摘要
本文利用边界层函数法、多元缝接法、隐函数定理以及其他方法,构造了四类时滞奇摄动问题解的渐近表达式,得到原问题解存在的充分条件。探讨了内部层出现的区域及原因,证明了形式渐近解的对原问题精确解的一致有效性.并给出其在系统控制理论、化学反应动力学、半导体理论等领域的应用模型.对已有的一些结论做了一定程度的推广
全文共分五章.第一章叙述了奇摄动问题的发展过程,引入了与本文研究内容相关的一些概念与定理.着重介绍了本文工作的特色和创新之处.
第二章,第五章分别讨论了一类拟线性时滞奇摄动初边值问题和一类具有时滞的捕食-食饵种群问题.利用边界层函数构造了问题的形式渐近解,并利用逐次逼近和压缩映照不动点定理证明了解的存在性,最后结合数值模拟验证了所得结果.
第三章着重研究高维时滞奇摄动初边值问题.本章共分两节.我们首先对一类快系统的内部层问题进行研究.再对具有快慢变量的Tikhonov系统的内部层问题进行探讨.利用边界层函数法分别得到了两个系统的近似解,再运用多元缝接法对轨道进行了光滑缝接.不同于第二、五章,本章我们运用隐函数定理证明了在退化解附近,对于充分小的参数μ,原问题解的存在性和渐近解的一致有效性.
第四章致力于探论了临界情况下的时滞奇摄动问题.我们分别对弱非线性初值问题的内部层和高维方程组初边值问题的内部层进行研究.构造了原文的形式渐近解.并利用逐次逼近和压缩映照不动点定理得到解的存在性及其近似解的一致有效性.最后举例说明所得结果.
By applying the boundary layer function method, patching connections, implicit function theorem and other methods, this dissertation mainly aims to construct the asymptotic expansion of the solution for four kinds of singu-larly perturbed problems. The sufficient conditions to obtain the existence of the solution are also found. We consider the region and reason for the inner layer, and give the proof of the uniformly valid asymptotic expansion. Meanwhile, we give the corresponding application models in the system of control theory, chemical kinetics, semiconductor theory and the other fields. The result obtained in the thesis generalizes and extends the corresponding known results.
The whole thesis contains fives chapters. The first chapter introduces concisely the background of the subjects relevant to this dissertation and some definitions and theorems, emphatically introducing the main work to be done in this thesis.
Chapter2and Chapter5mainly consider a kind of quasi-linear singu-larly perturbed differential difference equation with initial values and bound-ary values and a class of predator-prey models. By using the boundary layer function method, the asymptotic solutions of these problems are constructed. The existence of the solution to the original problem has also been proved by successive approximation method and the contraction mapping theorem.
Chapter3is devoted to the problems for vector singularly perturbed delay-differential equations. This Chapter contains two sections. The first section narrate the internal layer problem of a kind of fast system. In the sec-ond section, we study the singularly perturbed delayed systems of Tikhonov's type with fast and slow variables. By means of the boundary layer function method, patching connections and implicit function theorem we prove the existence of solution of our problems near the degenerate solution for suffi-ciently small μ and obtain the uniformly valid solution in the whole interval.
Chapter4is focused on singularly perturbed delayed boundary value problems in the critical case. We consider the internal layers of a weakly nonlinear initial value problem and high dimensional differential equations respectively. We not only construct the asymptotic expansion of the solu-tion for the original equation but also give the proof of the uniformly valid asymptotic expansion by using successive approximations and the contrac-tion mapping theorem. Meanwhile, we give examples to demonstrate our results.
引文
[1]倪明康,林武忠,奇异摄动问题中的渐近理论,高等教育出版社,北京,2009。
[2]L. Prandtl, Fluid motions with very small friction, Proceedings of the 3rd International Mathematical Congress, Heidelberg:H. Schlichting,1904:484-491.
[3]F. P. Brethenton, Slow viscous motion round a cylinder in a simple shear, J.Fluid Mech.,12,1962:591-613.
[4]M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York,1964.
[5]W. A. Wasow, Asymptotic Expansions for Ordinary Differential Equations, John, Wiley, New York,1965.
[6]J. D. Cole and J. Kevorkian, niformly valid asymptotic approximations for certain differential equations, Symp. on Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York,1963:113-120.
[7]A. B. Vasileva and V. F. Butuzov, Asymptotic Expansions of the Solutions of Singularly Perturbed Equations, Nauka Moscow,1973.
[8]A. B. Vasileva and V. F. Butuzov, The Singular Perturbation of ODE in the Critical Case, Nauka Moscow,1978.
[9]章国华,F.A.侯斯,非线性奇异摄动现象：理论和应用,林宗池等译,福建科学技术出版社,1989.
