上海大学学报(自然科学版) ›› 2020, Vol. 26 ›› Issue (6): 853-883.doi: 10.12066/j.issn.1007-2861.2259
• 特邀综述 • 下一篇
收稿日期:
2020-10-11
出版日期:
2020-12-31
发布日期:
2020-12-29
通讯作者:
倪明康
E-mail:xiaovikdo@163.com
作者简介:
倪明康(1964—), 男, 教授, 博士生导师, 博士, 研究方向为奇摄动理论和方法. E-mail: xiaovikdo@163.com基金资助:
NI Mingkang1,2(), PAN Yafei3, WU Xiao1
Received:
2020-10-11
Online:
2020-12-31
Published:
2020-12-29
Contact:
NI Mingkang
E-mail:xiaovikdo@163.com
摘要:
主要研究了右端不连续奇异摄动系统中空间对照结构研究状况.介绍了右端不连续的二阶非线性奇异摄动问题的空间对照结构的一系列工作,其中包 括半线性系统、拟线性系统和弱非线性系统的Dirichlet问题.同时, 介绍了右端不连续的一阶常微分方程组的齐次Neumann边值问题、一类分段光滑二阶Tikhonov系统Dirichlet边值问题和具有不连续项的奇异摄动抛物方程边值问题.
中图分类号:
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.
NI Mingkang, PAN Yafei, WU Xiao. Spatial contrast structure for singular perturbation problems with right end discontinuities[J]. Journal of Shanghai University(Natural Science Edition), 2020, 26(6): 853-883.
[1] | Tikhonov A N. Systems of differential equations containing a small parameter[J]. Matematicheskii Sbornik, 1952,31:575-586. |
[2] | Tikhonov A N. On the dependence of solutions of differential equations on a smallparameter[J]. Matematicheskii Sbornik, 1948,22:193-204. |
[3] |
Kamel A A. Perturbation method in the theory of nonlinear oscillations[J]. Celestial Mechanics, 1970,3(1):90-106.
doi: 10.1007/BF01230435 |
[4] | Vasil'eva A B, Butuzov V F. Asymptotic expansions of solutions of singularly perturbed equations[M]. Moscow: Nauka, 1973. |
[5] |
Zhu H P. The dirichlet problem for a singular singularly perturbed quasilinear second order differential system[J]. J Math Anal Appl, 1997,210(1):308-336.
doi: 10.1006/jmaa.1997.5405 |
[6] |
Butuzov V F, Nefedov N N. A singularly perturbed boundary value problem for a second-order equation in the case of variation of stability[J]. Math Notes, 1998,63(3):311-318.
doi: 10.1007/BF02317775 |
[7] |
Butuzov V F, Nefedov N N, Schneider K R. Singularly perturbed boundary value problems in case of exchange of stabilities[J]. J Math Anal Appl, 1999,229(2):543-562.
doi: 10.1006/jmaa.1999.6180 |
Larin A A. A boundary value problem for a second-order singular elliptic equation in a sector on the plane[J]. Differ Equ, 2000,36(12):1850-1858. | |
[9] |
Farrell P A, Hegarty A F, Miller J J H, et al. Singularly perturbed convection-diffusion problems with boundary and weak interior layers[J]. J Comput Appl Math, 2004,166(1):133-151.
doi: 10.1016/j.cam.2003.09.033 |
[10] | Vasil'eva A B, Kalachev L V. Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions[J]. Abstr Appl Anal, 2000,2006(2):1-21. |
[11] | Kuma V. Solving singularly perturbed reaction diffusion problems using wavelet optimized finite difference and cubic spline adaptive wavelet scheme[J]. Int J Numer Anal Model, 2008,5(2):270-285. |
[12] |
Martinez S, Wolanski N. A singular perturbation problem for a quasi-linear operator satisfying the natural growth condition of Lieberman[J]. SIAM J Math Anal, 2009,41(1):318-359.
doi: 10.1137/070703740 |
[13] |
Vasil'eva A B, Butuzov V F, Nefedov N N. Singularly perturbed problems with boundary and internal layers[J]. Proc Steklov Inst Math, 2010,268(1):258-273.
