变分迭代法及几个非线性方程（组）的近似解
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摘要
20世纪中后期,非线性科学迅速发展成为科学技术研究的前沿领域。在非线性科学的研究中,非线性方程的求解一直是研究的难点、热点。孤子方程、积分方程的求解是非线性科学的研究中的两个重要领域。经过数学家和物理学家不懈努力,已经找到一些孤子方程求解的有效方法,如反散射方法、Hirota双线性法、齐次平衡法、F-展开法、指数展开法等等,但是,由于孤子方程的多样性及复杂性,导致只有有限的孤子方程才能够得到精确解,因此寻找有效的近似方法就成为孤子方程、积分方程的一个重要研究方向。
变分迭代算法是何吉欢在Inokuti-Sekine-Mura方法基础上提出来的。变分迭代法自提出以后,得到了科技工作者的普遍关注,已经应用于非线性问题的研究中。首先,本文借助于变分迭代法求得了具强非线性、耦合孤子方程(组)的近似解,并对近似解进行了分析,讨论了强非线性项系数对近似解的影响;其次,利用变分迭代法,对FKPP方程进行近似求解;最后,研究了三个二维积分方程,求得了方程的近似解;借助Matlab对所得的近似解及误差进行近似模拟。
In the late 20th century, the nonlinear science is rapidly developing and become the frontier of science and technology. In the research of the nonlinear science, solving nonlinear equation is one of the most difficult and hottest topics. Solving soliton equations and integral equations are two of the most important fields of the nonlinear science. Many efficient methods for exploring the exact solutions of the soliton equations are presented by the mathematicians and physicists, such as inverse scattering method, Hirota bilinear methods, the homogeneous balance method, F-expansion method, exponential expansion method, etc. However, due to the variety and complexity of soliton equations, the only special soliton equations can be solved. Thus, looking for an effective approximate method of solving soliton equations and integral equations has become an important problem.
Based on the Inokuti-Sekine-Mura’s method, the variational iteration method is presented by Prof. He Ji-huan. Since the variational iteration method is introduced, more and more scientists have paid attention. As the rapid development of variational iteration method, it has been widely applied to the nonlinear problems. In this thesis, the approximate solutions of the soliton equation with high-order nonlinear terms, two coupled soliton equations are respectively presented with the aids of the variational iteration method and the approximate solutions are analyzed, the influence of coefficients of high-order nonlinear terms on the approximate solutions is discussed. And then the approximate solutions of FKPP equation are derived by variational iteration method. In the last, three two-dimensional integral equations are analyzed and the approximate solutions are obtained. The approximate solutions are simulated by Matlab.
变分迭代算法是何吉欢在Inokuti-Sekine-Mura方法基础上提出来的。变分迭代法自提出以后,得到了科技工作者的普遍关注,已经应用于非线性问题的研究中。首先,本文借助于变分迭代法求得了具强非线性、耦合孤子方程(组)的近似解,并对近似解进行了分析,讨论了强非线性项系数对近似解的影响;其次,利用变分迭代法,对FKPP方程进行近似求解;最后,研究了三个二维积分方程,求得了方程的近似解;借助Matlab对所得的近似解及误差进行近似模拟。
In the late 20th century, the nonlinear science is rapidly developing and become the frontier of science and technology. In the research of the nonlinear science, solving nonlinear equation is one of the most difficult and hottest topics. Solving soliton equations and integral equations are two of the most important fields of the nonlinear science. Many efficient methods for exploring the exact solutions of the soliton equations are presented by the mathematicians and physicists, such as inverse scattering method, Hirota bilinear methods, the homogeneous balance method, F-expansion method, exponential expansion method, etc. However, due to the variety and complexity of soliton equations, the only special soliton equations can be solved. Thus, looking for an effective approximate method of solving soliton equations and integral equations has become an important problem.
