Volterra积分微分方程高效谱配置方法研究

项目名称： Volterra积分微分方程高效谱配置方法研究

项目编号： No.11271145

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 陈艳萍

作者单位： 华南师范大学

项目金额： 60万元

中文摘要： Volterra积分方程和积分微分方程在物理、生物、化学与工程等许多领域中具有广泛的应用背景，由于这类方程具备记忆性质，对其数值求解更为困难。当前，利用具有高精度谱方法来研究Volterra积分微分方程的数值计算是国际上最热门的前沿研究领域之一，具有重要的学术意义和应用价值。本项目主要研究Volterra积分方程和积分微分方程的带光滑核以及带弱奇异核的谱配置方法，设计高效率和高精度算法。同时我们提出和分析带延迟项Volterra方程的谱方法。由于奇异方程的解的导数在端点具有某种奇异性，对其数值方法的研究在理论分析上相当复杂。通过变量替换和函数变换，利用正交多项式和Gauss数值积分逼近理论、Jacobi 加权Besov /Sobolev空间和紧算子理论、插值多项式的Lebesgue常数估计和一些重要的不等式等工具进行收敛性分析，得到谱精度的误差估计，并且进行大量的数值试验证实理论分析结果。

中文关键词： Volterra 积分方程；Volterra 积分微分方程；谱配置方法；高精度算法；收敛性分析

英文摘要： Volterra integral equations arise in great many branches of science like physics, biology, chemistry, engineering, and control theory. For example they arise from population dynamics, spread of epidemics, semi-conductor devices, inverse problems related to wave propagation, which frequently occur in connection with time dependent or evolutionary systems. They are particularly suitable to describe evolutionary phenomena with memory and this feature makes the theoretical study and the numerical treatment complicated. Spectral methods are a class of important numerical methods for differential equations. The earliest applications of the spectral collocation method to partial differential equations were made for spatially periodic problems. The study of convergence properties of collocation methods for Volterra integral equations and of methods for accelerating the convergence orders has received considerable attention. As we known, classical spectral methods were reasonably mature, and the research focus had clearly shifted to the use of high-order methods for problems on complex domains. Although spectral methods have attracted much attention in solving differential equations, little experience is available in applying spectral collocation type methods to solve Volterra integral equations. In this project, we prop

英文关键词： Volterra integral equations；Volterra integro-differential；spectral-collocation methods；high accuracy algorithms；convergence analysis