微分方程解析解及解析近似解的符号计算研究
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摘要
自然界中的很多现象都可用非线性微分方程来描述.非线性微分方程解析解的研究对洞察事物内部的结构,剖析事物之间的关系,并应用于解释各种物理现象都起到至关重要的作用.高性能计算机的诞生,极大地推动了非线性微分方程领域的符号计算研究,涌现出了许多构造非线性微分方程解析解的方法和算法.本文以非线性微分方程为研究对象,借助于非线性代数系统Maple,研究了多种构造非线性微分方程精确解及解析近似解的方法和算法.主要工作如下:
第一部分研究构造非线性演化方程精确解的方法和算法,具体包括两方面的内容:
对已有的构造非线性演化方程精确行波解的几种代数方法,如Riccati方程方法、耦合的Riccati方程方法、假设法、形变映射法等进行了推广和整合,提出了“椭圆方程方法”.并结合吴消元法的思想和方法,在计算机代数系统Maple上编写了推导非线性演化方程精确行波解的软件包RAEEM,该软件包可自动推导出输入方程一系列可能的精确行波解,其中包括多项式解、有理函数解、指数函数解、三角函数解、双曲函数解及Jacobi椭圆函数解、Weierstrass椭圆函数解等.
Bticldund变换研究对非线性微分方程的可积性及精确解的求解都有十分重要的意义.特别是,一旦从B(?)cldund变换推导出解的非线性叠加公式,则仅通过代数运算就可构造微分方程的新解.我们借鉴已有的构造B(?)cklund变换的方法,提出了构造1+1维非线性演化方程一类自B(?)cklund变换的机械化算法,并结合吴文俊数学机械化思想,在计算机代数系统Maple上实现了该算法,其中的软件包AutoBT不仅可自动推导出输入方程的可能的特定类型的自B(?)cklund变换及相应的参数约束条件,还可自动推导出解的非线性叠加公式.
第二部分研究非线性微分系统解析近似解的求解方法和算法.同伦分析方法是近几年发展起来的构造非线性系统解析近似解十分有效的方法.与摄动方法不同,同伦分析方法的有效性与所考虑的非线性问题是否含有小参数无关.此外,不同于所有其它传统的摄动方法和非摄动方法,如人工小参数法,δ展开方法和Adomian分解方法等,同伦分析方法本身提供了一种方便的途径来控制和调节解级数的收敛速度和收敛区域.同伦分析方法已被广泛应用于求解应用数学和力学中的许多问题.
复合介质在物理学和工程领域随处可见,因此,复合介质的实验与理论研究受到了广泛的重视.摄动方法是求解弱非线性复合介质问题的有效工具.求解强非线性复合介质问题仍然非常困难,同伦分析方法的提出为强非线性问题的求解提供了有效的工具.文[85]和[86]分别应用同伦分析方法构造了强非线性复合介质问题的解析近似解,然而,为了计算简单,他们首先应用模式展开法将原系统简化为常微分系统,且只截取到第一模式项,这使所得的常微分系统与原系统之间存在较大的误差.为了提高解的精度,本文选取线性算子为线性偏微分方程,直接应用同伦分析方法构造原系统的解析近似解.所获结果明显优于已有的摄动解及同伦分析解.另外,本文也将同伦分析方法推广应用到分数阶微分方程情形.
Many phenomena in nature can be described with nonlinear differential equations. The study of analytical solutions to nonlinear differential equations plays a very important role in penetrating the inner structure, in analyzing the relationship of things as Well as in interpreting various physical phenomena. The naissance of high performance computer greatly promotes the study on symbolic computation of nonlinear differential equations and then coming forth abundant algorithms and methods to construct the analytical solutions of nonlinear differential equations. Armed with the computer algebraic system Maple, this dissertation concentrates on the nonlinear differential equations and do much research on various algorithms and approaches to construct exact solutions and analytical approximation ones. Our main works are summarized bellow.
Part I is devoted to study the algorithm and method to construct exact solutions of nonlinear differential equations. We do research from the following two aspects.