Based on the Inokuti-Sekine-Mura’s method, the variational iteration method is presented by Prof. He Ji-huan. Since the variational iteration method is introduced, more and more scientists have paid attention. As the rapid development of variational iteration method, it has been widely applied to the nonlinear problems. In this thesis, the approximate solutions of the soliton equation with high-order nonlinear terms, two coupled soliton equations are respectively presented with the aids of the variational iteration method and the approximate solutions are analyzed, the influence of coefficients of high-order nonlinear terms on the approximate solutions is discussed. And then the approximate solutions of FKPP equation are derived by variational iteration method. In the last, three two-dimensional integral equations are analyzed and the approximate solutions are obtained. The approximate solutions are simulated by Matlab.
引文
[1] Ablowitz M J , Clarkson P A. Solitons nonlinear evolution equations and inverse scattering[M]. Cambridge University Press, Cambridge:1991, 284-288.
[2] Gu C H, et al. Soliton Theory and its Application[M]. Zhejiang :Publishing House of Science and Technology, 1990, 14-25.
[3] Hereman W Takaoka M.Solitary wave solutions of nonlinear evolution and wave equations using a direct method and macsyma[J]. Phys. A: Math. Gen, 1990, 23:4805-4807.
[4] Parkes E J Duffy B R. Travelling solitary wave solutions to a compound KdV-Burgers equation[J]. Physics Letters A, 1997, 229:217-220.
[5] Wang M L. Solitary wave solutions for variant Boussinesq equations[J]. Physics Letters A, 1995, 199: 169-172.
[6] Wang M L, Zhou Y B. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J]. Physics Letters A, 1996, 216:67-75.
[7] Liu S K, Fu Z T. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Physics Letters A, 2001, 289: 69–74.
[8] Wang M L, Zhou Y B. The periodic wave solutions for the Klein–Gordon–Schr?dinger equations[J]. Physics Letters A, 2003, 318:84–92.
[9]张金良,王明亮,王跃明.推广的F-展开法及变系数KdV和mKdV的精确解[J].数学物理学报, 2006, 26A(3):353-360.
[10] Wang M L, Li X Z, Zhang J L. The ( G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Physics Letters A, 2008, 372:417–423.
[11] Wang M L, Zhang J L, Li X Z. Application of the ( G′/G)-expansion to travelling wave solutions of the Broer–Kaup and the approximate long water wave equations[J]. Applied Mathematics and Computation, 2008, 206:321–326.
[12]吴东旭.非线性方程优化迭代方法[D].长春:吉林大学, 2010, 1-18.
[13]刘红梅.变分法在离子声波方程中的应用[D].上海:东华大学, 2005, 2-10.
[14] Ott E Sudan R N. Nonlinear Theory of Ion AcouStic Waves with landau Damping[J]. Phys. Fluids, 1969, 12:2388-2394.
[15] Kaup D J. A Perturbation Expansion for the Zakharov-Shabat Inverse Scattering Transform[J]. Slam Journal. Appl. Math, 1976, 31:121-125.
[16] Adomian G. Review of the decomposition method in applied mathematics[J]. Journal of Mathematical Analysis and Applications, 1988, 135(2):501-544
[17]程雪苹.直接微扰方法和非线性薛定锷方程族的近似解[D].杭州:浙江师范大学,2006:10-25.
[18]石兰芳,周先春.受迫广义Klein-Gordon方程的孤子近似解[J].南京信息工程大学学报, 2010,2(1):68-70
[19]王正岭,罗开基.用行波解法研究非线性偏振旋转控制光[J].孤子系统光子学报, 2003, 32(1):89-91.
[20] Kanwal R P, Liu K C. A Taylor expansion approach for solving integral equation[J]. Int. J. Math. Educ. Sci., 1989, 20:411-415.
