非线性Volterra型积分微分方程的高精度谱方法

Volterra integral-differential equations arise in many branches of science such as physics, biology, chemistry, engineering, and control theory. They are particularly suitable to describe evolutionary phenomena with memory, which feature makes the theoretical study and the numerical treatment complicated. Nonlinear Volterra integro-differential equations are more important. Due to the nonlinear term, numerical mehods are more challenging. As we know, spectral methods are the global and high accuracy methods, so spectral methods are more suitable to solve the integral-differential equations. Although spectral methods have attracted great attention in solving differential equations, little experience is available in applying spectral methods to solve nonlinear Volterra integral equations. In this project, we plan to propose and analyze the spectral collection methods to solve nonlinear Volterra integral-differential equations. We will also study spectral collocation methods for nonlinear Volterra delay integral-differential equations and nonlinear Volterra integral-differential equations with singular kernels. The main purpose of our work is to carry out the error estimates for the spectral method. The spectral rate of convergence for the proposed methods will be established. Numerical experiments will be carried out to verify the theoretical results. We hope taht our results provide the basis for highly accurate algorithms of integral-differential equations. The achievements of this project will be of great importance in both theory and practice.

Volterra型积分积分微分方程在物理、生物、化学与工程等许多领域具有广泛的应用背景，由于积分微分方程具备记忆性质，具有全局性，因此对其数值求解较为困难。非线性Volterra型积分微分方程是积分微分微分方程中最重要的一种，且非线性导致其数值求解更加困难。谱方法是一种高精度的全局方法，所以谱方法更适合于求解积分微分方程。本项目主要针对非线性Volterra型积分微分方程、带延迟项的非线性Volterra型积分微分方程和带弱奇异性的Volterra型积分微分方程，设计高精度谱配置方法，研究其误差估计、收敛性以及超收敛性；并将方法推广到高维的情况，进行大量的数值试验证实理论分析结果。本项目所获得研究成果对于积分微分方程的高精度数值方法具有重要的意义。

Volterra型积分微分方程因为具有非局部算子，能更好地刻画更具有记忆和遗传等行为，越来越多地被用于不同领域的建模，但是非局部算子的存在使得这类方程的理论研究和数值求解非常困难。谱方法是高精度的全局方法，在保证同样精度的前提下，谱方法比其它低阶方法所用的信息量要少，因此高阶的谱方法更适合处理非局部方程。本项目针对带有奇异核和带延迟项的Volterra型积分微分方程，以及对描述反常扩散的几类分数阶积分微分方程，利用谱方法的高精度和全局性，设计了几种高效数值方法，给出了算法的稳定性和敛性分析。这些问题都是学术上具有挑战性的研究热点，期望通过本项目的研究，提高这些非局部模型的计算效率，促进谱方法在分数阶积分微分方程的数值求解中的应用。