一类四阶线性微分方程的奇摄动问题
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摘要
具有奇性的常微分方程问题不仅在理论研究中占有非常重要的地位,而且在气体动力学、流体力学、边界层理论、经济学以及生物学等众多领域中都有非常重要的应用价值.长期以来,二阶奇摄动问题一直受到人们的重视,并深入而广泛的研究.本文研究一类四阶奇异摄动问题.我们应用重整化群方法构造出一类四阶线性奇异摄动初值问题的一致有效的渐近展开式.
本文共分三部分.第一部分主要介绍重整化群的相关背景及发展,以及一些已有重要结果;第二部分应用重整化群方法给出了一类线性四阶微分方程的渐近展开式.第三部分给出几个具体例子.
Most of the physical problems facing engineers,physicists,and applied mathematicians today exhibit certain essential features which preclude exact analytical solutions.Some of these features are nonlinearities,variable coefficients,complex boundary shapes,and nonlinearboundary conditions at known or,in some cases, unknown boundaries.Even if the exact solution of a problem can be found explicitly,it may be useless for mathematical and physicalinterpretation or numerical evaluation.Examples of such problems are Bessel functions of large argument and large-order and doubly periodic functions.Thus,in order to obtain informationabout solutions of equations,we are forced to resort to approximations,numerical solutions,or combinations of both.Foremost among the approximation methods are perturbation(asymptotic) methods.
According to these techniques,the solution is represented by the first few terms of an asymptotic expansion,usually not more than two terms.The expansions may be carried out in terms of a parameter(small or large)which appears naturally in the equations,or which may be artificially introduced for convenience.Such expansions are called parameter pertur-bations.Alternatively,the expansions may be carried out in terms of a coordinate(either small or large);these are called coordinate perturbations.
In parameter perturbations the quantities to be expanded can be functions of one or more variables besides the perturbation parameter.If we develop the asymptotic expansion of a function f(x;∈),where x is a scalar or vector variable independent ofε,in terms of the asymptotic sequenceδ_m(∈),we have f(x;∈)～∑_(m=0)~∞as∈→0 where the coefficientsa_m are functions of x only.
Definition 1 This expansion is said to be uniformly valid ifOtherwise the expansion is said to be nonuniformly valid.
For the uniformity conditions to hold,a_m(x)δ_m(∈)must be small compared to the precedingterm a_(m-1)(x)δ_(m-1)(∈)for each m.Sinceδ_m(∈) = o[δ_(m-1)(∈)]as∈→0,we require that a_m(x)be no more singular than a_(m-1)(x)for all values of x of interest if the expansion is to be uniform.In other words,each term must be a small correction to the preceding term irrespective of the value of x.
Construction of such approaches solutions,people have developed many effective methods,such as asymptotic matching,averaging,multiple scales,stretched coordinate and WKB methods.
But in the use of these methods,in order to ensure expansion of solutions uniformly valid. To irregular part,such as the Boundary Layer,or by the location and thickness of surfacemay appear small fraction of the market parameters,such as the need to have some understanding.There is also a need for closer asymptotic matching necessary.This makes them much more restricted applications.
Recently,a perturbative renormalization group method was developed by Chen,Goldenfeld,and Oono as unified tool for asymptotic analysis.
This paper consist of two parts.
In the first part,we use the perturbative renormalization group method to study a Class of Forth-order Linear Differential EquationsWe have:
Theorem 1 Assume thatp(x),q(x)are second continuous differentiable functiions in[0,1],p(0)＞0.Let u(t) be the solution of initial value problem (1)-(2),then wherewhere a_1(0),a_2(0),a_3(0),a_4(0) are determined by the initial value conditions (2).
In the second part,we give some examples to illustrate the application of the theorem.
本文共分三部分.第一部分主要介绍重整化群的相关背景及发展,以及一些已有重要结果;第二部分应用重整化群方法给出了一类线性四阶微分方程的渐近展开式.第三部分给出几个具体例子.
Most of the physical problems facing engineers,physicists,and applied mathematicians today exhibit certain essential features which preclude exact analytical solutions.Some of these features are nonlinearities,variable coefficients,complex boundary shapes,and nonlinearboundary conditions at known or,in some cases, unknown boundaries.Even if the exact solution of a problem can be found explicitly,it may be useless for mathematical and physicalinterpretation or numerical evaluation.Examples of such problems are Bessel functions of large argument and large-order and doubly periodic functions.Thus,in order to obtain informationabout solutions of equations,we are forced to resort to approximations,numerical solutions,or combinations of both.Foremost among the approximation methods are perturbation(asymptotic) methods.