[21]赵华敏,谢远涵.线性Fredholm积分方程的近似解法[J].西安矿业学院学报, 1995, 15: 374-375
[22] M. G. T. Shaidurow. Difference methods and their Extrapolations[M]. New York Springer-Verlag, 1983
[23]刘英,郑克旺.第二类弱奇异Volterra积分方程的求解[J].河北轻化工学院学报, 1995, 16:11-18
[24] Inkuti M, Sekine H, Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics, in variational method in the mechanics of solids[M]. Pergamon Press, 1978, 156-162.
[25] He Jihuan. A New Approach to Nonlinear Partial Differential Equations[J]. Communication in Nonlinear Science & numerical Simulation, 1997, 2:230-235.
[26]徐建平,桂子鹏.变分方法[M].上海:同济大学出版社, 1999:5-35.
[27] He Ji-Huan. Variational iteration method for autonomous ordinary differential systems[J]. Applied Mathematics and Computation, 2000, 114:115-123
[28] Abdou M.A. Soliman.New applications of variational iteration method[J]. Physica D, 2005, 211:1-8.
[29] Zayed E M E, Zedan Ha, Gepreel K A. On the solitary wave solutions for nonlinear Hirota–Satsuma coupled KdV equations[J]. Chaos Solitons and Fractals, 2004, 22(2): 285-303.
[30] Abdul-Majid Wazwaz .The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations[J]. Journal of Computational and Applied Mathematics, 2007, 207:18-23
[31] Abdul-Majid Wazwaz. The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations[J]. Computers and Mathematics with Applications, 2007, 54:926-932.
[32] Wang Shu-qiang,He Ji-huan .Variational iteration method for solving integro-differential equations variational iteration method for solving integro-differential equations[J]. Physics Letters A, 2007, 36 :188-191
[33] Odibat Z M, Momani S. Application of variational iteration method to nonlinear differential equations of fractal order[J] . International Nonlinear Science and Numerical Simulation, 2006, 7:27-34.
[34]何吉欢. Schrodinger方程的变分迭代解法[J].数学物理学报, 2001, 21A(增刊):577-583.
[35]莫嘉琪,林万涛,朱江.厄尔尼诺-拉尼娜—南方海涛模型的变分迭代解法[J].数学进展, 2006, 35(2):232-236.
[36]冯茂春.海-气振ENSO模型的变分迭代解法[J].中山大学学报(自然科学版), 2005, 44:152-153
[37]莫嘉琪.厄尔尼诺大气物理机理的变分迭代解法[J].物理学报, 2005, 54:1081-1083.
[38]莫嘉琪.赤道东太平洋SST的海-气振子模型[J].物理学报, 2008, 55:6-8.
[39] Wazwaz A M. A comparison between the variational iteration method and Adomian decomposition method[J]. Journal of Computational and Applied mathematics, 2007, 207(1):129-136.
[40] Abassy T A, EI-Tawil M A, EI-Zoheiry H. Toward a modified variational iteration method[J]. Journal of Computational and Applied Mathematics, 2007, 207:137一147.
[41] Abassy T A, EI-TawiM A l. Modified variational iteration method for Boussinesq equation[J]. Computers and Mathematics with Applications, 2007, 54: 955-965.
[42] Abassy T A. Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and thepade technique[J].Computers & Mathematics with Applications,2007, 54:950-954.
[43]黄得建.变分迭代法及其应用[D].长沙:长沙理工大学, 2009, 19-20.
[44] Sweilam N.H. Variational iteration method for coupled nonlinear Schrodinger Equations[J]. Computers & Mathematics with Applications, 2007, 54:993-999.
[45] He J H. Approximate solution of nonlinear differential equations with convolution product nonlinearities[J]. Computer Methods in Applied Mechanics and Engineering, 1998, 167:69-73.
[46] He J H. Variational iteration method—a kind of non-linear analytical technique: Some examples[J]. International Journal of Non-Linear Mechanics, 1999, 34:699-708
[47]何吉欢.流体力学广义变分原理[M].中国科学文化出版社,2003,15-40.
[48] Su C H, Gardener C S. Derivation of the Korteweg-de Vries and Burgers' equation[J]. Journal of Mathmatic Physics, 1969, 10(3):536-539.
