同伦摄动方法在求解非线性微分方程中的应用
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摘要
非线性现象出现在现代科学技术的各领域,其数学模型通常由非线性方程所描述,因而非线性方程的求解具有非常重要的理论和实践意义.近年来,人们结合了同伦理论和摄动方法提出了一种新的求解非线性问题近似解析解方法,即同伦摄动方法.同伦摄动方法的基本思想,实质上是把复杂的非线性问题转化成若干个简单的线性问题来处理.
本文应用同伦摄动方法从两个方面研究了非线性方程的求解.
1.求解了一些非线性常微分方程的近似解.求解了包括一类燃烧方程、Rayleigh方程和非线性边界层问题.
2.求解了一些非线性偏微分方程的近似解.求解了包括非线性平流方程、非线性水波方程以及Fisher方程.
本文所做工作充分表明了同伦摄动方法一些显著的特点,如不依赖于非线性方程中的小参数以及初始猜测解可以任意选取等等,同时突出了该方法求解非线性问题时的操作方便、适用性广、灵活度高等优点.
Nonlinear phenomena appear in various fields of modern science and technology, whose mathematical model is usually described by nonlinear equations, and thus solving non-linear equations has important theoretical and practical significance. In recent years ,a new approximate analytic solution method for nonlinear equations is proposed in some papers by combining the homotopy theory with the perturbation method , namely the homotopy perturbation method.The essence of the homotopy perturbation method to solve problems lies in to change the complicated nonlinear problem into several simple linear problems to deal with.
This paper applied homotopy perturbation method to solve nonlinear equations from two aspects.
1.Some approximate solutions of nonlinear ordinary differential equations were achieved,including a burning equation, Rayleigh equation and nonlinear boundary problem.
2.Some approximate solutions of nonlinear partial differential equations achieved,including nonlinear advection equation, nonlinear wave equation and Fisher equation.
This paper worked fully to demonstrate the homotopy perturbation method some notable features, such as not dependent on nonlinear equation of small parameter and the initial guess solution can be freely choosen, etc.The method has the characteristics of convenient operation, suitability, high flexibility for solving nonlinear problem.
本文应用同伦摄动方法从两个方面研究了非线性方程的求解.
1.求解了一些非线性常微分方程的近似解.求解了包括一类燃烧方程、Rayleigh方程和非线性边界层问题.
2.求解了一些非线性偏微分方程的近似解.求解了包括非线性平流方程、非线性水波方程以及Fisher方程.
本文所做工作充分表明了同伦摄动方法一些显著的特点,如不依赖于非线性方程中的小参数以及初始猜测解可以任意选取等等,同时突出了该方法求解非线性问题时的操作方便、适用性广、灵活度高等优点.
Nonlinear phenomena appear in various fields of modern science and technology, whose mathematical model is usually described by nonlinear equations, and thus solving non-linear equations has important theoretical and practical significance. In recent years ,a new approximate analytic solution method for nonlinear equations is proposed in some papers by combining the homotopy theory with the perturbation method , namely the homotopy perturbation method.The essence of the homotopy perturbation method to solve problems lies in to change the complicated nonlinear problem into several simple linear problems to deal with.
This paper applied homotopy perturbation method to solve nonlinear equations from two aspects.
1.Some approximate solutions of nonlinear ordinary differential equations were achieved,including a burning equation, Rayleigh equation and nonlinear boundary problem.
2.Some approximate solutions of nonlinear partial differential equations achieved,including nonlinear advection equation, nonlinear wave equation and Fisher equation.
This paper worked fully to demonstrate the homotopy perturbation method some notable features, such as not dependent on nonlinear equation of small parameter and the initial guess solution can be freely choosen, etc.The method has the characteristics of convenient operation, suitability, high flexibility for solving nonlinear problem.
引文
[1]楼森岳,唐晓艳.非线性数学物理方法[M].北京:科学出版社,2006.
[2]刘式适,刘式达.物理学中的非线性方程[M].北京:北京大学出版社,2000.
[3]范恩贵.可积系统与计算机代数[M].北京:科学出版社,2004.
[4]周凌云,王瑞丽,吴光敏,段良和.非线性物理理论及应用[M].北京:科学出版社,2000.
[5] Wang Mingliang, Zhou Yubin,Li Zhibin. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathem atical physics[J].Physics Letters A.1996,216: 67-75.
[6]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法[J].物理学报,1998,3.