We improve and integrate several algebraic methods of constructing the exact solitary wave solutions to nonlinear evolution equations such as the Riccati equation method, the coupled Riccati equation method, the deformed mapping method, and the ansats making method and then propose an approach named "elliptic equation method." Based on Wu elimination method, we provide an software package RAEEM in computer algebraic system Maple for seeking for exact traveling wave solutions to nonlinear differential equations, which can output automatically a series possible exact traveling wave solutions for inputted equation including the polynomial type solutions, rational function solutions, exponential function solutions, triangular function solutions, hyperbolic function solutions, Jacobi elliptic function solutions and the Weierstrass elliptic function solutions etc..
The study of Backlund transformation method is very important in the sense of integrability and the solving of exact solutions to nonlinear differential equations. Especially, once the nonlinear superposition formula of solutions is obtained starting from the Backlund transformation, one can construct new solutions of differential equations only by algebraic computation. Benefiting from the existed constructing Backlund transformation method, we propose an mechanization algorithm to build a kind of self-Backlund transformations for 1+1 nonlinear evolution equations. Similarly, utilizing the idea of Wu Wentsun mechanization, we give a corresponding implementation software package AutoBT in Maple, which can not only output the self-Backlund transformations of all possible specified types and the corresponding parameters constraints, but also can output the nonlinear superposition formula of solutions.
Part II is devoted to study the algorithm and method to construct the analytic and approximating solutions of nonlinear differential systems. The homotopy analysis method is effective in constructing the analytic and approximating solutions of nonlinear differential systems, which is developed in recent several years. Differing from the perturbed method, the effectiveness of the homotopy analysis method is independent of whether or not the considering nonlinear problem having small parameter. Moreover, It is differ from all other traditional perturbed and unperturbed methods, such as the manual small parameter method, the 5 expansion method and the Adomian decomposition method etc. The homotopy analysis method itself provides a kind of convenient tool in controlling and adjusting the convergence speed and region of the solution series. This method has been widely used in solving many problems of applied mathematics and mechanics.
The experimental and theoretical researches concerning composite media have received much attention because of their potentially wide applications in engineering and physics. The perturbation method is a powerful tool for dealing with weakly nonlinear problems of composite media. However, it is still very difficult to solve strongly nonlinear problems of composite media. The new homotopy analysis method is a powerful tool for solving strongly nonlinear problems. The authors in [85] and [86] have constructed the analytic approximating solutions of strongly nonlinear composite media by the homotopy analysis method. However, for the simplicity of computation, they first predigest the original system into an ordinary system with the help of the mode expansion method and just keep the first mode, this may evoke great errors between the original system with the reduced ordinary differential system. To get the higher precision, in this paper we choose linear partial differential equations as linear operators, and directly construct the homotopy analysis solutions for the original system. Our obtained solutions are obviously superior to the known ones. Another innovation is that we extend the homotopy analysis method to solving fractional differential systems.
第一部分研究构造非线性演化方程精确解的方法和算法,具体包括两方面的内容:
对已有的构造非线性演化方程精确行波解的几种代数方法,如Riccati方程方法、耦合的Riccati方程方法、假设法、形变映射法等进行了推广和整合,提出了“椭圆方程方法”.并结合吴消元法的思想和方法,在计算机代数系统Maple上编写了推导非线性演化方程精确行波解的软件包RAEEM,该软件包可自动推导出输入方程一系列可能的精确行波解,其中包括多项式解、有理函数解、指数函数解、三角函数解、双曲函数解及Jacobi椭圆函数解、Weierstrass椭圆函数解等.
Bticldund变换研究对非线性微分方程的可积性及精确解的求解都有十分重要的意义.特别是,一旦从B(?)cldund变换推导出解的非线性叠加公式,则仅通过代数运算就可构造微分方程的新解.我们借鉴已有的构造B(?)cklund变换的方法,提出了构造1+1维非线性演化方程一类自B(?)cklund变换的机械化算法,并结合吴文俊数学机械化思想,在计算机代数系统Maple上实现了该算法,其中的软件包AutoBT不仅可自动推导出输入方程的可能的特定类型的自B(?)cklund变换及相应的参数约束条件,还可自动推导出解的非线性叠加公式.
