两类非线性微分方程的多点边值问题研究

发布时间：2018-05-18 00:14

本文选题：二阶非线性微分方程 + 多点边值问题　；参考：《广西大学》2017年硕士论文

【摘要】：在实际的工程应用中存在着许多非线性振动问题的数学模型可以化成duffing方程来研究.对于一些较为复杂的振动情况,其非线性项不仅仅是运动函数及其立方形式的复合,为此我们考虑一般化的Duffing方程.另外分数阶微积分是微积分的重要分支,一般的分数阶变系数Bagley-Torvik方程更是具有深刻的物理背景以及良好的应用价值.本文考虑这两类重要的非线性微分方程,研究它们的多点边值条件下解的存在性和唯一性及逼近方法.主要内容有:(1)采用积分方法将一般化Duffing方程多点边值问题转转化为第二类非线性Hammerstein积分方程,根据Banach空间上的不动点原理在平方可积函数空间研究了其解的唯一性.然后提出了采用积分中值定理和分段逼近求解第二类非线性Hammerstein积分方程的思想,获得了近似解的表达式并进行收敛性分析和误差估计,和已有方法进行比较,通过数值实例验证了数值方法的可行性和有效性.(2)将分数阶变系数非线性Bagley-Torvik方程四点边值问题转化为含有弱奇异核或者连续核的第二类Fredholm-Hammerstein积分方程,利用不动点原理在连续函数空间研究了解的唯一性.然后提出了积分型分段Taylor级数展开法,给出了相应的充分条件.得到了含有弱奇异核的第二类非线性Fredholm-Hammerstein积分方程的近似解.通过对近似解的收敛性、误差估计和数值实例进行计算分析,验证了获得的定理.以上研究成果丰富了微分方程边值问题理论及数值求解方法,为实际物理力学过程的模拟提供了一定的理论依据.
[Abstract]:In practical engineering applications, there are many mathematical models of nonlinear vibration problems which can be transformed into duffing equations to be studied. For some more complex vibration cases, the nonlinear term is not only a combination of the motion function and its cubic form, so we consider the generalized Duffing equation. In addition, fractional calculus is an important branch of calculus, and the general fractional variable coefficient Bagley-Torvik equation has a profound physical background and good application value. In this paper, we consider these two important nonlinear differential equations, and study the existence, uniqueness and approximation methods of solutions under multi-point boundary value conditions. In this paper, the multipoint boundary value problem of the generalized Duffing equation is transformed into the nonlinear Hammerstein integral equation of the second kind by using the integral method. The uniqueness of the solution is studied in the square integrable function space according to the fixed point principle on the Banach space. Then the idea of solving the nonlinear Hammerstein integral equation of the second kind by using integral mean value theorem and piecewise approximation is presented. The expression of approximate solution is obtained, and the convergence analysis and error estimation are carried out, which are compared with the existing methods. The feasibility and validity of the numerical method are verified by numerical examples. The four-point boundary value problem of fractional variable coefficient nonlinear Bagley-Torvik equation is transformed into the second kind of Fredholm-Hammerstein integral equation with weak singular kernel or continuous kernel. The uniqueness of solution in continuous function space is studied by using the fixed point principle. Then the integral type piecewise Taylor series expansion method is proposed and the corresponding sufficient conditions are given. The approximate solutions of the second kind of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels are obtained. The convergence, error estimation and numerical examples of the approximate solution are calculated and analyzed, and the obtained theorem is verified. The above research results enrich the theory of boundary value problem of differential equation and numerical solution method, and provide a certain theoretical basis for the simulation of practical physical and mechanical process.
【学位授予单位】：广西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.8

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