几类奇摄动边值问题解的渐近分析
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摘要
奇异摄动理论及方法是一门应用非常广泛的学科,常用于求解非线性、高阶或变系数的数学物理方程的解析近似解,且对于摄动参数ε比较小的情况,经典的数值方法给不出令人满意的数值结果.目前关于奇异摄动方法的研究非常活跃且在不断拓展,许多奇摄动方法得到进一步发展,包括如匹配渐近展开法,多变量展开法,边界层函数法和多重尺度法.
本文的内容如下:
1、在适当条件下研究具有边界摄动的非线性反应扩散方程的奇摄动Robin问题,并运用微分不等式理论,讨论原问题解的存在性、唯一性及其一致有效的渐近估计.
2、讨论了一类具有间断的拟线性奇异摄动边值问题,运用边界层函数法分段构造微分方程的形式渐近解,并用缝补法实现解的连续性,最后获得了解的存在性及渐近解的一致有效性.
3、讨论了一类含时滞的奇异摄动边值问题,用边界层函数法和多重尺度法分段构造形式渐近解,并用缝补法把内部层连接起来,用微分不等式理论证明解的存在性,同时给出解的一致有效估计.
The theory and method of application for singular perturbation problem is a very broad range of subject. The singularly perturbed method is used to find approximate analytical solutions of nonlinear, high order, or variable coefficients mathematical physical equations. Classical numerical methods usually give unsatisfactory numerical results when the singular perturbation parameterεis small. The current research is very active and constantly expanding .Recently, many approximate methods have been developed, including the method of matched asymptotic expansion, multi-variable expansion method, the boundary layer function method and the method of multiple scales.
The main contents of this paper are outlined as follows:
1. The nonlinear singularly perturbed Robin problems for reaction diffusion equations with boundary perturbation are considered in the paper. By using the method of differential inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed under some suitable conditions.
2. A class of quasi-linear singularly perturbed boundary value problems with discontinuous of differential systems is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. At the same time, the solution is continuous under some suitable conditions. At last, the existence of the solution is proved and the uniformly efficiency of this asymptotic solution is obtained.
3. A class of singularly perturbed boundary value problems with delay is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. Based on differential inequality techniques and the method of multiple scales variables, the existence of the solution is proved and the uniform validity of the asymptotic expansion is proved.
本文的内容如下:
1、在适当条件下研究具有边界摄动的非线性反应扩散方程的奇摄动Robin问题,并运用微分不等式理论,讨论原问题解的存在性、唯一性及其一致有效的渐近估计.
2、讨论了一类具有间断的拟线性奇异摄动边值问题,运用边界层函数法分段构造微分方程的形式渐近解,并用缝补法实现解的连续性,最后获得了解的存在性及渐近解的一致有效性.
3、讨论了一类含时滞的奇异摄动边值问题,用边界层函数法和多重尺度法分段构造形式渐近解,并用缝补法把内部层连接起来,用微分不等式理论证明解的存在性,同时给出解的一致有效估计.
The theory and method of application for singular perturbation problem is a very broad range of subject. The singularly perturbed method is used to find approximate analytical solutions of nonlinear, high order, or variable coefficients mathematical physical equations. Classical numerical methods usually give unsatisfactory numerical results when the singular perturbation parameterεis small. The current research is very active and constantly expanding .Recently, many approximate methods have been developed, including the method of matched asymptotic expansion, multi-variable expansion method, the boundary layer function method and the method of multiple scales.
The main contents of this paper are outlined as follows:
1. The nonlinear singularly perturbed Robin problems for reaction diffusion equations with boundary perturbation are considered in the paper. By using the method of differential inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed under some suitable conditions.
2. A class of quasi-linear singularly perturbed boundary value problems with discontinuous of differential systems is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. At the same time, the solution is continuous under some suitable conditions. At last, the existence of the solution is proved and the uniformly efficiency of this asymptotic solution is obtained.
3. A class of singularly perturbed boundary value problems with delay is considered. Using the method of boundary functions, the formal asymptotic expansions of the solution is constructed with step-wise method. Based on differential inequality techniques and the method of multiple scales variables, the existence of the solution is proved and the uniform validity of the asymptotic expansion is proved.