According to these techniques,the solution is represented by the first few terms of an asymptotic expansion,usually not more than two terms.The expansions may be carried out in terms of a parameter(small or large)which appears naturally in the equations,or which may be artificially introduced for convenience.Such expansions are called parameter pertur-bations.Alternatively,the expansions may be carried out in terms of a coordinate(either small or large);these are called coordinate perturbations.
In parameter perturbations the quantities to be expanded can be functions of one or more variables besides the perturbation parameter.If we develop the asymptotic expansion of a function f(x;∈),where x is a scalar or vector variable independent ofε,in terms of the asymptotic sequenceδ_m(∈),we have f(x;∈)～∑_(m=0)~∞as∈→0 where the coefficientsa_m are functions of x only.
Definition 1 This expansion is said to be uniformly valid ifOtherwise the expansion is said to be nonuniformly valid.
For the uniformity conditions to hold,a_m(x)δ_m(∈)must be small compared to the precedingterm a_(m-1)(x)δ_(m-1)(∈)for each m.Sinceδ_m(∈) = o[δ_(m-1)(∈)]as∈→0,we require that a_m(x)be no more singular than a_(m-1)(x)for all values of x of interest if the expansion is to be uniform.In other words,each term must be a small correction to the preceding term irrespective of the value of x.
Construction of such approaches solutions,people have developed many effective methods,such as asymptotic matching,averaging,multiple scales,stretched coordinate and WKB methods.
But in the use of these methods,in order to ensure expansion of solutions uniformly valid. To irregular part,such as the Boundary Layer,or by the location and thickness of surfacemay appear small fraction of the market parameters,such as the need to have some understanding.There is also a need for closer asymptotic matching necessary.This makes them much more restricted applications.
Recently,a perturbative renormalization group method was developed by Chen,Goldenfeld,and Oono as unified tool for asymptotic analysis.
This paper consist of two parts.
In the first part,we use the perturbative renormalization group method to study a Class of Forth-order Linear Differential EquationsWe have:
Theorem 1 Assume thatp(x),q(x)are second continuous differentiable functiions in[0,1],p(0)＞0.Let u(t) be the solution of initial value problem (1)-(2),then wherewhere a_1(0),a_2(0),a_3(0),a_4(0) are determined by the initial value conditions (2).
In the second part,we give some examples to illustrate the application of the theorem.
引文
[1]SUN Q. R. Uniformly Convergent Differfence Method For a Class of Singular Pertrbation Problem of Fourth-order Quasilinear Ordinary Differential Equation[M]. Proceedings of MMM, Shanghai, 1987.
[2]王国英、陈明伦.四阶常微分方程奇异摄动向题的二阶精度差分分解[J].应用数学和力学,1990,11(5):31-43.
[3]岑仲迪.四阶奇异摄动边值问题在自适应网格上的一致收敛分析[J].高等学校计算数学学报2003,25(4).
[4]JIANG XIUFEN, YAO QINGLIU. An Existence Theorem of Positive Solutions for Elastic Beam Equation with Both Fixed End-points[J ]. Appl. Math. J. Chinese Univ. ,2001,163(3): 237-240.
[5]JIANG XIUFEN, YAO QINGLIU. An Existence Theorem of Positive Solutions For Fourth-order Superlinear Semipostione Eigenvaalue Problems[J]. 数学季刊, 2001, 16(2): 64-68.
[6]马巧珍,马和平.四阶非线性特征值问题正解的存在性[J].西北师范大学学报(自然科学版),2001,37(2):1-4.
[7]吴红萍.四阶非线性特征值问题正解的存在性[J].应用泛函分析学报,2002,4(3):229-232.
[8]吴红萍.一类弹性梁方程三个正解的存在性[J].甘肃科学学报,2002,14(2):9-11.
[9]马巧珍.一类四阶半边值问题正解的存在性[J].工程数学学报,2002,19(3):133-136.
[10]马如云,吴红萍.一类四阶两点边值问题多个正解的存在性[J].数学物理学报,2002,22A(2):244-249.
[11]宋灵宇.一类四阶两点边值问题解的存在性[J].陕西师范大学学报(自然科学版),2003,31(2):29-31.
[12]吴红萍,马如云.一类四阶两点边值问题正解的存在性[J].应用泛函分析学报,2002,2(4):342-348.
[13]万阿英,许晓婕,蒋达清.奇异非线性四阶两点边值问题正解[J].东北师范大学学报(自然科学版),2003,35(3):1-8.
[14]孔令彬,张仲毅.奇异非线性四阶边值问题正解[J].吉林大学学报(自然科学版),2002,1:40-43.
[15]吴红萍.一类四阶奇异非线性边值问题的正解[J].山西大学学报(自然科学版),2001,24(4):286-288.
[16]姚庆六,江秀芬.非线性四阶两点边值问题的一个正解存在定理[J].数学杂志,2004,24(1):35-38.
