【作者】 李娟；

【导师】 陈景润； 杜锐；

【作者基本信息】 苏州大学 ， 计算数学， 2019， 硕士

【摘要】 本文提出了一种求解一阶含时振荡系数微分方程(均匀化)解的配点方法.由于均匀化方程涉及到Volterra积分微分方程,Volterra积分方程,以及一个微分公式和积分公式,故该方法包括四个部分:数值积分求解积分公式;配点法求解Volterra积分方程;数值微分求解微分公式;配点法求解Volterra积分微分方程.该方法的误差来源于以上四个步骤中数值逼近产生的误差.通过数值分析,我们得到了该方法的理论收敛阶,并由数值算例加以验证.此外,通过控制这些误差,我们得到了均匀化方程解与微分方程解关于方程中小参数ε之间的关系.由于已有的均匀化理论没有解之间的收敛性结果,因此我们的结果为均匀化理论提供了一个很好的补充.更多还原

【Abstract】 In this thesis,we propose a collocation method for solving the(homogenization)solution of first-order differential equations with time-dependent oscillation coefficient.Due to that the homogeniza.tion equation involves Volterra integral differential equa?tion,Volterra integral equation,and a,differential formula and integral formula,the method includes four parts:numerical integration to solve integral formula;collocation methocl to solve Volterra integral equation;numerical differential to solve differential formula;collocation method to solve Volterra integral differential equation.The error of this method is derived from the numerical approximation in the above four steps.By numerical analysis,we obtain the theoretical convergence order of the method and verify it by numerical examples.In addition,by controlling these errors,we obtain the relationship between the homogenization equation solution and the differential e-quation solution for the small parameter ε.Since the existing homogenization theory has no convergence results between homogenization equation solution and the differen-tial equation solution,our results provide a good complement to the homogenization theory.更多还原