引文
[1]章本照,印建安,张宏基.流体力学数值方法[M].北京:机械工业出版社,2003
[2] A.B.Vasil’eva,V.F.Butuzov著,倪明康,林武忠译.奇异摄动方程解的渐近展开[M].北京:高等教育出版社,2008
[3] Nayfeh A. H.著,宋家魕编译.摄动方法导引[M].上海翻译出版公司,1990
[4] O’Malley,Jr.R.E.On the asymptotic solution of the singularly perturbed boundary va-lue problems posed by Bohé[J].Math Anal Appl.,2000(242):18-38
[5] Vasil’eva A.B,Butuzov V.F,Kalachev L.V.The Boundary Function Method for Singular Perturbation Problems[M].SIAM Philadelphia,1995
[6] Kelley W.G.A singular perturbation problem of Carrier and Pearson[J].Math Anal. Appl,2001(255):678-697
[7] MizoguchiN,Yanagida,E.Life.Span of solutions for a semilinear parabolic problem with small diffusion[J].Math.Anal.Appl,2001(261):350-368.
[8] Hamouda.Interior layer for second-older singular equations[J].Applicable Anal,2003 (81):837-866
[9] Akhmetov D R.,Lavrentievm,Spigler R..Singular perturbations for certain partial di-fferential equations without boundary-layers[J].Asymptotic Anal,2003(35):65-89
[10] Bell D.C.,Deng B.Singular perturbation of N-front traveling waves in the Fitzhugh-Nagumo equations[J].Nonlinear Anal.Real World Appl,2003(3):515-541
[11] Ni W M,Wei J C.On positive solution concentrating on spheres for the Gierer-Me-inhardt system[J].Differential Equations,2006(221):158-189
[12] Zhang F.Coexistence of a pulse and multiple spikes and transition layers in the sta-nding waves of a reaction-diffusion systems[J].Differential Equations,2004(205):77-155
[13]钱伟长.Asympotic behavior of a thin clamped circular under uniform normal pressu-re at very large deflection[J].清华大学理科报告,1948(5):1-24
[14] Kuo Y.H.On the flow of an in compressible viscous fluid past a flat plate at mode-rate Reynolds numbers[J].Math.Phys,1953(2):83-101
[15] Tsien H.S.The Poincare-Lighthill-Kuo Methods[J].Advan Appl Mech,1956(4):281-349
[16] Lin C.C.On a perturbation theory based on the method of characteristic[J].Math.Ph-ys,1954(33):117-134
[17] Mo Jiaqi.The problem for singularly perturbed combustion reaction diffusion[J].Mat-hematica Applicata,2000(13):85-88
[18]莫嘉琪,韩祥临.一类奇摄动非线性边值问题[J].数学研究与评论,2002(22):626-630
[19]莫嘉琪.一类拟线性Robin问题的激波解[J].数学物理学报,2008(28A):818-822
[20] Mo Jiaqi,Lin Wantao,ZhuJiang.The perturbed solution of sea-air oscillator for E-NSO model[J].Prog.Nat.Sci,2004(14):550-552
[21] Mo Jiaqi,Wang Hui,Lin Wantao,Lin Yihua.Variational iteration method the me-chanism of the equatorial eastern Pacific EI Nino-Southern Oscillation[J].Chin.Phys,2006(15):671-675.
[22] Mo Jiaqi,Lin Wantao,Wang Hui.Variational iteration solution of a sea-air oscillat-or model for the ENSO[J].Progress in Natural Science,2007(17):230-232.
[23] Mo Jiaqi,Lin Wantao,Wang Hui.Variational iteration method for solving perturbe-d mechanism of wastern boundary undercurrents in the Pacific[J].Chin.Phys,2007(16):951-964.
