非线性微分系统解析解的符号计算研究
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摘要
数学机械化研究是我国数学家吴文俊先生于上世纪70年代末开始倡导的一个研究领域.国际上在上世纪80年代就积极推进基于符号的计算机处理方法,发展利用计算机进行分析、演算和推理的理论与实践,随之也先后诞生了几个优秀的符号计算软件,如Reduce、MACSYMA、Mathematica和Maple等.特别是Mathematica和Maple已经在数学和工程领域中被广泛使用.我国在该领域的应用研究(尤其是计算软件)起步较晚,目前水平也远远落后于西方发达国家.因而,国家十二五发展规划将计算软件列为重点支持的研究方向.
科学研究和工程技术中很多问题的研究,最终都可以归结为非线性微分方程的求解问题.因此,非线性微分方程的解法研究始终是数理科学中核心的课题.本文以微分方程为研究对象,在吴文俊数学机械化思想的指引下,主要研究构造非线性微分方程(特别是非线性微分初、边值问题)解析解的机械化算法,进而研发自动推导非线性微分系统特定类型解析解的软件包.本文的创新之处在于首次将双重分解法及二步分解法等嵌入到经典的Adomian分解法中,发展出构造非线性微分初、边值问题解析近似解的新算法,并研制出相应的符号推演软件包.具体工作如下：
1.解析近似解Adomian分解法是构造非线性微分系统解析近似解的有效方法之
.该方法因其计算过程简单且能求解强非线性问题,被广泛应用于各种非线性问题的求解中.在经典Adomian分解法的基础上,Adomian及其合作者还提出了改进的Adomian分解法、加速的Adomian分解法、二步分解法、双重分解法等.特别是由Adomian和Rach发展起来的双重分解法,可大大简化求解非线性微分边值问题的计算过程.二步分解法的实质是在经典分解法的基础上增加了尝试构造微分方程精确解的环节.2008年Rach借助于截断算子重新定义了Adomian多项式,其新算法不仅涵盖了已有的Adomian多项式计算算法,而且新算法的计算效率明显提高.本文基于Rach的新算法、二步分解法和Pade近似技术,提出了构造非线性微分初值问题解析近似解的ADM-Pade新算法；并将双重分解法嵌入到求解初值问题的ADM-Pade算法中,进而提出了构造非线性微分边值问题解析近似解的新算法.然后将这两个算法推广到分数阶微分方程情形.在上述三个新算法的基础上,本文还在计算机代数系统Maple平台下研发了软件包ADMP该软件包可自动推导出非线性微分初、边值问题(包括分数阶非线性微分初、边值问题)的解析近似解.该软件包对具有Robin等复杂边界条件的非线性微分系统和具有分数阶初始条件的非线性微分系统同样也有效.
2.精确解：不变子空间方法是构造非线性微分方程精确解的有效方法之一.本文应用不变子空间方法构造了一个一维反应扩散方程的精确解,并深入分析了其行为特征.在一维方程不变子空间的基础上,由二维反应扩散方程的特征,进一步构造出二维方程的不变子空间,从而获得了二维反应扩散方程的精确解.最后,通过斑图和时空序列图,成功解释了一系列自然现象.不变子空间方法的原理虽然简单,但是其计算过程相当繁复.本文也在Maple平台下完全实现了不变子空间方法,其中包括软件包ISM.它可以自动推导出输入方程的精确解和相应的参数约束条件,现己成功求解了三十多个非线性微分方程.需要注意的是,利用该软件包还可推导出输入方程一系列的特解,其中包括多项式解、有理函数解、三角函数解、指数函数解及不同函数的混合型解,如文中(4.72)表示的complexitions解就是由指数函数与三角函数混合表达的,又如文中(4.25)表示的positons解就是由不同三角函数混合表达的解等等.
In the70's of last century, the Chinese mathematician Wentsun Wu advocated a new field of research-the mathematical mechanization. International scholars actively promot-ed the symbol computation in the1980s, and they also developed the theory and practice of computer-aided analysis, calculations and reasoning. So some excellent computer algebra systems are developed, such as Reduce, MACSYMA, Mathematica and Maple. Especial-ly, Mathematica and Maple have been widely used in mathematics and engineering. It is late that the Chinese scholars start to study on the symbol computation in the field of application research, particularly for the computing software. Furthermore, the applica-tion research of symbol computation is far behind that of the western developed countries. Therefore, computing software is highlighted to be a promising research direction in the support of the National Twelfth Five-Year Development Plan.
A lot of problems ultimately are attributed to nonlinear differential equations in science and engineering. Therefore, the studies of nonlinear differential equations are always impor-tant scientific topic. With the mathematical mechanization, this dissertation concentrates on the nonlinear differential equations and do much research on various mechanization al-gorithms to construct analytical solutions of nonlinear differential equations (especially the initial value or the boundary value problems in nonlinear differential equations). Further-more, a software package is developed to automatically construct some special types of analytical solutions. The innovation of this dissertation is that the double decomposition method and the two-step decomposition method are embedded into the classic Adomian decomposition mehtod for the first time, then we propose a new algorithm to construct an-alytical approximate solutions of nonlinear differential equations with initial or boundary conditions. Besides, we develop the corresponding software package. Our main works are summarized bellow.
