收稿日期: 2020-10-11
网络出版日期: 2020-11-10
基金资助
国家自然科学基金资助项目(11871217);上海市科委基金资助项目(A18DZ2271000)
Spatial contrast structure for singular perturbation problems with right end discontinuities
Received date: 2020-10-11
Online published: 2020-11-10
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 . DOI: 10.12066/j.issn.1007-2861.2259
This paper surveys recent developments in spatial contrast structure solutions to singularly perturbed problems with discontinuous right-hand sides. Studies on second-order non-linear singularly perturbed problems, including semi-linear, quasi-linear, and weakly non-linear system Dirichlet problems, are reviewed. In addition, the first-order ordinary differential equations under homogeneous Neumann conditions are discussed. A type of piecewise-continuous second-order Dirichlet problems of the Tikhonov system and boundary value problem of a singularly perturbed parabolic equation with a discontinuous term is also included.
| [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. |
| [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. |
| [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. |
| [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. |
| [7] | 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [42] | Lin J Q. On a perturbation theory based on the method of characteristics[J]. J Math Phys, 1954,33:117-134. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
/
| 〈 |
|
〉 |