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[25] Jackson L K.Subfunction and second-order ordinary differential inequalities[J].Adva-nce in Mathematics,1967(10):307-363
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[28]江福汝.关于边界层方法[J].应用数学和力学,1981(5):461-473
[29] Jiang Fu Ru.On the boundary layer methods[J].Appl. Math. And Mech(English edit-ion),1981(2):505-518
[30] Jiang Fu Ru.On singular perturbations of nonlinear boundary value problems for or-dinary differential equations[J].Scientia Sinicar(Ser.A.),1987(30):588-600
[31] Jiang Fu Ru.Asymptotic solution of boundary value problem for semilinear ordinarydifferential equations with turning points[J].Computational Fluid Dynamics Journal,1994(2):471-484
[32] Jiang Fu Ru.On the asymptotic solution of boundary value problems for a class ofsystem of nonlinear differential equations[J].Appl.Math. and Mech,2001(3):239-249
[33] M.K. Jain,R.K. Jain,R.K. Mohanty.A fourth order difference method for the one dimensional general quasi-linear parabolic partial differential equation[J].Numer Meth-ods Partial Differen. Equat.,1990(6):311–319
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[35] R.K. Mohanty.A family of variable mesh methods for the estimates of (du/dr)andthe solution of non-linear two point boundary value problems with singularity[J].Co-mput Appl Math,2005(182):173–187
[36] R.K. Mohanty,Swarn Singh.Non-uniform mesh arithmetic average discretization forparabolic initial boundary value problems[J].Neural Parallel Sci. Comput,2005(13):401–416
[37] R.K. Mohanty.An O(k2 + h4) finite difference method for one space Burgers’equat-ion in polar coordinates[J].Numer.Methods Partial Differen. Equat,1996(12):579–583
[38] R.K. Mohanty,N. Jha,D.J. Evans.Spline in compression method for the numerica-l solution of singularly perturbed two point singular boundary value problems[J].Int.J.Comput. Math,2004(81):615–627
[39] R.K. Mohanty,D.J. Evans,U. Arora.Convergent spline in tension methods for sin-gularly perturbed two point singular boundary value problems[J].Comput. Math.,2005(82):55–66
[40] J. Rashidinia,R. Mohammadi,R. Jalilian.The numerical solution of non-linear sing-ular boundary value problems arising in physiology[J].Appl. Math. Comput,2007(18
[41] J. Rashidinia,R. Mohammadi,R. Jalilian.The numerical solution of non-linear sing-ular boundary value problems arising in physiology[J].Appl. Math. Comput,2007(185):360–367
[42] Kumar Manoj,Aziz Tariq.A uniform mesh finite difference method for a class of singular two-point boundary-value problems[J].Appl Math Comput,2006(180):173–177
[43] Kumar Manoj.A difference method for singular two-point boundary-value problems[J].Appl Math Comput,2003(146):879–884
[44]胡键伟,汤怀民著.微分方程数值方法[M].北京:科学出版社,2007
[45]李德茂,王刚著.奇异微分方程有限元法[M].呼和浩特:内蒙古大学出版社,2004
[46]李荣华著.偏微分方程数值解法[M].北京:高等教育出版社,2005
[47] Burden,R.L等著.数值分析[M].北京:高等教育出版社,2001
[48]郭乙木等著.线性与非线性有限元及其应用[M].北京:机械工业出版社,2003
[49] Aziz Tariq,Khan Arshad.A spline method for second-order singularly perturbed bo-undary-value problems[J].Comput Appl Math,2002(47):445–52
[50] Kadalbajoo MK,Bawa RK.Variable mesh difference scheme for singularly perturbe-d boundary-value problems using splines[J].Optim Theory Appl,1996(90):405–416
[51] Vigo-Aguiara J,Natesan S.An efficient numerical method for singular perturbation problems[J].Comput Appl Math,2006(192):132–41
[52] Reddy YN,Pramod Chakravarthy P.Method of reduction of order for solving singu-larly perturbed two-point boundary-value problems[J].Appl Math Comput,2003(136):27–45
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[73]莫嘉琪.具有边界摄动的非线性反应扩散方程奇摄动问题[J].系统科学与数学,2006(26):95-100
[74]莫嘉琪.具有边界摄动弱非线性反应扩散方程的奇摄动[J].应用数学与力学,2008 (29):1003-1008
[75] Mo Jiaqi,Ouyang Cheng.A class of nonlocal boundary value problems of nonline-ar elliptic systems in unbounded domains[J].Acta Math Sci,2001(21):93-97
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[2] A.B.Vasil’eva,V.F.Butuzov著,倪明康,林武忠译.奇异摄动方程解的渐近展开[M].北京:高等教育出版社,2008
[3] Nayfeh A. H.著,宋家魕编译.摄动方法导引[M].上海翻译出版公司,1990
[4] O’Malley,Jr.R.E.On the asymptotic solution of the singularly perturbed boundary va-lue problems posed by Bohé[J].Math Anal Appl.,2000(242):18-38
[5] Vasil’eva A.B,Butuzov V.F,Kalachev L.V.The Boundary Function Method for Singular Perturbation Problems[M].SIAM Philadelphia,1995
[6] Kelley W.G.A singular perturbation problem of Carrier and Pearson[J].Math Anal. Appl,2001(255):678-697
[7] MizoguchiN,Yanagida,E.Life.Span of solutions for a semilinear parabolic problem with small diffusion[J].Math.Anal.Appl,2001(261):350-368.
