兰州大学机构库 >数学与统计学院
几类常差分方程精确解的研究
Alternative TitleStudies of Exact Solutions to Several Kinds of the Ordinary Difference Equations
龚东山
Thesis Advisor李自珍
2009-05-27
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword变上限定和分 精确解 线性差分方程(组) 特征函数法 非线性差分方程 独立通解
Abstract差分方程研究的主要内容包括两个问题:差分方程的精确解和方程解的定性分析。其中对于解的定性分析研究,一般以差分模型为基础,具体讨论差分方程的稳定性、有界性、振动性、渐近性、周期性和概周期性等问题,相关的研究在过去几十年中取得了许多重要的成果,然而对于差分方程精确解的研究则相对滞后。尽管许多学者在寻找差分方程的精确解方面取得了一定的进展,但仍有许多结论还不太成熟:在线性差分方程中,有些论证还不是很完善,一般函数的和分不易得到;常系数齐次线性方程的多重特征根所对应特解的线性无关性并非显然;变系数线性方程的主要结论还停留在解的结构与形式上,缺乏一个普适性的方法;在非线性差分方程中,即使是一阶的离散黎卡提方程,也很难得到精确解。 本文重点讨论了定义在整数集Z或它的子集D上的几类常差分方程精确求解方法与解的显式表达。所得的主要结论如下: 1 对于线性齐次差分方程,可以得到它的解结构。并通过运用参数待定法(阮炯, 2002),论证了常系数线性齐次差分方程的多重特征根所对应的全部特解存在线性无关性,并推导出该非齐次差分方程的通解表达式。 2 对于线性非齐次差分方程,证明了其通解等于它的一个特解与相应的齐次方程的通解之和。如果已知齐次方程的一个基本解组,可运用常数变异法与函数和分计算,得到非齐次方程的一个特解的形式表达。 2.1 对于常系数线性非齐次差分方程,利用特征函数法(李自珍,龚东山, 2009)得到非齐次项为多项式、指数函数、三角函数、多项式与指数函数的乘积、对数函数以及它们的线性组合时的公式化特解。该方法简便易行,克服了常数变异法(Paul Mason Batchelder, 1927)、比较系数法(Saber Elaydi, 2005)及拉普拉斯变换法(张广,张高英, 2001)等传统方法计算工作量过大的缺陷,且特解形式非常直观。 2.2 对于几类可精确求解的变系数线性差分方程,利用构造函数法,将某些变系数方程化为常系数差分方程;通过引入变上限定和分,给出了一阶变系数线性差分方程的通解;运用函数积分法,得到了系数为线性函数时方程的通解;利用观察法找到方程的一个特解,并以此特解为基础,得出二阶变系数线性齐次方程的通解;借助降阶法,当差分方程的系数函数满足一定条件时,将复合差分方程问题化成若干个一阶变系数线性差分方程的求解问题。 3 对于线性差分方程组,得到了该方程组有解的一个充要条件,加强了王联(1991)关于一阶线性差分方程组与高阶线性差分方程同解的充分性结论,完善了线性差分方程精确解的理论体系,并利用线性代数与矩阵理论,推导出常系数线性差分方程组精确解的显式表达。 4 对于两类非线性差分方程,通过引入独立通解(龚东山, 2008),得到了相应齐次方程的通解表示。研究表明:这两类齐次差分方程的通解由若干个独立通解共同构成,且独立通解的个数与差分方程的阶数无关;当非齐次项为某些特殊函数时,给出了方程的若干个特解,且特解的个数与方程的次数有关,并论证了这些特解的线性相关性。
Other AbstractThe study of difference equations is main about two problems :exact solutions of the equations and qualitative analysis of the solutions. Qualitative analysis of the solutions generally bases on difference models and concretely talks about some problems of the equations, such as stability, boundedness, vibratility, asymptotic property, periodicity and almost periodicity. People have got many important results related above problems, but researching for exact solutions of the equations comparatively lag. In spite of some people have got along in this way, there are still many immature results. For examples: some argumentations are not consummate and the anti-difference of general equation can not be got easily in linear difference equations; the independent linearity between the multi-ply latent roots of homogenous linear equations with constant coefficients and their responded special solutions isn’t apparent.; the main results of difference equations with variable coefficients are still on the construction and form of the solutions, there still isn’t a universal method; it is hard to find the exact solutions of the non-linear equations even an order discrete Riccalti equation. In the article, methods of finding the exact solution of several kinds of difference equations which are defined in set of integer Z or its subset D and the apparent expression of the solutions are mostly talked about. Main results are listed as follow : 1 For linear homogenous difference equations , We get their general solution ,also prove the independent linearity on the multi-ply latent roots of homogenous linear equations with constant coefficients and their responded special solutions and deduce the expression of the general solution of the difference equations by using method of undetermined parameters (Ruan J., 2002). 2 For linear non-homogenous equations, we have proved that their general solution equal to the sum of one special solution and the general solution of homogenous equation. If a group of base solutions of a homogenous equation are known, the form of a special solution of non-homogenous equation can be got by method of variation of constant and computing anti-difference . 2.1 For linear non-homogenous difference equation with constant coefficients, we can use method of eigenfunction(Li Zi-zhen and Gong Dong-shan, 2009)to get the formula of special solution when the non-homogenous item is polynomial functions, exponential functions, trigonometric functi...
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225056
Collection数学与统计学院
Recommended Citation
GB/T 7714
龚东山. 几类常差分方程精确解的研究[D]. 兰州. 兰州大学,2009.
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