[10]A. H. Nayfeh,摄动方法导引.宋家骕(译),上海翻译出版公司,1990.
[11]钱伟长,奇异摄动理论及其在力学中的应用,科学出版社,1981.
[12]Y. H. Kou, On the flow of an incompressoble viscous fluid past a flat plate at moderate Reynolds number, J. Math. And phys.,32,1953:83-101.
[13]H. S. Tsien, The Poincare-Lighthill-Kuo method, Advan. Appl. Mech.,4, 1956:281-349.
[14]E. M. DeJager and F. R. Jiang, the theory of singular perturbations, NORTH-HOLLAND,1996.
[15]R. E. O'Malley, Introduction to Singular Perturbations, Academic Press. New York.1974.
[16]P. Germain, Recent evolution in problems and methods in aerodynamics, J. Roy, Aeron Soc.,71,1967:673-691.
[17]G. F. Carrier, Singular Perturbation theory and geophysics, SIAM Rew.,12, 1970:175-193.
[18]郑祖庥,泛函微分方程理论,安徽教育出版社,1994.
[19]汪娜,倪明康,经典物理中的扰动时滞模型解,物理学报,2011,60(5)：050203。
[20]J. K. Hale and W. Z. Huang, Periodic solutions of singularly perturbed delay equations. Z angew Math Phys,47,1996.
[21]J. Mallet-Paret and R. D. Nussbaumb, Boundary layer phenomena for differential-delay equations withstate-dependent time lags:III, J. Differen-tial Equations,189,2003:640-692.
[22]H.-O. Walther, The solution manifold and C1-smoothness for differential equations with state-dependent delay, J. Differential Equations,195,2003: 46-65.
[23]V. Y. Glizer, NovelControllability Conditions for a Class of Singularly Per-turbed Systems with Small State Delays, J Optim Theory Appl,137,2008: 135-156.
[24]M. K. Kadalbajoo and D. Kumar, A computational method for singularly perturbed nonlinear differential-difference equations with small shift, Ap-plied Mathematical Modelling,2009.
[25]M. S. Alwan, X. Z. Liu, Stability of singularly perturbed switched systems with time delay and impulsive effects, Nonlinear Analysis,71,2009:4297-4308.
[26]M. H. Adhikari, E. A. Coutsias and J. K. Mclver, Periodic solutions of a singularly perturbed delay differential equation, Physica D,237,2008:3307-3321.
[27]G. Grammel, Robustness of exponential stability to singular perturbations and delays. Systems & Control Letters,57,2008:505-510.
[28]A. R. Ansari, S. A. Bakr and G. I. Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, Journal of Computational and Applied Mathematics,205,2007: 552-566.
[29]M. Gulsu and M. Sezer, Approximations to the solution of linear Fredholm Integrodifferential-difference equation of high order, Journal of the Franklin Institute.343,2006:720-737.
[30]H. J. Tian, Asymptotic expansion for the solution of singularly perturbed delay differential equations, J. Math. Anal. Appl.281,2003:678-696.
[31]D. Duehring and W. Z. Huang, Periodic traveling wave solutions for a reaction-diffusion equation with time delay and non-local response, Journal of Dynamics and Differential Equations,19,2006(2):457-477.
[32]M. K. Ni and Z. M. Wang, On step-like contrast structure of singularly perturbed systems. Boundary Value Problems,2009.
[33]A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problem, American Mathematical Society Providence, Rhode Island, 1991.
[34]J. Q. Mo and W.T. Lin, A class of singularly perturbed nonlinear boundary value problem, Appl. Math. J. Chinese Univ. Ser. B,20,2005(2):159-164.
[35]J.1. Ren and W. G. GE, Singularly perturbed problems for semi-linear retarded equations, Applied Mathematics and Mechanics (English Edition), 24,2003(12):1450-1455.
[36]J. Q. Mo, Quasilinear singularly perturbed problem with boundary pertur-bation. Journal of Zhejiang University SCIENCE,5,2004(9):1144-1147.
[37]J. Q. Mo, H. Wang and W. T. L, The solvability for a class of singularly perturbed quasi-linear differential system, Acta Mathematica Scientia,28B, 2008(3):495-500.
[38]J. Q. Mo, Singularly perturbed differential-difference reaction diffusion equa-tions with time delay, J.Shanghai Jiaotong Univ.(Sci.),14,2009(5):629-631.
[39]Y. Kuang, Delay differential equations with applications in population dy-namical system. New York:Academic Press,1993.
[40]V. Kolmanovskii and A. Myshkis, Applied theory of functional differential equations, Kluwer,1999.
[41]R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIMA Journal on Control and Optimization,26,1988:697-713.
[42]H. Logemann, R. Rebarber and G. Weiss, Conditions for robustness and nonbustness of the stability of feedback systems with respect to small de-lays in the feedback loop, SIAM Journal on Control and Optimization,34, 1996(2):572-600.