[17]江秀芬,姚庆六.渐进非线性四阶两点边值问题的一个存在定理[J].甘肃科学学报,2002,14(1):1-3.
[18]吴红萍,马如云.一类四阶两点边值问题正解的存在性[J].应用泛函分析学报,2002,2(4):342-348.
[19]姚庆六.一类四阶边值问题的n个正解的存在性(英文)[J].南京大学学报,2004,40(1):83-88.
[20]BENDER C. M. and ORSZAY S. A. Advanced Mathematical Methods for Scientisis and Engineers, McGraw-Hill[M]. New York, 1978.
[21]HINCH E.J. Perturbation Methods[M]. Cambridge University Press, Cambridge, 1991.
[22]BRETHERTON F. P. Slow Viscous Motion Round a Cylinder In a Simple Shear[J]. J. Fluid Mech., 1962(12): 591-613.
[23] CHEN L. Y, GOLDENFELD N. and OONO Y., Renormalization Group Theory for Globel Asymptotic Analysis[J]. Phys. Rev. Lett., 1994(73): 1311-1315.
[24] CHEN L. Y. GOLDENFELD N. and OONO Y., Renormalization Group And Singular Perturbations:Multiple Scales, Boundary Layers, and Perturbation Theory[J]. Phys. Rev. E, 1996(54): 376-394.
[25] CARRIER G. F. Singular Perturbation Theory and Geophysics[J]. SIAM Rev., 1970(12): 179-193.
[26] COMSTOCK C. Singular Perturbations of Elliptic Equations I[J]. SIAM J. Appl. Math., 1971(16): 491-502.
[27] ECKSTEIN M. C. and SHI Y. Y. Low-thrust Elliptic Spiral Trajectories of a Satellite of Variable Mass[J]. AIAA J., 1967(5): 1491-1494.
[28] FOWKES N. D. A Singular Perturbation Method, Part II[J]. Quart. Appl. Math., 1968(26): 71-85.
[29] GERMAIN P. Recent Evolution in Problems and Methods in Aerodynamics[J]. J. Roy. Aeron. Soc. 1967(71): 673-691.
[30] HAN L. S. On The Free Vibration of A Beam On A Nonlinear Elastic Foundation[J]. J. Appl. Mech., 1968(32): 445-447.
[31] KELLER J. B. Perturbation Theory, Lecture Notes, Mathematics Department[M]. Michigan State University, 1968.
[32] LINDSTEDT A. Ueber Die Integration Einer fur ie strorungstheorie wichtigen differ-entialgleichung[J]. Astron. Nach., 1882(103): 211-220.
[33] MURDOCK J. A. Perturbations Theory and Methods[M]. Wiley, New York, 1991.
[34] CARR J. Applications of Certer Manifold Theory[M]. Springer, Berlin, 1981.
[35] KEVORKIAN J. and COLE J. D. Perturbation Methods in Applied Mathematics[M]. Springer, New York, 1981.
[36] NAYFEN A. H. Perturbation Method[M]. Wiley-interscience, New York, 1973.
[37] NAYFEN A. H. A Comparison of Three Perturbation Methods for the Earthmoon-spaceship Problem[J]. ALAA J., 1965(3): 1682-1687.
[38] POINVARE H. New Methods of Celestial Mechanics[M]. Vol I-III, NASA TTF, 1967.
[39] RAMNATH R. V. Transition Dynamics of VTOL Aircraft[J]. AIAA J., 1970(8): 1214-1221.
[40] ZIANE M. On a Certain Renormaliztion Group Method[J]. J. Math. Phys., 2000(41): 3290-3299.
[2]王国英、陈明伦.四阶常微分方程奇异摄动向题的二阶精度差分分解[J].应用数学和力学,1990,11(5):31-43.
[3]岑仲迪.四阶奇异摄动边值问题在自适应网格上的一致收敛分析[J].高等学校计算数学学报2003,25(4).
[4]JIANG XIUFEN, YAO QINGLIU. An Existence Theorem of Positive Solutions for Elastic Beam Equation with Both Fixed End-points[J ]. Appl. Math. J. Chinese Univ. ,2001,163(3): 237-240.
[5]JIANG XIUFEN, YAO QINGLIU. An Existence Theorem of Positive Solutions For Fourth-order Superlinear Semipostione Eigenvaalue Problems[J]. 数学季刊, 2001, 16(2): 64-68.
[6]马巧珍,马和平.四阶非线性特征值问题正解的存在性[J].西北师范大学学报(自然科学版),2001,37(2):1-4.
[7]吴红萍.四阶非线性特征值问题正解的存在性[J].应用泛函分析学报,2002,4(3):229-232.
[8]吴红萍.一类弹性梁方程三个正解的存在性[J].甘肃科学学报,2002,14(2):9-11.