[49]钟秋平,丁宣浩. KdV-Burgers方程的小波Galerkin法数值解[J].桂林电子科技大学学报,2010, 30(4):359-362.
[50]谢元喜,唐驾时.用试探函数法求KdV-Burgers方程的精确解析解[J].湖南大学学报(自然科学版), 2005, 32(6):118-120.
[51] Pelinovsky D, Grimshaw R. An asymptotic approach to solitary wave instability and critical collapse in long-wave KdV-type cvolutions, Equations[J]. Physica D, 1996, 98:139-155.
[52] Pelinovsky D, Afanasjev V, Kivshar Y. Nonlinear theory of oscillating, decaying and collapsing solitons in the generalized nonlinear Schrodinger equation[J]. Math. Rev., 1996, 53 (2):1940-1953.
[53] Parkes E J, Duffy B R. An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equation[J]. Computer Physics Communications, 1996, 98:288-300.
[54] Inc M. Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method[J]. Chaos Solitons and Fractals, 2007, 34 (4):1075-1081.
[55]谢长珍.变分迭代法求解量子力学中的微扰问题[J].江西科学学报, 2004, 22(5):317-322.
[56] Wang Mingliang, Zhou Yubin .A nonlinear transformation of variant shallow water wave equations and its applications[J].数学进展, 1999, 28(1):71-74.
[57]刘春平.双参数假设与变形浅水波方程组的精确解[J].应用数学, 2000, 13(1):15- 18.
[58] Khuri S A,Sayfy A. A numerical approach for solving an extended Fisher-Kolomogrov-Petrovslcii-Pislcunov equation[J]. Journal of Computational and Applied Mathematics 2010, 233:2081-2089.
[59] Babolian E. A numerical method to solve Fredholm-Volterra integral equations in two dimensional spaces using Block Pulse Functions and perational matrix[J]. Journal of Computational and Applied Mathematics, DOI: S0377-0427(10)00594-7
[60]张宋宋.变分迭代法的若干研究[D].杭州:浙江大学,2008, 12-13.
[61] Afroozi G A. Solving a class of two—dimensional linear and nonlinear Volterra Integral equations by means of the homotopy analysis method[J]. International Journal of Nonlinear Science, 2010, 9(2):213-219.
[2] Gu C H, et al. Soliton Theory and its Application[M]. Zhejiang :Publishing House of Science and Technology, 1990, 14-25.
[3] Hereman W Takaoka M.Solitary wave solutions of nonlinear evolution and wave equations using a direct method and macsyma[J]. Phys. A: Math. Gen, 1990, 23:4805-4807.
[4] Parkes E J Duffy B R. Travelling solitary wave solutions to a compound KdV-Burgers equation[J]. Physics Letters A, 1997, 229:217-220.
[5] Wang M L. Solitary wave solutions for variant Boussinesq equations[J]. Physics Letters A, 1995, 199: 169-172.
[6] Wang M L, Zhou Y B. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J]. Physics Letters A, 1996, 216:67-75.
[7] Liu S K, Fu Z T. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations[J]. Physics Letters A, 2001, 289: 69–74.
[8] Wang M L, Zhou Y B. The periodic wave solutions for the Klein–Gordon–Schr?dinger equations[J]. Physics Letters A, 2003, 318:84–92.
[9]张金良,王明亮,王跃明.推广的F-展开法及变系数KdV和mKdV的精确解[J].数学物理学报, 2006, 26A(3):353-360.
[10] Wang M L, Li X Z, Zhang J L. The ( G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics[J]. Physics Letters A, 2008, 372:417–423.
[11] Wang M L, Zhang J L, Li X Z. Application of the ( G′/G)-expansion to travelling wave solutions of the Broer–Kaup and the approximate long water wave equations[J]. Applied Mathematics and Computation, 2008, 206:321–326.