[43]S. Nicaise and C. Pignotti, Stability and instability results of the wave eqution with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization,45,2006(5):1561-1585.
[44]A.B.瓦西里耶娃,V.F.布图索夫,奇异摄动方程解的渐近展开,倪明康,林武忠译.高等教育出版社,北京,2008.
[45]M. K. Ni, Asymptotics of the rung-type contrast structure for a class of variational problems, Automation and Remote Control,69,2008(4):708-715.
[46]倪明康,林武忠,具有阶梯状空间对照结构的奇摄动,高校应用数学学报24,2009(3)：290-300.
[47]冯茂春,二次方程Robin题奇摄动群的估计,应用数学,20,2007(3)：467-472.
[48]陈怀军,莫嘉琪,一类燃烧奇摄动问题的渐近估计,系统科学与数学,30,2010(1)：114-117.
[49]R. R. Tang, The shock problems for a class of nonlinear singularly perturbed equations. Advances in Mathematics,34,2005(2):233-240.
[50]倪明康,V.I. Gurman,奇摄动问题中阶梯状解的缝接法,数学的实践与认识,4，2011：202-208。
[51]A. B. Vasil' eva, V. F. Butuzov and N. N. Nefedov, Contrast structures in sin-gularly perturbed problems, Pure Mathematics and Applied Mathematics, 4,1998(3):799-851.
[52]林苏榕,林宗池,对角化方法在非线性积分微分方程组奇摄动边值问题中的应用,应用数学学报,23,2000(4)：543-550。
[53]K. W. Chang, Diagonalization method for a vector boundary problem of singular perturbation type, Journal of Mathematical Analysis and Applica-tions,48,1974:652-665.
[54]K. W. Chang, Singular perturbations of a boundary problem for a vector second order differential equation, SIAM J. Appl. Math.30,1976(1):42-54.
[55]M. K. Ni and Z. M. Wang, On Step-like Contrast Structure of Singularly Perturbed Systems, Boundary Value Problems, Article ID 634324,17 pages doi:10.1155/2009/634324.
[56]倪明康,林武忠,具有内部层的奇摄动微分差分方程的渐近解,数学物理学报,30A,2010(6)：1413-1423.
[57]K. L. Cooke, Asymptotic theory for the delay-differential equation u' (t)=-au(t-r(u(t))), J. Math. Anal. Appl.,19,1967:160-173.
[58]R. D. Driver and M. J. Norris, Note on uniqueness for a one-dimensional two-body problem of classical electrodynamics, Ann. Physics,42,1967:347-351.
[59]W. Alt, Some periodicity criteria for functional differential equations, Manuscripta Math,23,1978:295-318.
[60]R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl.,101,1974:263-306.
[61]W. G. Aiello, H. I. Preedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math.,52,1992:855-869.
[62]P. Brunovsky, A. Erdelyi and H.-O. Walther, On a model of a currency exchange rate-local stability and periodic solutions, J. Dynam. Differential Equations,16,2004:393-432. Erratum to:"On a model of a currency ex-change rate-local stability and periodic solutions", J. Dynam. Differential Equations,20,2008:271-276.
[63]H. Muller-Krumbhaar and J. P. van der Eerden, Some properties of simple recursive differential equations, Z. Phys. B,67,1987:239-242.
[64]T. Krisztin, C1-smoothness of center manifolds for differential equations with state-dependent delay, in:H. Brunner, X.-Q. Zhao, X. Zou (Eds.), Nonlinear Dynamics and Evolution Equations, in:Fields Inst. Commun., vol.48, Amer. Math. Soc., Providence, RI,2006, pp.213-226.
[65]J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymp-totics in singularly perturbed state-dependent delay-differential equations, Journal of Differential Equations,250,2011:4037-4084.
[66]V. F. Butuzov and N. N. Nefedov, Singularly Perturbed Boundary Value Problems for Systems of Tichonov's Type in Case of Exchange of Stabilities, Journal of Differential Equations,159,1999:427-446.
[67]V. F. Butuzov and N. N. Nefedov, Singularly Perturbed Boundary Value Problems in Case of Exchange of Stabilities, Journal of Mathematical Anal-ysis and Applications,229,1999:543-562.
[68]R. E. O' Malley, Singular perturbations for Ordinary Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York,1991.
[69]康永海,关于二阶线性时变系统在第一临界情况下的稳定性,数学研究,1998，31：308-311。
[70]倪明康,临界情况奇摄动边值问题,华东师范大学学报(自然科学版),2,1989：1-10.
[71]倪明康,临界情况一般非线性边值问题,华东师范大学学报(自然科学版),3,1991：13-18.