第二部分研究非线性微分系统解析近似解的求解方法和算法.同伦分析方法是近几年发展起来的构造非线性系统解析近似解十分有效的方法.与摄动方法不同,同伦分析方法的有效性与所考虑的非线性问题是否含有小参数无关.此外,不同于所有其它传统的摄动方法和非摄动方法,如人工小参数法,δ展开方法和Adomian分解方法等,同伦分析方法本身提供了一种方便的途径来控制和调节解级数的收敛速度和收敛区域.同伦分析方法已被广泛应用于求解应用数学和力学中的许多问题.
复合介质在物理学和工程领域随处可见,因此,复合介质的实验与理论研究受到了广泛的重视.摄动方法是求解弱非线性复合介质问题的有效工具.求解强非线性复合介质问题仍然非常困难,同伦分析方法的提出为强非线性问题的求解提供了有效的工具.文[85]和[86]分别应用同伦分析方法构造了强非线性复合介质问题的解析近似解,然而,为了计算简单,他们首先应用模式展开法将原系统简化为常微分系统,且只截取到第一模式项,这使所得的常微分系统与原系统之间存在较大的误差.为了提高解的精度,本文选取线性算子为线性偏微分方程,直接应用同伦分析方法构造原系统的解析近似解.所获结果明显优于已有的摄动解及同伦分析解.另外,本文也将同伦分析方法推广应用到分数阶微分方程情形.
Many phenomena in nature can be described with nonlinear differential equations. The study of analytical solutions to nonlinear differential equations plays a very important role in penetrating the inner structure, in analyzing the relationship of things as Well as in interpreting various physical phenomena. The naissance of high performance computer greatly promotes the study on symbolic computation of nonlinear differential equations and then coming forth abundant algorithms and methods to construct the analytical solutions of nonlinear differential equations. Armed with the computer algebraic system Maple, this dissertation concentrates on the nonlinear differential equations and do much research on various algorithms and approaches to construct exact solutions and analytical approximation ones. Our main works are summarized bellow.
Part I is devoted to study the algorithm and method to construct exact solutions of nonlinear differential equations. We do research from the following two aspects.
We improve and integrate several algebraic methods of constructing the exact solitary wave solutions to nonlinear evolution equations such as the Riccati equation method, the coupled Riccati equation method, the deformed mapping method, and the ansats making method and then propose an approach named "elliptic equation method." Based on Wu elimination method, we provide an software package RAEEM in computer algebraic system Maple for seeking for exact traveling wave solutions to nonlinear differential equations, which can output automatically a series possible exact traveling wave solutions for inputted equation including the polynomial type solutions, rational function solutions, exponential function solutions, triangular function solutions, hyperbolic function solutions, Jacobi elliptic function solutions and the Weierstrass elliptic function solutions etc..
The study of Backlund transformation method is very important in the sense of integrability and the solving of exact solutions to nonlinear differential equations. Especially, once the nonlinear superposition formula of solutions is obtained starting from the Backlund transformation, one can construct new solutions of differential equations only by algebraic computation. Benefiting from the existed constructing Backlund transformation method, we propose an mechanization algorithm to build a kind of self-Backlund transformations for 1+1 nonlinear evolution equations. Similarly, utilizing the idea of Wu Wentsun mechanization, we give a corresponding implementation software package AutoBT in Maple, which can not only output the self-Backlund transformations of all possible specified types and the corresponding parameters constraints, but also can output the nonlinear superposition formula of solutions.