doi: 10.1134/S0081543810010189 |
[14] |
Butuzov V F, Nefedov N N, Recke L, et al. On a singularly perturbed initial value problem in the case of a double root of the degenerate equation[J]. Nonlinear Anal Theor Methods Appl, 2013,83:1-11.
doi: 10.1016/j.na.2013.01.013 |
[15] | Bogoliubov N N, Mitropolski Y A. Asymptotic methods in the theory of non-linear oscillations[M]. New York: Gordon and Breach, 1961. |
[16] | Rellich F. Perturbation theory of eigenvalue problems[M]. New York: Gordon and Breach, 1969. |
[17] | de Bruijn N G. Asymptotic methods in analysis[M]. New York: Dover Publication, 2010. |
[18] | Vasil'eva A B, Butuzov V F, Kalachev L V. The boundary function method for singular perturbation problems[M]. Philadelphia: SIAM, 1995. |
[19] | de Jager E M, Furu J F. The theory of singular perturbations[M]. North Holland: Elsevier, 1996. |
[20] | Wasow W. Asymptotic expansions for ordinary differential equations[M]. New York: Dover Publication, 2002. |
[21] | Bellman R E. Perturbation techniques in mathematics, engineering and physics[M]. New York: Dover Publication, 2003. |
[22] | Copson E T. Asymptotic expansions [M]. Cambridge: Cambridge University Press, 2004. |
[23] | Chang K W, Howes F A. Nonlinear singular perturbation phenomena: theory andapplications[M]. New York: Springer-Verlag, 2013. |
[24] | Grasman J. Asymptotic methods for relaxation oscillations and applications[M]. New York: Springer-Verlag, 2013. |
[25] | Mishchenko E. Differential equations with small parameters and relaxation oscillations[M]. New York: Springer-Verlag, 2012. |
[26] | 瓦西里耶娃АБ, 布图索夫ВФ. 奇异摄动方程解的渐近展开 [M]. 倪明康, 林武忠, 译. 北京: 高等教育出版社, 2008. |
[27] | Nayfeh A H. Perturbation methods[M]. New York: John Wiley & Sons, 2008. |
[28] | Nayfeh A H. Introduction to perturbation techniques[M]. New York: John Wiley & Sons, 1993. |
[29] | van Dyke M. Perturbation methods in fluid mechanics [M]. New York: Academic Press, 1964. |
[30] | Coppel W A. Dichotomies in stability theory[M]. Berlin: Springer-Verlag, 1978. |
[31] | Lagerstrom P A. Matched asymptotic expansions: ideas and techniques[M]. New York: Springer-Verlag, 1988. |
[32] | Il'in A M. Matching of asymptotic expansions of solutions of boundary value problems[M]. Providence R I: Amer Math Soc, 1992. |
[33] | O'malley R E. Singular perturbation methods for ordinary differential equations[M]. New York: Springer-Verlag, 1991. |
[34] | Holmes M. Introduction to perturbation methods[M]. New York: Springer-Verlag, 1999. |
[35] | Kevorkian J, Cole J D. Perturbation methods in applied mathematics[M]. New York: Springer-Verlag, 2013. |
[36] | Krylov N M, Bogoliubov N N. Introduction to non-linear mechanics [M]. New Jersey: Princeton University Press, 1943. |
[37] | Mickens R E. An introduction to nonlinear oscillations [M]. Cambridge: Cambridge University Press, 1981. |
[38] | Robert J E. Singular perturbation methods for ordinary differential equations[M]. New York: Springer-Verlag, 2012. |
[39] | Howes F A. Boundary-interior layer interactions in nonlinear singular perturbation theory[M]. Providence R I: Amer Math Soc, 1978. |
[40] | Chien W Z. Asymptotic behavior of a thin clamped circular plate under uniform normal pressure at very large defection[J]. The Science Reports of National Tsing Hua University, 1948,5(1):71-94. |
[41] |
Guo Y H. On the flow of an incompressible viscous fluid past a flat plate at moderate reynolds numbers[J]. J Math Phys, 1953,32(1):83-101.