[8] Hamouda.Interior layer for second-older singular equations[J].Applicable Anal,2003 (81):837-866
[9] Akhmetov D R.,Lavrentievm,Spigler R..Singular perturbations for certain partial di-fferential equations without boundary-layers[J].Asymptotic Anal,2003(35):65-89
[10] Bell D.C.,Deng B.Singular perturbation of N-front traveling waves in the Fitzhugh-Nagumo equations[J].Nonlinear Anal.Real World Appl,2003(3):515-541
[11] Ni W M,Wei J C.On positive solution concentrating on spheres for the Gierer-Me-inhardt system[J].Differential Equations,2006(221):158-189
[12] Zhang F.Coexistence of a pulse and multiple spikes and transition layers in the sta-nding waves of a reaction-diffusion systems[J].Differential Equations,2004(205):77-155
[13]钱伟长.Asympotic behavior of a thin clamped circular under uniform normal pressu-re at very large deflection[J].清华大学理科报告,1948(5):1-24
[14] Kuo Y.H.On the flow of an in compressible viscous fluid past a flat plate at mode-rate Reynolds numbers[J].Math.Phys,1953(2):83-101
[15] Tsien H.S.The Poincare-Lighthill-Kuo Methods[J].Advan Appl Mech,1956(4):281-349
[16] Lin C.C.On a perturbation theory based on the method of characteristic[J].Math.Ph-ys,1954(33):117-134
[17] Mo Jiaqi.The problem for singularly perturbed combustion reaction diffusion[J].Mat-hematica Applicata,2000(13):85-88
[18]莫嘉琪,韩祥临.一类奇摄动非线性边值问题[J].数学研究与评论,2002(22):626-630
[19]莫嘉琪.一类拟线性Robin问题的激波解[J].数学物理学报,2008(28A):818-822
[20] Mo Jiaqi,Lin Wantao,ZhuJiang.The perturbed solution of sea-air oscillator for E-NSO model[J].Prog.Nat.Sci,2004(14):550-552
[21] Mo Jiaqi,Wang Hui,Lin Wantao,Lin Yihua.Variational iteration method the me-chanism of the equatorial eastern Pacific EI Nino-Southern Oscillation[J].Chin.Phys,2006(15):671-675.
[22] Mo Jiaqi,Lin Wantao,Wang Hui.Variational iteration solution of a sea-air oscillat-or model for the ENSO[J].Progress in Natural Science,2007(17):230-232.
[23] Mo Jiaqi,Lin Wantao,Wang Hui.Variational iteration method for solving perturbe-d mechanism of wastern boundary undercurrents in the Pacific[J].Chin.Phys,2007(16):951-964.