[9]马巧珍.一类四阶半边值问题正解的存在性[J].工程数学学报,2002,19(3):133-136.
[10]马如云,吴红萍.一类四阶两点边值问题多个正解的存在性[J].数学物理学报,2002,22A(2):244-249.
[11]宋灵宇.一类四阶两点边值问题解的存在性[J].陕西师范大学学报(自然科学版),2003,31(2):29-31.
[12]吴红萍,马如云.一类四阶两点边值问题正解的存在性[J].应用泛函分析学报,2002,2(4):342-348.
[13]万阿英,许晓婕,蒋达清.奇异非线性四阶两点边值问题正解[J].东北师范大学学报(自然科学版),2003,35(3):1-8.
[14]孔令彬,张仲毅.奇异非线性四阶边值问题正解[J].吉林大学学报(自然科学版),2002,1:40-43.
[15]吴红萍.一类四阶奇异非线性边值问题的正解[J].山西大学学报(自然科学版),2001,24(4):286-288.
[16]姚庆六,江秀芬.非线性四阶两点边值问题的一个正解存在定理[J].数学杂志,2004,24(1):35-38.
[17]江秀芬,姚庆六.渐进非线性四阶两点边值问题的一个存在定理[J].甘肃科学学报,2002,14(1):1-3.
[18]吴红萍,马如云.一类四阶两点边值问题正解的存在性[J].应用泛函分析学报,2002,2(4):342-348.
[19]姚庆六.一类四阶边值问题的n个正解的存在性(英文)[J].南京大学学报,2004,40(1):83-88.
[20]BENDER C. M. and ORSZAY S. A. Advanced Mathematical Methods for Scientisis and Engineers, McGraw-Hill[M]. New York, 1978.
[21]HINCH E.J. Perturbation Methods[M]. Cambridge University Press, Cambridge, 1991.
[22]BRETHERTON F. P. Slow Viscous Motion Round a Cylinder In a Simple Shear[J]. J. Fluid Mech., 1962(12): 591-613.
[23] CHEN L. Y, GOLDENFELD N. and OONO Y., Renormalization Group Theory for Globel Asymptotic Analysis[J]. Phys. Rev. Lett., 1994(73): 1311-1315.
[24] CHEN L. Y. GOLDENFELD N. and OONO Y., Renormalization Group And Singular Perturbations:Multiple Scales, Boundary Layers, and Perturbation Theory[J]. Phys. Rev. E, 1996(54): 376-394.
[25] CARRIER G. F. Singular Perturbation Theory and Geophysics[J]. SIAM Rev., 1970(12): 179-193.
[26] COMSTOCK C. Singular Perturbations of Elliptic Equations I[J]. SIAM J. Appl. Math., 1971(16): 491-502.
[27] ECKSTEIN M. C. and SHI Y. Y. Low-thrust Elliptic Spiral Trajectories of a Satellite of Variable Mass[J]. AIAA J., 1967(5): 1491-1494.
[28] FOWKES N. D. A Singular Perturbation Method, Part II[J]. Quart. Appl. Math., 1968(26): 71-85.
[29] GERMAIN P. Recent Evolution in Problems and Methods in Aerodynamics[J]. J. Roy. Aeron. Soc. 1967(71): 673-691.
[30] HAN L. S. On The Free Vibration of A Beam On A Nonlinear Elastic Foundation[J]. J. Appl. Mech., 1968(32): 445-447.
[31] KELLER J. B. Perturbation Theory, Lecture Notes, Mathematics Department[M]. Michigan State University, 1968.
[32] LINDSTEDT A. Ueber Die Integration Einer fur ie strorungstheorie wichtigen differ-entialgleichung[J]. Astron. Nach., 1882(103): 211-220.
[33] MURDOCK J. A. Perturbations Theory and Methods[M]. Wiley, New York, 1991.
[34] CARR J. Applications of Certer Manifold Theory[M]. Springer, Berlin, 1981.
[35] KEVORKIAN J. and COLE J. D. Perturbation Methods in Applied Mathematics[M]. Springer, New York, 1981.
[36] NAYFEN A. H. Perturbation Method[M]. Wiley-interscience, New York, 1973.
[37] NAYFEN A. H. A Comparison of Three Perturbation Methods for the Earthmoon-spaceship Problem[J]. ALAA J., 1965(3): 1682-1687.
[38] POINVARE H. New Methods of Celestial Mechanics[M]. Vol I-III, NASA TTF, 1967.
[39] RAMNATH R. V. Transition Dynamics of VTOL Aircraft[J]. AIAA J., 1970(8): 1214-1221.
[40] ZIANE M. On a Certain Renormaliztion Group Method[J]. J. Math. Phys., 2000(41): 3290-3299.