[7]范恩贵,张鸿庆.齐次平衡法若干新的应用[J].物理学报1999,3.
[8]范恩贵.齐次平衡法、WTC法及CK约化法之间的联系[J].物理学报.2000,49(8):1409- 1414.
[9] E J. Parkes and B. R duffy. Travelling solitary wave solutions to acompound Kdv-Burgers equation [J].Phys Lett.1997, A229: 217-220.
[10] Fan guizhong.Extended tanh-function method and its applications to nonlinear equations[J]. Phys Lett .2000,A277: 212-218.
[11]李志斌.张善卿.非线性波方程准确孤立波解的符号计算[J].数学物理学报.1997,17(1): 81-89.
[12] Li zhibin Exact Solitary wave solutions of nonlinear evolution equations, in: X.S.Gao,d.M. Wang ( Eds) , Mathematics Mechanization and Application. Academic Press, 2000.
[13] N.A.Kudryashow.Exact solutions of the generalized kuramo to Sivash in sky equations[J]. Phy Lett.1990, A 147: 287-292.
[14]刘式适等.求某此非线性偏微分力程特解的一个简洁力法[J].应用数学和力学.2001, 22 (3) : 281-286.
[15] Z T. Fu et al .New solutions to generalized m-kdv eqution[J].Commun Theor Phys, 2004, 41: 25-28.
[16] M. Otwinow ski, R. Paul and W. G Laid law. Exact travelling wave solutions of a class of nonlinear diffusion equations by reduction to a quadrature[J].Phys Lett, 1988,A 128:483-489.
[17] Chun Tao Yan. A simple transformation for nonlinear waves[J].Phys Lett .1996, A224: 77-84.
[18] Liu S. K, et al. Expansion method about the Jacobi elliptic function and its applications to nonlinear wave equations [J].Acta Phys Sin. 2001.50: 2062-2067.
[19]张善卿,李志斌.Jacobi椭圆函数展开法的新应用[J].物理学报.2003,52(7):1565-1570.
[20]刘官厅,范天佑.一般变换下的Jacobi椭圆函数展开法及应用[J].物理学报.2004,53(3):676-679.
[21]刘成仕.试探方程法及其在非线性发展方程中的应用[J].物理学报,2005,54(06):2505- 2508.
[22]刘成仕.用试探方程法求变系数非线性发展方程的精确解[J].物理学报,2005:54 (10):4506-4510.
[23]詹杰民,李毓湘.一维Burgers方程和Kdv方程的广义有限谱方法[J].应用数学和力学,2006, 27 (12):1431-1438.
[24]龚玉飞,许传炬.一类非线性Schrodinger方程的守恒差分法与Fourier谱方法.数学研究[J]. 2006, 39 (4):360-369.
[25] J.H.He.The homotopy perturbation method for nonlinear oscillators with discontinuities[J]. Appl. Math. Comput,2004,151(1):287.
[26] J.H.He. Comparison of homotopy perturbation method and homotopy analysis method [J].Appl.Math .Comput,2004,156(2):527.
[27] J.H.He. Application of homotopy perturbation method to nonlinear wave equations [J].Chaos Soltons Fracta1s,2005,26(3):695.
[28]廖世俊.超越摄动:同论分析方法导论[M].陈晨,徐航译.北京:科学出版社,2006.
[29] Liao Shi-jun. On the homotopy analysis method for nonlinear problems[J]. Appl Math and Comput.2004, 147(2): 499-513.
[30] Liao S J. CHEUNG K F. Homotopy analysis of nonlinear progressive waves in deep water[J]. J Eng·Math, 2003, 45(2): 105-116.
[31] Liao S J. A simple way to enlarge the convergence region of perturbation approximations [J]. Int J of Nonlinear Dynamics,1999, 19(2): 93110.
[32] Liao S J.A kind of approximate solution technique which does not depend upon small parameters ( 2):an application in fluid mechanics[J], Int .J .Non-Linear Mechanics,32(5)(1997),815 -822.
[33] Wang Z, Zou L, Zhang H. Applying homotopy analysis method for solving differential-difference equation [J].Physics Letters A, 2007, 370: 287-294.
[34] Wang C, Wu Y Y, Wu W. Solving the nonlinear periodic wave problems with the homotopy analysis method [J]. Wave Motion, 2004, 41: 329-337.