1. Analytical approximate solutions:The Adomian decomposition method is one of the most effective methods to construct analytical approximate solutions of nonlinear dif-ferential equations. The Adomian decomposition method is widely applied to solve various nonlinear differential equations, because of the simplicity of calculations and the ability of solving strongly nonlinear problems. Based on the classic Adomian decomposition method, Adomian and his colleagues delivered the modified Adomian decomposition method, the two-step decomposition method, and even the double decomposition method which can solve boundary value problems in nonlinear differential equations with less computation. An essence is that the two-step decomposition method attempts at constructing exact so-lutions of differential equations on the basis of the classic Adomian decomposition method. In2008, Rach developed the novel notion to derive the unifying formula for the family of classes of the Adomian polynomials, but also the new definition of Adomian polynomials is more efficient. On the basis of the Rach's new definitions of the Adomian polynomi-als, two-step decomposition method and Pade technique, a new algorithm is proposed to construct analytical approximate solutions of nonlinear differential equations with initial conditions. Then combined with the double decomposition method, a new algorithm is presented to construct analytical approximate solutions of nonlinear differential equations with boundary conditions. Furthermore, we extend these algorithms to construct analyti-cal approximate solutions of fractional differential equations. Finally, we develop a Maple software package ADMP, which automatically construct analytical approximate solutions of nonlinear differential equations (including fractional differential equations) with initial or boundary conditions. ADMP also can solve nonlinear differential equations with Robin boundary conditions or fractional initial conditions.
2. Exact solutions:The invariant subspaces method is one of the most effective methods to construct exact solutions for nonlinear differential equations. We apply the invariant subspace method to construct exact solutions of the one-dimensional reaction-diffusion equation and analyze its inner behavior. Furthermore, based on the invariant subspaces of the one-dimensional reaction-diffusion equation, we construct exact solutions of two-dimensional reaction-diffusion equation. Finally, natural phenomena are successfully interpreted by pattern and temporal or spatial sequence diagrams. The theory of the invari-ant subspace method is simple, but its calculation is rather complicated. By the invariant subspace method, we develop a Maple software package ISM, which can output automat-ically a series possible exact solutions for system input and the corresponding parameters constraints. Applying it, we have succeeded in solving many nonlinear differential equation-s, especially for constructing exact solutions, including polynomial solutions, trigonometric solutions, exponential-trigonometric solutions, etc. We note that Eq.(4.72) is complexi-tions consisting of trigonometric functions and exponential functions. While Eq.(4.25) is positions consisting of different trigonometric functions.
科学研究和工程技术中很多问题的研究,最终都可以归结为非线性微分方程的求解问题.因此,非线性微分方程的解法研究始终是数理科学中核心的课题.本文以微分方程为研究对象,在吴文俊数学机械化思想的指引下,主要研究构造非线性微分方程(特别是非线性微分初、边值问题)解析解的机械化算法,进而研发自动推导非线性微分系统特定类型解析解的软件包.本文的创新之处在于首次将双重分解法及二步分解法等嵌入到经典的Adomian分解法中,发展出构造非线性微分初、边值问题解析近似解的新算法,并研制出相应的符号推演软件包.具体工作如下：
1.解析近似解Adomian分解法是构造非线性微分系统解析近似解的有效方法之
.该方法因其计算过程简单且能求解强非线性问题,被广泛应用于各种非线性问题的求解中.在经典Adomian分解法的基础上,Adomian及其合作者还提出了改进的Adomian分解法、加速的Adomian分解法、二步分解法、双重分解法等.特别是由Adomian和Rach发展起来的双重分解法,可大大简化求解非线性微分边值问题的计算过程.二步分解法的实质是在经典分解法的基础上增加了尝试构造微分方程精确解的环节.2008年Rach借助于截断算子重新定义了Adomian多项式,其新算法不仅涵盖了已有的Adomian多项式计算算法,而且新算法的计算效率明显提高.本文基于Rach的新算法、二步分解法和Pade近似技术,提出了构造非线性微分初值问题解析近似解的ADM-Pade新算法；并将双重分解法嵌入到求解初值问题的ADM-Pade算法中,进而提出了构造非线性微分边值问题解析近似解的新算法.然后将这两个算法推广到分数阶微分方程情形.在上述三个新算法的基础上,本文还在计算机代数系统Maple平台下研发了软件包ADMP该软件包可自动推导出非线性微分初、边值问题(包括分数阶非线性微分初、边值问题)的解析近似解.该软件包对具有Robin等复杂边界条件的非线性微分系统和具有分数阶初始条件的非线性微分系统同样也有效.
