摘要
本文基于数学机械化思想,借助于符号计算软件,以非线性方程为对象,系统地研究了适用于强非线性问题的解析近似方法：Adomian分解方法(ADM)和同伦分析方法(HAM)的应用和机械化实现。
第一章是与本文相关的研究背景。简要综述了计算机代数和孤立子理论的发展进程,针对性地介绍了近年来解析近似方法的研究成果和现状。
第二章改进了Adomian分解方法,能够获得修正Korteweg-de Vries (mKdV)方程和Kadomtsev-Petviashvili (KP)方程的双孤子解。通过引入自变量变换和行波变换,将Degasperis-Procesi (DP)方程短波模型化为常微分方程,应用Adomian分解方法求解之,获得其闭合形式的解析解,再经过反变换,能够获得其环状孤子解。以上结果表明了Adomian分解方法在求解方程特殊孤子解方面的有效性。对Adomian分解方法进行了推广,解决了方程中离散变量不同于连续方程中的变量问题,并与Pade近似结合,能够获得几个经典的非线性微分差分方程组的孤子解,显著提高了方程解析近似解的精度。同时,我们还讨论了Pade有理近似中出现的伪极点问题,给出了合适选择Pade近似阶数的指导原则。获得的解析近似解与精确解符合得很好,表明了Adomian分解方法对复杂强非线性问题的有效性。
第三章通过引入自变量变换和行波变换,将偏微分方程化为常微分方程,通过同伦分析方法求解之,再经过反变换,能够获得DP方程短波模型的环状孤子解和Camassa-Holm (CH)方程短波模型的尖状孤子解,结果表明了同伦分析方法在求解方程特殊孤子解方面的有效性。对同伦分析方法进行了推广,解决了方程中离散变量不同于连续方程中的变量问题,改进了同伦分析方法选择初始猜测解的方法,能够获得离散修正KdV方程的亮孤子解,获得的解析近似解与精确解符合得很好,表明了同伦分析方法对复杂强非线性问题的有效性。
第四章在计算机代数系统Maple上实现了Biazar提出的求解Adomian多项式的算法,编制了构造微分方程(组)和积分方程(组)解析近似解的自动推导软件包,这个算法避免了Adomian多项式的计算膨胀问题,降低了计算难度并显著提高了计算速度,通过大量实例说明了该软件包的有效性和实用性。
In the dissertation, under the guidance of mathematical mechanization, two kinds of analytical approximation methods, which are Adomian decomposition method (ADM) and homotopy analysis method (HAM) for strong nonlinear problems around the non-linear equations, are investigated by means of symbolic computation. The application and mechanization of them are discussed, respectively.
Chapter 1 is the research background related to the dissertation. The development of computer algebra and the theory of solitons are briefly outlined. Subsequently, the recent development and achievement of analytical approximation methods are summarized at home and abroad.
In Chapter 2, the two-soliton solutions of modified Korteweg-de Vries (mKdV) equation and Kadomtsev-Petviashvili (KP) equation can be obtained by the modified ADM, respectively. By means of the transformation of the independent variables and trav-eling wave transformation, the short-wave model for Degasperis-Procesi (DP) equation is reduced to an ordinary differential equation the solution of which in closed form can be obtained by ADM. Then by means of the transformations back to the original variables, the loop-soliton solution of the short-wave model for DP equation can be derived. The results indicate the validity of ADM for constructing the special type of soliton solution of nonlinear differential equations. The discrete variable in nonlinear differential-difference equation is successfully overcome and ADM is extended to solving some classical sys-tems of differential-difference equations. The soliton solutions of them can be obtained with high accuracy by combining ADM and Pade approximants. Meanwhile, the pos-sibility of spurious poles of rational approximation is discussed and a criterion for the choice of the order of Pade approximants is given. The obtained results degree well with the exact solutions. This demonstrates the validity of ADM in strong nonlinear problems.
In Chapter 3, By means of the transformation of the independent variables and trav-eling wave transformation, the partial differential equation is reduced to an ordinary dif-ferential equation, which can be solved by HAM. Then by means of the transformations back to the original variables, the solution of the original equation is obtained. The one-loop soliton solution of the short-wave model for DP equation and one-cusp soliton for Camassa-Holm (CH) equation can be obtained. This indicates the validity of HAM for constructing the special type of soliton solution of nonlinear differential equations. The discrete variable in nonlinear differential-difference equation is successfully overcome and HAM is extended to solving the discrete mKdV equation. The bright soliton solution can be obtained. A technique for choosing the initial guess is also shown. The obtained results degree well with the exact solution. This demonstrates the validity of HAM in strong nonlinear problems.
In Chapter 4, Based on the existed algorithm for the calculation of ADM polynomials proposed by Biazar, an software package is developed to construct approximate analytic solutions of differential equations and integral equations automatically in computer alge-braic system Maple. Avoiding the huge size of the calculation of ADM polynomials, the algorithm needs less time without any need to formulas other than elementary operations than that based on ADM. Many examples are presented to illustrate the implementation of the package.
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