doi: 10.1002/sapm195332183 |
[42] |
Lin J Q. On a perturbation theory based on the method of characteristics[J]. J Math Phys, 1954,33:117-134.
doi: 10.1002/sapm1954331117 |
[43] | Qian X S. The poincare-lighthill-kuo method[J]. Adv Appl Mech, 1956,4:281-349. |
[44] | Lin W Z, Wang Z M. A singular singularly perturbed boundary value problem[J]. Ann Diff Eqs, 1996,12:70-82. |
[45] | Mo J Q. The nonlocal boundary value problems of nonlinear elliptic systems in unbounded domains[J]. Appl Math Comput, 1997,86(2):115-121. |
[46] |
Chen X, Yu P, Han M, et al. Canard solutions of two-dimensional singularly perturbed systems[J]. Chaos, Solitons and Fractals, 2005,23(3):915-927.
doi: 10.1016/S0960-0779(04)00339-X |
[47] |
Du Z, Ge W, Zhou M. Singular perturbations for third-order nonlinear multi-point boundary value problem[J]. J Differ Equ, 2005,218(1):69-90.
doi: 10.1016/j.jde.2005.01.005 |
[48] |
Ni M K, Vasil'eva A B, Dmitriev M G. Equivalence of two sets of transition points corresponding to solutions with interior transition layers[J]. Matematicheskie Zametki, 2006,79(1):120-126.
doi: 10.4213/mzm |
[49] |
Mo J Q, Lin W T. Asymptotic solution for a class of EL-Ni(n)o oscillator model for El Ni(n)o-southern oscillation[J]. Chinese Physics B, 2008,17(2):370-372.
doi: 10.1088/1674-1056/17/2/002 |
[50] | Zhou Z, Shen J. Delayed phenomenon of loss of stability of solutions in a second-order quasi-linear singularly perturbed boundary value problem with a turning point[J]. Boundary Value Problems, 2011(1):1-13. |
[51] |
Xie F. On a class of singular boundary value problems with singular perturbation[J]. J Differ Equ, 2012,252(3):2370-2387.
doi: 10.1016/j.jde.2011.10.003 |
[52] | Shen J, Han M. Delayed bifurcation in first-order singularly perturbed problems with a nongeneric turning point[J]. Int J Bifur Chaos, 2012,22(12):1973-1982. |
[53] |
Zheng Y G, Bao L J. Slow-fast dynamics of tri-neuron hopfield neural network with two timescales[J]. Commun Nonlinear Sci Numer Simulat, 2014,19(5):1591-1599.
doi: 10.1016/j.cnsns.2013.09.001 |
[54] | 王爱峰, 倪明康. 一类拟线性奇摄动方程的无穷大初值问题[J]. 吉林大学学报(理学版), 2010,48:52-56. |
[55] | 钱伟长. 奇异摄动理论及其在力学中的应用 [M]. 北京: 科学出版社, 1981. |
[56] | 林宗池, 周明儒. 应用数学中的摄动方法 [M]. 南京: 江苏教育出版社, 1995. |
[57] | 刘树德, 鲁世平. 奇异摄动边界层和内部层理论 [M]. 北京: 科学出版社, 2012. |
[58] | 周明儒, 杜增吉. 奇异摄动中的微分不等式理论 [M]. 北京: 科学出版社, 2012. |
[59] | 倪明康, 林武忠. 奇异摄动问题中的渐近理论 [M]. 北京: 高等教育出版社, 2009. |
[60] | 倪明康, 林武忠. 奇异摄动问题中的空间对照结构理论 [M]. 北京: 科学出版社, 2013. |
[61] |
Kinkaid N M, O'reilly O M, Papadopoulos P. Automotive disc brake squeal[J]. J Sound Vibration, 2003,267(1):105-166.
doi: 10.1016/S0022-460X(02)01573-0 |
[62] |
Chen H J. Social status human capital formation and super-neutrality in a two sector monetary economy[J]. Economic Modeling, 2011,28:785-794.