[35]石玉仁,汪映海,杨红娟,段文山,吕克璞.KdV方程的同伦分析法求解[J].兰州大学学报(自然科学版),2007,43(6) .
[36]石玉仁,许新建,吴枝喜等.同伦分析法在求解非线性演化方程中的应用[J].物理学报,2006, 55(4): 1555-1560.
[37] D.D.GANJLThe application of Homotopy perturbation method to nonlinear equation arising in heat transfer [J].Physics Letter A, 2006, 355:337-341.
[38] J.BIAZAR,H.GHAZVINI. Homotopy perturbation method for solving hyperbolic partial differential equations [J]. Computers and Mathematics with Applications, 2008,56:453-458.
[39] J.BIAZAR,H.GHAZVINI. Exact solutions for nonlinear schrodinger equations by Homotopy perturbation method [J]. Physics Letter A, 2007, 366:79-84.
[40] A.GOLBABAI,B.B. Modified Homotopy perturbation method for solving Fredholm integral equations [J].Chao,soliton snd Fractsls, 2008, 3:1528-1537.
[41] SMITH. Numerical Methods for partial Differential Equation [M]. Oxford press,1978.
[42] S.ABBASBANDY. Application of homotopy perturbation method for Laplace transform [J]. Chao,solitons and Fractsls, 2006, 30:1206-1212.
[43]吴钦宽.一类燃烧模型的同伦分析解法[J].物理学报,2008,57:2654-2657.
[44]刘明姬.同伦方法求解非线性微分方程边值问题[D].吉林大学博士论文.2009.
[45]季仲贞,王斌,曾庆存.大气海洋环境数值模拟中的若干计算问题[J].气候与环境研究.1999,4:135-151.
[46]邹丽.某些非线性水波问题的同伦分析方法研究[D].大连理工大学博士论文.2008.
[47]包树蕊.若干非线性发展方程普遍性差分格式的构造与分析[D].华北电力大学硕士论文.2009.
[2]刘式适,刘式达.物理学中的非线性方程[M].北京:北京大学出版社,2000.
[3]范恩贵.可积系统与计算机代数[M].北京:科学出版社,2004.
[4]周凌云,王瑞丽,吴光敏,段良和.非线性物理理论及应用[M].北京:科学出版社,2000.
[5] Wang Mingliang, Zhou Yubin,Li Zhibin. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathem atical physics[J].Physics Letters A.1996,216: 67-75.
[6]范恩贵,张鸿庆.非线性孤子方程的齐次平衡法[J].物理学报,1998,3.
[7]范恩贵,张鸿庆.齐次平衡法若干新的应用[J].物理学报1999,3.
[8]范恩贵.齐次平衡法、WTC法及CK约化法之间的联系[J].物理学报.2000,49(8):1409- 1414.
[9] E J. Parkes and B. R duffy. Travelling solitary wave solutions to acompound Kdv-Burgers equation [J].Phys Lett.1997, A229: 217-220.
[10] Fan guizhong.Extended tanh-function method and its applications to nonlinear equations[J]. Phys Lett .2000,A277: 212-218.
[11]李志斌.张善卿.非线性波方程准确孤立波解的符号计算[J].数学物理学报.1997,17(1): 81-89.
[12] Li zhibin Exact Solitary wave solutions of nonlinear evolution equations, in: X.S.Gao,d.M. Wang ( Eds) , Mathematics Mechanization and Application. Academic Press, 2000.
[13] N.A.Kudryashow.Exact solutions of the generalized kuramo to Sivash in sky equations[J]. Phy Lett.1990, A 147: 287-292.
[14]刘式适等.求某此非线性偏微分力程特解的一个简洁力法[J].应用数学和力学.2001, 22 (3) : 281-286.
[15] Z T. Fu et al .New solutions to generalized m-kdv eqution[J].Commun Theor Phys, 2004, 41: 25-28.
[16] M. Otwinow ski, R. Paul and W. G Laid law. Exact travelling wave solutions of a class of nonlinear diffusion equations by reduction to a quadrature[J].Phys Lett, 1988,A 128:483-489.
[17] Chun Tao Yan. A simple transformation for nonlinear waves[J].Phys Lett .1996, A224: 77-84.
[18] Liu S. K, et al. Expansion method about the Jacobi elliptic function and its applications to nonlinear wave equations [J].Acta Phys Sin. 2001.50: 2062-2067.