[12]吴东旭.非线性方程优化迭代方法[D].长春:吉林大学, 2010, 1-18.
[13]刘红梅.变分法在离子声波方程中的应用[D].上海:东华大学, 2005, 2-10.
[14] Ott E Sudan R N. Nonlinear Theory of Ion AcouStic Waves with landau Damping[J]. Phys. Fluids, 1969, 12:2388-2394.
[15] Kaup D J. A Perturbation Expansion for the Zakharov-Shabat Inverse Scattering Transform[J]. Slam Journal. Appl. Math, 1976, 31:121-125.
[16] Adomian G. Review of the decomposition method in applied mathematics[J]. Journal of Mathematical Analysis and Applications, 1988, 135(2):501-544
[17]程雪苹.直接微扰方法和非线性薛定锷方程族的近似解[D].杭州:浙江师范大学,2006:10-25.
[18]石兰芳,周先春.受迫广义Klein-Gordon方程的孤子近似解[J].南京信息工程大学学报, 2010,2(1):68-70
[19]王正岭,罗开基.用行波解法研究非线性偏振旋转控制光[J].孤子系统光子学报, 2003, 32(1):89-91.
[20] Kanwal R P, Liu K C. A Taylor expansion approach for solving integral equation[J]. Int. J. Math. Educ. Sci., 1989, 20:411-415.
[21]赵华敏,谢远涵.线性Fredholm积分方程的近似解法[J].西安矿业学院学报, 1995, 15: 374-375
[22] M. G. T. Shaidurow. Difference methods and their Extrapolations[M]. New York Springer-Verlag, 1983
[23]刘英,郑克旺.第二类弱奇异Volterra积分方程的求解[J].河北轻化工学院学报, 1995, 16:11-18
[24] Inkuti M, Sekine H, Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics, in variational method in the mechanics of solids[M]. Pergamon Press, 1978, 156-162.
[25] He Jihuan. A New Approach to Nonlinear Partial Differential Equations[J]. Communication in Nonlinear Science & numerical Simulation, 1997, 2:230-235.
[26]徐建平,桂子鹏.变分方法[M].上海:同济大学出版社, 1999:5-35.
[27] He Ji-Huan. Variational iteration method for autonomous ordinary differential systems[J]. Applied Mathematics and Computation, 2000, 114:115-123
[28] Abdou M.A. Soliman.New applications of variational iteration method[J]. Physica D, 2005, 211:1-8.
[29] Zayed E M E, Zedan Ha, Gepreel K A. On the solitary wave solutions for nonlinear Hirota–Satsuma coupled KdV equations[J]. Chaos Solitons and Fractals, 2004, 22(2): 285-303.
[30] Abdul-Majid Wazwaz .The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations[J]. Journal of Computational and Applied Mathematics, 2007, 207:18-23
[31] Abdul-Majid Wazwaz. The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations[J]. Computers and Mathematics with Applications, 2007, 54:926-932.
[32] Wang Shu-qiang,He Ji-huan .Variational iteration method for solving integro-differential equations variational iteration method for solving integro-differential equations[J]. Physics Letters A, 2007, 36 :188-191
[33] Odibat Z M, Momani S. Application of variational iteration method to nonlinear differential equations of fractal order[J] . International Nonlinear Science and Numerical Simulation, 2006, 7:27-34.
[34]何吉欢. Schrodinger方程的变分迭代解法[J].数学物理学报, 2001, 21A(增刊):577-583.
[35]莫嘉琪,林万涛,朱江.厄尔尼诺-拉尼娜—南方海涛模型的变分迭代解法[J].数学进展, 2006, 35(2):232-236.
[36]冯茂春.海-气振ENSO模型的变分迭代解法[J].中山大学学报(自然科学版), 2005, 44:152-153
[37]莫嘉琪.厄尔尼诺大气物理机理的变分迭代解法[J].物理学报, 2005, 54:1081-1083.
[38]莫嘉琪.赤道东太平洋SST的海-气振子模型[J].物理学报, 2008, 55:6-8.
