一类具有转点的右端不连续奇摄动边值问题

帅欣; 倪明康

doi:10.21656/1000-0887.440353

一类具有转点的右端不连续奇摄动边值问题

doi: 10.21656/1000-0887.440353

帅欣,
倪明康^,

华东师范大学数学科学学院, 上海 200241

基金项目:

国家自然科学基金 12371168

上海市科学技术委员会基金 18dz2271000

详细信息

作者简介:
帅欣(1997—)，男，博士生(E-mail：994552419@qq.com)

通讯作者:
倪明康(1963—)，男，教授，博士，博士生导师(通讯作者. E-mail：xiaovikdo@163.com)

中图分类号: O175.14
计量
- 文章访问数: 69
- HTML全文浏览量: 24
- PDF下载量: 12
- 被引次数: 0
出版历程
- 收稿日期: 2023-12-11
- 修回日期: 2024-01-12
- 刊出日期: 2024-04-01

A Class of Right-Hand Discontinuous Singularly Perturbed Boundary Value Problems With Turning Points

SHUAI Xin,
NI Mingkang^,

School of Mathematical Sciences, East China Normal University, Shanghai 200241, P.R.China

摘要

摘要: 研究了一类具有转点的右端不连续二阶半线性奇摄动边值问题解的渐近性. 首先，在间断处将原问题分为左右两个问题，通过修正左问题退化问题的正则化方程，提高了左问题渐近解的精度，并利用Nagumo定理证明了左问题光滑解的存在性. 其次，证明了右问题具有空间对照结构的解，并通过在间断点的光滑缝接，得到了原问题的渐近解. 最后，通过一个算例验证了结果的正确性.
- 奇摄动 /
- 边值问题 /
- 转点 /
- 右端不连续 /
- 空间对照 /
- 渐近解
Abstract: The asymptotic behavior of solutions to a class of right-hand discontinuous 2nd-order semilinear singularly perturbed boundary value problems with turning points was studied. Firstly, the original problem was divided into left and right problems at the discontinuity, the accuracy of the asymptotic solution to the left problem was improved through modification of the regularization equation for the left problem degradation problem, and the existence of the smooth solution to the left problem was proved by means of the Nagumo theorem. Secondly, the solution to the right problem was proved to have a spatial contrast structure, and the asymptotic solution to the original problem was obtained through smooth joints at the discontinuity points. Finally, the correctness of the results was verified by an example.
- singular perturbation /
- boundary value problem /
- turning point /
- discontinuous right-hand side /
- spatial contrast structure /
- asymptotic solution estimation

HTML全文

图 1 问题(1)退化解的示意图及真解u(x)的示意图

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. Schematic diagram of the problem (1) degenerate solution and true solution u(x)

下载: 全尺寸图片幻灯片

图 2 u(x)与φ(x, μ)的示意图

Figure 2. Schematic diagram of u(x) and φ(x, μ)

下载: 全尺寸图片幻灯片

图 3 问题(1)的零阶渐近解(μ=0.01)

Figure 3. The zero-order asymptotic solution of problem (1) (μ=0.01)

下载: 全尺寸图片幻灯片

参考文献(18)