[24] Nagumo M.Uber die differential gleichung y′′= f (t , y , y′)[J].Proc Math Soc,1937(10):861-866
[25] Jackson L K.Subfunction and second-order ordinary differential inequalities[J].Adva-nce in Mathematics,1967(10):307-363
[26] Chang K W,Howes F A.Nonlinear singular perturbation phenomenon:theory and a-pplication[M].New York:Spring Verlag,1984
[27] Howes F A.非线性奇异摄动现象:理论与应用[M].福州:福建省科技出版社,1998
[28]江福汝.关于边界层方法[J].应用数学和力学,1981(5):461-473
[29] Jiang Fu Ru.On the boundary layer methods[J].Appl. Math. And Mech(English edit-ion),1981(2):505-518
[30] Jiang Fu Ru.On singular perturbations of nonlinear boundary value problems for or-dinary differential equations[J].Scientia Sinicar(Ser.A.),1987(30):588-600
[31] Jiang Fu Ru.Asymptotic solution of boundary value problem for semilinear ordinarydifferential equations with turning points[J].Computational Fluid Dynamics Journal,1994(2):471-484
[32] Jiang Fu Ru.On the asymptotic solution of boundary value problems for a class ofsystem of nonlinear differential equations[J].Appl.Math. and Mech,2001(3):239-249
[33] M.K. Jain,R.K. Jain,R.K. Mohanty.A fourth order difference method for the one dimensional general quasi-linear parabolic partial differential equation[J].Numer Meth-ods Partial Differen. Equat.,1990(6):311–319
[34] R.K. Mohanty,M.K. Jain,Dinesh Kumar.Single cell finite difference approximatio-ns of O(kh2 + h4) for (ou/ox) for one space dimensional non-linear parabolic equati-ons[J].Numer Methods Partial Differen Equat,2000(16):408–415
[35] R.K. Mohanty.A family of variable mesh methods for the estimates of (du/dr)andthe solution of non-linear two point boundary value problems with singularity[J].Co-mput Appl Math,2005(182):173–187
[36] R.K. Mohanty,Swarn Singh.Non-uniform mesh arithmetic average discretization forparabolic initial boundary value problems[J].Neural Parallel Sci. Comput,2005(13):401–416
[37] R.K. Mohanty.An O(k2 + h4) finite difference method for one space Burgers’equat-ion in polar coordinates[J].Numer.Methods Partial Differen. Equat,1996(12):579–583
[38] R.K. Mohanty,N. Jha,D.J. Evans.Spline in compression method for the numerica-l solution of singularly perturbed two point singular boundary value problems[J].Int.J.Comput. Math,2004(81):615–627
[39] R.K. Mohanty,D.J. Evans,U. Arora.Convergent spline in tension methods for sin-gularly perturbed two point singular boundary value problems[J].Comput. Math.,2005(82):55–66
[40] J. Rashidinia,R. Mohammadi,R. Jalilian.The numerical solution of non-linear sing-ular boundary value problems arising in physiology[J].Appl. Math. Comput,2007(18
[41] J. Rashidinia,R. Mohammadi,R. Jalilian.The numerical solution of non-linear sing-ular boundary value problems arising in physiology[J].Appl. Math. Comput,2007(185):360–367
[42] Kumar Manoj,Aziz Tariq.A uniform mesh finite difference method for a class of singular two-point boundary-value problems[J].Appl Math Comput,2006(180):173–177
[43] Kumar Manoj.A difference method for singular two-point boundary-value problems[J].Appl Math Comput,2003(146):879–884
[44]胡键伟,汤怀民著.微分方程数值方法[M].北京:科学出版社,2007
[45]李德茂,王刚著.奇异微分方程有限元法[M].呼和浩特:内蒙古大学出版社,2004
[46]李荣华著.偏微分方程数值解法[M].北京:高等教育出版社,2005
[47] Burden,R.L等著.数值分析[M].北京:高等教育出版社,2001
[48]郭乙木等著.线性与非线性有限元及其应用[M].北京:机械工业出版社,2003
[49] Aziz Tariq,Khan Arshad.A spline method for second-order singularly perturbed bo-undary-value problems[J].Comput Appl Math,2002(47):445–52
[50] Kadalbajoo MK,Bawa RK.Variable mesh difference scheme for singularly perturbe-d boundary-value problems using splines[J].Optim Theory Appl,1996(90):405–416
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