Part II is devoted to study the algorithm and method to construct the analytic and approximating solutions of nonlinear differential systems. The homotopy analysis method is effective in constructing the analytic and approximating solutions of nonlinear differential systems, which is developed in recent several years. Differing from the perturbed method, the effectiveness of the homotopy analysis method is independent of whether or not the considering nonlinear problem having small parameter. Moreover, It is differ from all other traditional perturbed and unperturbed methods, such as the manual small parameter method, the 5 expansion method and the Adomian decomposition method etc. The homotopy analysis method itself provides a kind of convenient tool in controlling and adjusting the convergence speed and region of the solution series. This method has been widely used in solving many problems of applied mathematics and mechanics.
The experimental and theoretical researches concerning composite media have received much attention because of their potentially wide applications in engineering and physics. The perturbation method is a powerful tool for dealing with weakly nonlinear problems of composite media. However, it is still very difficult to solve strongly nonlinear problems of composite media. The new homotopy analysis method is a powerful tool for solving strongly nonlinear problems. The authors in [85] and [86] have constructed the analytic approximating solutions of strongly nonlinear composite media by the homotopy analysis method. However, for the simplicity of computation, they first predigest the original system into an ordinary system with the help of the mode expansion method and just keep the first mode, this may evoke great errors between the original system with the reduced ordinary differential system. To get the higher precision, in this paper we choose linear partial differential equations as linear operators, and directly construct the homotopy analysis solutions for the original system. Our obtained solutions are obviously superior to the known ones. Another innovation is that we extend the homotopy analysis method to solving fractional differential systems.
引文
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55 范恩贵.齐次平衡法、Weiss-Tabor-Carnevale法与Clarkson-Kruskat约化法之间的联系,物理学报,2000,49:1409.
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61 Fu Z T,Liu S K,Liu S D.New kinds of solutions to Gardner equation,Chaos,Solitons and Fractals,2004,20:301.
62 Kichenassamy S,Oliver P J.Existence and Nonexistence of Solitary Wave Solutions to Higher Order Model Evolution Equations,SIAM J.Math.Anal.,1992,23:1141.
63 李志斌,潘素起,广义五阶KdV方程的孤波解与孤子解,物理学报,2001,50:402.
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73 Liu Y P,Li Z B.The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation,Chaos;Soliton & Fractals,(in press).
74 Liu Y P,Li Z B.Homotopy analysis method for nonlinear differential equations with fractional orders,Zeitschrift fur Naturforschung A,2008,63a:1.
75 Liu Y P,Li Z B.As an application of homotopy analysis method to nonlinear composites,Journal of Physics A:Mathematical and Theoretical,(accepted).
76 MAINARDI F. Fractional calculus: some basic problems in continuum and statistical mechanics, CISM lecture notes , Udine, Italy, 1996.
77 SAKAKIBARA S. Properties of vibration with fractional derivative damping of order 1/2, JSME International Joural, 1997,40: 393.
78 PETRASAND I, VINAGRE B M. Practical application of digital fractional order controller to temperature control, Porto, Portugal: Proc, of the European Control Conference (ECC2001), 2001,1764.
79 Kaya D. An application for the higher order modified KdV equation by decomposition method, Commun. Non. Sci. Numer. Simul., 2005,10: 693.
80 Orsingher E, Zhao X. The space-fractional telegraph equation and the related fractional telegraph process, Chinese Ann. Math., 2003,24: 1.
81 Orsingher E, Beghin L. Time-fractional telegraph equation and telegraph processes with brownian time, Prob. Theory Relat. Fields, 2004,128: 141.
82 Stroud D, Hui P M. Nonlinear susceptibilities of granular matter, Phys. Rev. B, 1988, 37: 8719.
83 Zeng X C, Bergman D J, Hui P M and Stroud D. Effective medium theory for weakly nonlinear composites, Phys. Rev. B, 1988,38: 10970.
84 Gu G Q and Yu K W. Effective conductivity of nonlinear composites, Phys. Rev. B, 1992,46:4502.
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85 Gu G Q, Yu K W. Homotopy Continuation Approach to Electrostatic Boundary Value Problems of Nonlinear Media, Commun. Theor. Phys., 1998, 29: 523.
86 Liu Y P and Li Z B, Analytic solutions of the nonlinear composite media, Chaos, Solitons & Fractals (in press).