[19]张善卿,李志斌.Jacobi椭圆函数展开法的新应用[J].物理学报.2003,52(7):1565-1570.
[20]刘官厅,范天佑.一般变换下的Jacobi椭圆函数展开法及应用[J].物理学报.2004,53(3):676-679.
[21]刘成仕.试探方程法及其在非线性发展方程中的应用[J].物理学报,2005,54(06):2505- 2508.
[22]刘成仕.用试探方程法求变系数非线性发展方程的精确解[J].物理学报,2005:54 (10):4506-4510.
[23]詹杰民,李毓湘.一维Burgers方程和Kdv方程的广义有限谱方法[J].应用数学和力学,2006, 27 (12):1431-1438.
[24]龚玉飞,许传炬.一类非线性Schrodinger方程的守恒差分法与Fourier谱方法.数学研究[J]. 2006, 39 (4):360-369.
[25] J.H.He.The homotopy perturbation method for nonlinear oscillators with discontinuities[J]. Appl. Math. Comput,2004,151(1):287.
[26] J.H.He. Comparison of homotopy perturbation method and homotopy analysis method [J].Appl.Math .Comput,2004,156(2):527.
[27] J.H.He. Application of homotopy perturbation method to nonlinear wave equations [J].Chaos Soltons Fracta1s,2005,26(3):695.
[28]廖世俊.超越摄动:同论分析方法导论[M].陈晨,徐航译.北京:科学出版社,2006.
[29] Liao Shi-jun. On the homotopy analysis method for nonlinear problems[J]. Appl Math and Comput.2004, 147(2): 499-513.
[30] Liao S J. CHEUNG K F. Homotopy analysis of nonlinear progressive waves in deep water[J]. J Eng·Math, 2003, 45(2): 105-116.
[31] Liao S J. A simple way to enlarge the convergence region of perturbation approximations [J]. Int J of Nonlinear Dynamics,1999, 19(2): 93110.
[32] Liao S J.A kind of approximate solution technique which does not depend upon small parameters ( 2):an application in fluid mechanics[J], Int .J .Non-Linear Mechanics,32(5)(1997),815 -822.
[33] Wang Z, Zou L, Zhang H. Applying homotopy analysis method for solving differential-difference equation [J].Physics Letters A, 2007, 370: 287-294.
[34] Wang C, Wu Y Y, Wu W. Solving the nonlinear periodic wave problems with the homotopy analysis method [J]. Wave Motion, 2004, 41: 329-337.
[35]石玉仁,汪映海,杨红娟,段文山,吕克璞.KdV方程的同伦分析法求解[J].兰州大学学报(自然科学版),2007,43(6) .
[36]石玉仁,许新建,吴枝喜等.同伦分析法在求解非线性演化方程中的应用[J].物理学报,2006, 55(4): 1555-1560.
[37] D.D.GANJLThe application of Homotopy perturbation method to nonlinear equation arising in heat transfer [J].Physics Letter A, 2006, 355:337-341.
[38] J.BIAZAR,H.GHAZVINI. Homotopy perturbation method for solving hyperbolic partial differential equations [J]. Computers and Mathematics with Applications, 2008,56:453-458.
[39] J.BIAZAR,H.GHAZVINI. Exact solutions for nonlinear schrodinger equations by Homotopy perturbation method [J]. Physics Letter A, 2007, 366:79-84.
[40] A.GOLBABAI,B.B. Modified Homotopy perturbation method for solving Fredholm integral equations [J].Chao,soliton snd Fractsls, 2008, 3:1528-1537.
[41] SMITH. Numerical Methods for partial Differential Equation [M]. Oxford press,1978.
[42] S.ABBASBANDY. Application of homotopy perturbation method for Laplace transform [J]. Chao,solitons and Fractsls, 2006, 30:1206-1212.
[43]吴钦宽.一类燃烧模型的同伦分析解法[J].物理学报,2008,57:2654-2657.
[44]刘明姬.同伦方法求解非线性微分方程边值问题[D].吉林大学博士论文.2009.
[45]季仲贞,王斌,曾庆存.大气海洋环境数值模拟中的若干计算问题[J].气候与环境研究.1999,4:135-151.
[46]邹丽.某些非线性水波问题的同伦分析方法研究[D].大连理工大学博士论文.2008.
[47]包树蕊.若干非线性发展方程普遍性差分格式的构造与分析[D].华北电力大学硕士论文.2009.