[72]M. Alwan and X. Z. Liu, Stability of singularly perturbed switched systems with time delay and impulsive effects, Nonlinear Analysis,71,2009:4297-4308.
[73]M. H. Adhikari, E. A. Coutsias and J. K. McIver, Periodic solutions of a singularly perturbed delay differential equation, Physica D,237,2008:3307-3321.
[74]M. K. Kadalbajoo and V. P. Ramesh, Hybrid method for numerical solution of singularly perturbed delay differential equations, Applied Mathematics and Computation,187,2007:797-814.
[75]M. Guulsu and M. Sezer, Approximations to the solution of linear Fredholm integrodifferential-difference equation of high order, Journal of the Franklin Institute,343,2006:720-737.
[76]K. K. Sharma and A. Kaushik, A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations, Applied Mathematics and Computation,181,2006:756-766.
[77]A. B. Vasili'eva, V. F. Butuzov and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia,1995.
[78]C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential difference equations V. Small shift with layer behavior, SIAM Journal on Applied Mathe-matics.54,1994:249-272.
[79]A. S. Lodge, J. B. Mcleod and J. A. Nohel, A nonlinear singularly per-turbed volterra integro-differential equation occurring in polymer rheology, Proc.Roy.Soc.EdinburghSect.A,1978(80):99-137.
[80]A. B. Vasil'eva, A. F. Kardosysoev and V. G. Stelmakh, Boundary layers in the thoery of p-n junctions, Physics and Technology of semiconductors,10, 1976:1321-1329.
[81]P. A. Markowich, A theory for the apporoximation of solutions of boundary value problems on infinite intervals, SIMA J. Math. Anal.13,1982:484-513.
[82]A. B. Vasil'eva and V. G. Stelmakh, Singularly perturbed systems in the theory of transistors, USSR Comp. Math. Phys.,17,1977:48-58.
[83]A. B. Vasil'eva and V. F. Butuzov, Singularly perturbed equations in the critical case, Moscow State University, translated as MRC-Technical Sum-mary Report 2039, Mathematics Resarch Center, Madison, WI.1978.
[84]D. R. Smith, On a singularly perturbed boundary value problem arising in the physical theory of semiconductors, Techn. Univ. Miinchen, Techn. Rep. TUM-M8021.1980
[85]A. B. Vasil'eva, A. F. Kardosysoev and V. G. Stelmakh, Boundary layers in the thoery of p-n junctions, Physics and Technology of semiconductors,10, 1976:1321-1329.
[86]C. Schmeiser and R. Weiss, Asymptotic analysis of singular singularly per-turbed boundary value problems, SIAM J. Math. Anal.,17,1986(3):560-579.
[87]A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, MD,1925.
[88]V. Volterra, Variazionie futtuaziono del numero di individui in specie animali conviventi, Mem. Accad. Lincei,2,1926:31-113.
[89]A.L. Koch, Competitive coexistence of two predators utilizing the same prey under constant environmental conditions, J. Theoret. Biol.,44,1974:373-386.
[90]S.-B. Hsu, S.P. Hubbell and P.Waltman, Competing predators, SIAM J. Appl.Math.,35,1978:617-625.
[91]S.-B. Hsu, S.P. Hubbell and P. Waltman, A contribution to the theory of competing predators, Ecological Monographs,48,1978:337-349.
[92]H. L. Smith, The interaction of steady state and Hopf bifurcation in a two-predator-one-prey competition model, SIAM J. Appl. Math.,42,1982:27-43.
[93]J. P. Keener, Oscillatory coexistence in the chemostat:a codimension two unfolding, SIAM J. Appl. Math.43,1983:1005-1018.
[94]G. J. Butler and P. Waltman, Bifurcation from a limit cycle in a two preda-tor-one prey ecosystem modeled on a chemostat, J. Math. Biol.,12,1981: 295-449.
[95]H. L. Smith, Competitive coexistence in an oscillating chemostat, SIAM J. Appl. Math.40,1981:498-522.
[96]R. P. Wang, Competition in a patchy environment with cross-diffusion in a three-dimensional Lotka-Volterra system, Nonlinear Analysis:RealWorld Applications,11,2010:2726-2730.
[97]K.O. Yukio, Existence and Instability of Neumann Layer Solutions for a 3-component Lotka-Volterra Model with Diffusion, Journal of Mathematical Analysis and Applications,243,2000:357-372.
[98]J. L. Qiu and J. D. Cao, Exponential stability of a competitive Lotka-Volterra system with delays, Applied Mathematics and Computa-tion,201,2008:819-829.
[99]S. Muratori and S. Rinaldi, Remarks on competitive coexistence, SIAM J. Appl. Math.49,1989:1462-1472.
[100]W. S. Liu, D. M. Xiao and Y. F. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differential Equations,188,2003:306-331.