doi: 10.1016/j.econmod.2010.10.010 |
[63] |
Hargrove J W, Humphrey J H, Mahomva A, et al. Declining HIV prevalence and incidence in perinatal women in Harare[J]. Zimbabwe Epidemics, 2011,3:88-94.
doi: 10.1016/j.epidem.2011.02.004 pmid: 21624779 |
[64] | Masri S F. Analytical and experimental studies of multiple-unit impact dampers[J]. J Acoust Soci Amer, 1969,45(5):1111-1117. |
[65] | Shi H, Bai Y, Han M. On the maximum number of limit cycles for a piecewise smooth differential system[J]. Bulletin Des Ences Mathématiques, 2020,163:102887. |
[66] |
Liang H H, Li S M, Zhang X. Limit cycles and global dynamics of planar piecewise linear refracting systems of focus-focus type[J]. Nonlinear Anal Real World Appl, 2021,58:103228.
doi: 10.1016/j.nonrwa.2020.103228 |
[67] |
Kadalbajoo M K, Reddy Y N. Asymptotic and numerical analysis of singular perturbation problems: a survey[J]. Applied Mathematics and Computation, 1989,30(3):223-259.
doi: 10.1016/0096-3003(89)90054-4 |
[68] |
Kadalbajoo M K, Patidar K C. A survey of numerical techniques for solving singularly perturbed ordinary differential equations[J]. Applied Mathematics and Computation, 2002,130(2):457-510.
doi: 10.1016/S0096-3003(01)00112-6 |
[69] |
Kadalbajoo M K, Gupta V. A brief survey on numerical methods for solving singularly perturbed problems[J]. Applied Mathematics and Computation, 2016,217(8):3641-3716.
doi: 10.1016/j.amc.2010.09.059 |
[70] |
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 |
[71] |
Ni M K, Nefedov N N. Internal layers in the one-dimensional reation-diffusion equation with a discontinuous reactive term[J]. Comput Math Math Phys, 2015,55(12):2001-2007.
doi: 10.1134/S096554251512012X |
[72] |
Levashova N T, Nefedov N N, Orlov A O. Time-independent reation-diffusion equation with a discontinuous reactive term[J]. Comput Math Math Phys, 2017,57(5):854-866.
doi: 10.1134/S0965542517050062 |
[73] | Volkov V, Nefedov N N. Asymptotic-numerical investigation of generation and motion of fronts in phase transition models[M]. Berlin: Springer-Verlag, 2012: 524-531. |
[74] |
Nefedov N N, Recke L, Schneider K R. Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations[J]. J Math Anal Appl, 2013,405(1):90-103.
doi: 10.1016/j.jmaa.2013.03.051 |
[75] |
Ni M K, Pan Y F, Levashova N T, et al. Internal layer for a singularly perturbed second-order quasi-linear differential equation with discontinuous right-hand side[J]. Differ Equ, 2017,53(12):1616-1626.
doi: 10.1134/S0012266117120096 |
[76] |
Pan Y F, Ni M K, Davydova M A. Contrast structures in problems for a stationary equation of reaction-diffusion-advection type with discontinuous nonlinearity[J]. Mathematical Notes, 2018,104(5/6):735-744.
doi: 10.1134/S0001434618110159 |
[77] |
Pang Y F, Ni M K, Levashova N T. Internal layer for a system of singularly perturbed equations with discontinuous right-hand side[J]. Differential Equations, 2018,54(12):1583-1594.
doi: 10.1134/S0012266118120054 |
[78] |
Qi X T, Ni M K. On the asymptotic solution to a type of piecewise-continuous second-order dirichlet problems of Tikhonov system[J]. Journal of Applied Analysis and Computation, 2019,9(1):105-117.
doi: 10.11948/2019.105 |
[79] | 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, 2020. DOI: 10.3934/dcdss.2020341. |
[80] |
Wu X, Ni M K. Existence and stability of periodic contrast structure in reaction-advection-diffusion equation with discontinuous reactive and convective terms[J]. Commun Nonlinear Sci Numer Simulat, 2020,91:105457.
doi: 10.1016/j.cnsns.2020.105457 |
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