[1]	KASHCHENKO S. Asymptotics of regular and irregular solutions in chains of coupled Van der Pol equations[J]. Mathematics, 2023, 11(9): 2047. doi: 10.3390/math11092047
[2]	ARZANI A, CASSEL K W, D'SOUZA R M. Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation[J]. Journal of Computational Physics, 2023, 473: 111768. doi: 10.1016/j.jcp.2022.111768
[3]	CENGIZCI S, KUMAR D, ATAY M T. A semi-analytic method for solving singularly perturbed twin-layer problems with a turning point[J]. Mathematical Modelling and Analysis, 2023, 28(1): 102-117. doi: 10.3846/mma.2023.14953
[4]	SAY F. On the asymptotic behavior of a second-order general differential equation[J]. Numerical Methods for Partial Differential Equations, 2022, 38(2): 262-271. doi: 10.1002/num.22774
[5]	NISHIURA Y, SUZUKI H. Matched asymptotic expansion approach to pulse dynamics for a three-component reaction-diffusion system[J]. Journal of Differential Equations, 2021, 303: 482-546. doi: 10.1016/j.jde.2021.09.026
[6]	NEFEDOV N N. Development of methods of asymptotic analysis of transition layers in reaction-diffusion-advection equations: theory and applications[J]. Computational Mathematics and Mathematical Physics, 2021, 61: 2068-2087. doi: 10.1134/S0965542521120095
[7]	包立平, 李瑞翔, 吴立群. 一类KdV-Burgers方程的奇摄动解与孤子解[J]. 应用数学和力学, 2021, 42(9): 948-957. doi: 10.21656/1000-0887.420011 BAO Liping, LI Ruixiang, WU Liqun. Singularly perturbed and soliton solutions to a class of KdV-Burgers equations[J]. Applied Mathematics and Mechanics, 2021, 42(9): 948-957. (in Chinese) doi: 10.21656/1000-0887.420011
[8]	朱红宝, 陈松林. 一类二阶双参数非线性时滞问题的奇摄动[J]. 应用数学和力学, 2020, 41(11): 1292-1296. doi: 10.21656/1000-0887.410082 ZHU Hongbao, CHEN Songlin. A class of 2nd-order singularly perturbed time delay nonlinear problems with 2 parameters[J]. Applied Mathematics and Mechanics, 2020, 41(11): 1292-1296. (in Chinese) doi: 10.21656/1000-0887.410082
[9]	包立平, 胡玉博, 吴立群. 具有初值间断的Burgers方程奇摄动解[J]. 应用数学和力学, 2020, 41(7): 807-816. doi: 10.21656/1000-0887.400270 BAO Liping, HU Yubo, WU Liqun. Singularly perturbed solutions of Burgers equations with initial value discontinuities[J]. Applied Mathematics and Mechanics, 2020, 41(7): 807-816. (in Chinese) doi: 10.21656/1000-0887.400270
[10]	ACKERBERG R C, O'MALLEY R E. Boundary layer problems exhibiting resonance[J]. Studies in Applied Mathematics, 1970, 49(3): 277-295. doi: 10.1002/sapm1970493277
[11]	SHARMA K K, RAI P, PATIDAR K C. A review on singularly perturbed differential equations with turning points and interior layers[J]. Applied Mathematics and Computation, 2013, 219(22): 10575-10609. doi: 10.1016/j.amc.2013.04.049
[12]	KARALI G, SOURDIS C. Resonance phenomena in a singular perturbation problem in the case of exchange of stabilities[J]. Communications in Partial Differential Equations, 2012, 37(9): 1620-1667. doi: 10.1080/03605302.2012.681333
[13]	KUMAR D. A collocation method for singularly perturbed differential-difference turning point problems exhibiting boundary/interior layers[J]. Journal of Difference Equations and Applications, 2018, 24(12): 1847-1870. doi: 10.1080/10236198.2018.1543417
[14]	BUTUZOV V F, NEFEDOV N N, SCHNEIDER K R. Singularly perturbed problems in case of exchange of stabilities[J]. Journal of Mathematical Sciences, 2004, 121(1): 1973-2079. doi: 10.1023/B:JOTH.0000021571.21423.52
[15]	赵敏, 倪明康. 一类具有转点的二阶半线性奇摄动边值问题[J]. 华东师范大学学报(自然科学版), 2023(2): 26-33. https://www.cnki.com.cn/Article/CJFDTOTAL-HDSZ202302005.htm ZHAO Ming, NI Mingkang. A class of second-order semilinear singularly perturbed boundary value problems with turning points[J]. Journal of East China Normal University (Natural Science), 2023(2): 26-33. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-HDSZ202302005.htm
[16]	WU X, NI M K. Solution of contrast structure type for a reaction-diffusion equation with discontinuous reactive term[J]. Discrete and Continuous Dynamical Systems: S, 2021, 14(9): 3249-3266. doi: 10.3934/dcdss.2020341
[17]	倪明康, 潘亚飞, 吴潇. 右端不连续奇异摄动问题的空间对照结构[J]. 上海大学学报(自然科学版), 2020, 26(6): 853-883. https://www.cnki.com.cn/Article/CJFDTOTAL-SDXZ202006001.htm NI Mingkang, PAN Yafei, WU Xiao. Spatial contrast structure for singular perturbation problems with right end discontinuities[J]. Journal of Sanghai University (Natural Science), 2020, 26(6): 853-883. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-SDXZ202006001.htm
[18]	LIUBAVIN A, NI M, WU X. Spatial contrast structural solution of a class of piecewise-smooth critical semilinear singularly perturbed differential equation[J]. Journal of Jilin University (Science Edition), 2021, 59(6): 1303-1309.