[39] Wazwaz A M. A comparison between the variational iteration method and Adomian decomposition method[J]. Journal of Computational and Applied mathematics, 2007, 207(1):129-136.
[40] Abassy T A, EI-Tawil M A, EI-Zoheiry H. Toward a modified variational iteration method[J]. Journal of Computational and Applied Mathematics, 2007, 207:137一147.
[41] Abassy T A, EI-TawiM A l. Modified variational iteration method for Boussinesq equation[J]. Computers and Mathematics with Applications, 2007, 54: 955-965.
[42] Abassy T A. Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and thepade technique[J].Computers & Mathematics with Applications,2007, 54:950-954.
[43]黄得建.变分迭代法及其应用[D].长沙:长沙理工大学, 2009, 19-20.
[44] Sweilam N.H. Variational iteration method for coupled nonlinear Schrodinger Equations[J]. Computers & Mathematics with Applications, 2007, 54:993-999.
[45] He J H. Approximate solution of nonlinear differential equations with convolution product nonlinearities[J]. Computer Methods in Applied Mechanics and Engineering, 1998, 167:69-73.
[46] He J H. Variational iteration method—a kind of non-linear analytical technique: Some examples[J]. International Journal of Non-Linear Mechanics, 1999, 34:699-708
[47]何吉欢.流体力学广义变分原理[M].中国科学文化出版社,2003,15-40.
[48] Su C H, Gardener C S. Derivation of the Korteweg-de Vries and Burgers' equation[J]. Journal of Mathmatic Physics, 1969, 10(3):536-539.
[49]钟秋平,丁宣浩. KdV-Burgers方程的小波Galerkin法数值解[J].桂林电子科技大学学报,2010, 30(4):359-362.
[50]谢元喜,唐驾时.用试探函数法求KdV-Burgers方程的精确解析解[J].湖南大学学报(自然科学版), 2005, 32(6):118-120.
[51] Pelinovsky D, Grimshaw R. An asymptotic approach to solitary wave instability and critical collapse in long-wave KdV-type cvolutions, Equations[J]. Physica D, 1996, 98:139-155.
[52] Pelinovsky D, Afanasjev V, Kivshar Y. Nonlinear theory of oscillating, decaying and collapsing solitons in the generalized nonlinear Schrodinger equation[J]. Math. Rev., 1996, 53 (2):1940-1953.
[53] Parkes E J, Duffy B R. An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equation[J]. Computer Physics Communications, 1996, 98:288-300.
[54] Inc M. Numerical simulation of KdV and mKdV equations with initial conditions by the variational iteration method[J]. Chaos Solitons and Fractals, 2007, 34 (4):1075-1081.
[55]谢长珍.变分迭代法求解量子力学中的微扰问题[J].江西科学学报, 2004, 22(5):317-322.
[56] Wang Mingliang, Zhou Yubin .A nonlinear transformation of variant shallow water wave equations and its applications[J].数学进展, 1999, 28(1):71-74.
[57]刘春平.双参数假设与变形浅水波方程组的精确解[J].应用数学, 2000, 13(1):15- 18.
[58] Khuri S A,Sayfy A. A numerical approach for solving an extended Fisher-Kolomogrov-Petrovslcii-Pislcunov equation[J]. Journal of Computational and Applied Mathematics 2010, 233:2081-2089.
[59] Babolian E. A numerical method to solve Fredholm-Volterra integral equations in two dimensional spaces using Block Pulse Functions and perational matrix[J]. Journal of Computational and Applied Mathematics, DOI: S0377-0427(10)00594-7
[60]张宋宋.变分迭代法的若干研究[D].杭州:浙江大学,2008, 12-13.
[61] Afroozi G A. Solving a class of two—dimensional linear and nonlinear Volterra Integral equations by means of the homotopy analysis method[J]. International Journal of Nonlinear Science, 2010, 9(2):213-219.