1104-1110.
[13]	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.
[14]	LIU Shuai, ZHOU Zhe-Yan, SHEN Jian-He. Asymptotic Behavior of Solutions for Second-Order Semilinear Singularly Perturbed Boundary Value Problem [J]. Acta mathematica scientia,Series A, 2014, 34(2): 437-444.
[15]	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.

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

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0