2.精确解：不变子空间方法是构造非线性微分方程精确解的有效方法之一.本文应用不变子空间方法构造了一个一维反应扩散方程的精确解,并深入分析了其行为特征.在一维方程不变子空间的基础上,由二维反应扩散方程的特征,进一步构造出二维方程的不变子空间,从而获得了二维反应扩散方程的精确解.最后,通过斑图和时空序列图,成功解释了一系列自然现象.不变子空间方法的原理虽然简单,但是其计算过程相当繁复.本文也在Maple平台下完全实现了不变子空间方法,其中包括软件包ISM.它可以自动推导出输入方程的精确解和相应的参数约束条件,现己成功求解了三十多个非线性微分方程.需要注意的是,利用该软件包还可推导出输入方程一系列的特解,其中包括多项式解、有理函数解、三角函数解、指数函数解及不同函数的混合型解,如文中(4.72)表示的complexitions解就是由指数函数与三角函数混合表达的,又如文中(4.25)表示的positons解就是由不同三角函数混合表达的解等等.
In the70's of last century, the Chinese mathematician Wentsun Wu advocated a new field of research-the mathematical mechanization. International scholars actively promot-ed the symbol computation in the1980s, and they also developed the theory and practice of computer-aided analysis, calculations and reasoning. So some excellent computer algebra systems are developed, such as Reduce, MACSYMA, Mathematica and Maple. Especial-ly, Mathematica and Maple have been widely used in mathematics and engineering. It is late that the Chinese scholars start to study on the symbol computation in the field of application research, particularly for the computing software. Furthermore, the applica-tion research of symbol computation is far behind that of the western developed countries. Therefore, computing software is highlighted to be a promising research direction in the support of the National Twelfth Five-Year Development Plan.
A lot of problems ultimately are attributed to nonlinear differential equations in science and engineering. Therefore, the studies of nonlinear differential equations are always impor-tant scientific topic. With the mathematical mechanization, this dissertation concentrates on the nonlinear differential equations and do much research on various mechanization al-gorithms to construct analytical solutions of nonlinear differential equations (especially the initial value or the boundary value problems in nonlinear differential equations). Further-more, a software package is developed to automatically construct some special types of analytical solutions. The innovation of this dissertation is that the double decomposition method and the two-step decomposition method are embedded into the classic Adomian decomposition mehtod for the first time, then we propose a new algorithm to construct an-alytical approximate solutions of nonlinear differential equations with initial or boundary conditions. Besides, we develop the corresponding software package. Our main works are summarized bellow.
1. Analytical approximate solutions:The Adomian decomposition method is one of the most effective methods to construct analytical approximate solutions of nonlinear dif-ferential equations. The Adomian decomposition method is widely applied to solve various nonlinear differential equations, because of the simplicity of calculations and the ability of solving strongly nonlinear problems. Based on the classic Adomian decomposition method, Adomian and his colleagues delivered the modified Adomian decomposition method, the two-step decomposition method, and even the double decomposition method which can solve boundary value problems in nonlinear differential equations with less computation. An essence is that the two-step decomposition method attempts at constructing exact so-lutions of differential equations on the basis of the classic Adomian decomposition method. In2008, Rach developed the novel notion to derive the unifying formula for the family of classes of the Adomian polynomials, but also the new definition of Adomian polynomials is more efficient. On the basis of the Rach's new definitions of the Adomian polynomi-als, two-step decomposition method and Pade technique, a new algorithm is proposed to construct analytical approximate solutions of nonlinear differential equations with initial conditions. Then combined with the double decomposition method, a new algorithm is presented to construct analytical approximate solutions of nonlinear differential equations with boundary conditions. Furthermore, we extend these algorithms to construct analyti-cal approximate solutions of fractional differential equations. Finally, we develop a Maple software package ADMP, which automatically construct analytical approximate solutions of nonlinear differential equations (including fractional differential equations) with initial or boundary conditions. ADMP also can solve nonlinear differential equations with Robin boundary conditions or fractional initial conditions.
2. Exact solutions:The invariant subspaces method is one of the most effective methods to construct exact solutions for nonlinear differential equations. We apply the invariant subspace method to construct exact solutions of the one-dimensional reaction-diffusion equation and analyze its inner behavior. Furthermore, based on the invariant subspaces of the one-dimensional reaction-diffusion equation, we construct exact solutions of two-dimensional reaction-diffusion equation. Finally, natural phenomena are successfully interpreted by pattern and temporal or spatial sequence diagrams. The theory of the invari-ant subspace method is simple, but its calculation is rather complicated. By the invariant subspace method, we develop a Maple software package ISM, which can output automat-ically a series possible exact solutions for system input and the corresponding parameters constraints. Applying it, we have succeeded in solving many nonlinear differential equation-s, especially for constructing exact solutions, including polynomial solutions, trigonometric solutions, exponential-trigonometric solutions, etc. We note that Eq.(4.72) is complexi-tions consisting of trigonometric functions and exponential functions. While Eq.(4.25) is positions consisting of different trigonometric functions.
引文
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