非线性及高维空间上的Volterra型积分方程谱配置法研究

Volterra integral equations are widely used in many scientific fields such as population growth, control theory, viscoelasticity, turbulent diffusion problems, American options valuation etc. The research of high-precision numerical methods for such equations has important theoretical value and application prospects. In recent years, there have been extensive studies in convergence properties of spectral collocation methods for the linear and one-dimensional Volterra integral equations. In this project, we will investigate the spectral collocation methods for some nonlinear and multidimensional Volterra integral equations. Combining the estimates for interpolation errors in Sobolev space norm, Gronwall inequality, Hardy inequality, maximum norm estimates of Lagrange basis functions as well as Lagrange mean value theorem, Newton iteration methods and orthogonal polynomials approximation theory in multidimensional space, we will obtain that the errors between exact solutions and approximate solutions obtained by the spectral collocation methods decay exponentially. We will also study Volterra integro-differential equations. These equations may be viewed formally as ordinary differential equations perturbed by a memory term given by a Volterra integral operator. In our theoretical analysis, the initial condition will be restated as an equivalent integral equation and the integral term will be approximated by using Legendre-Gauss quadrature formula. Some numerical experiments will be presented to demonstrate the effectiveness of the proposed method and confirm the theoretical results. A vigorous error analysis will be provided which can show that both the errors of approximated solutions and the errors of approximated derivatives of the solutions decay exponentially. This research will provide a theoretical basis for solving other nonlinear and multidimensional problems. This project is also the development and innovation of the theoretical system of spectral methods.

Volterra型积分方程广泛应用于许多科学领域中，如人口增长、控制理论、粘弹性研究、湍流扩散问题、美式期权定价问题等，研究此类方程的高精度数值计算方法具有重要的理论价值和应用前景。目前国内外有关Volterra型积分方程高效谱配置法的研究工作主要针对一维线性情形。本项目研究用谱配置法求解非线性以及高维空间上的Volterra型积分方程，结合Sobolev空间范数意义下的插值误差估计、Lagrange插值基函数和的最大模估计式、Gronwall不等式、Hardy不等式等，利用Lagrange中值定理、牛顿迭代法以及把正交多项式逼近理论中的一些误差估计结果推广到高维情形的方法与技巧，从理论和数值模拟两方面证明原方程的精确解与用谱配置法求得的近似解之间的误差具有指数衰减性。本项目的研究将为解决其它非线性及高维空间上的初值问题提供理论依据，也是对谱方法理论体系的发展和创新。

Volterra型积分方程在物理、天文、力学、工程、生物医学等学科都有着广泛和重要的应用，对于许多国民经济和社会发展中的重要问题，如人口增长、控制理论、粘弹性研究、湍流扩散问题等实际模型都可以通过转化为Volterra型积分方程而得到解决。虽然对于一维线性的Volterra型积分方程谱配置解法研究已经取得了很大进展，但这远远不能满足经济和社会发展的需要。本项目研究用谱配置法求解高维空间上的线性以及非线性Volterra型积分方程，包括带光滑核的Volterra型积分方程以及带弱奇异核的Volterra型积分方程，设计用谱配置法求解不同类型的Volterra型积分方程数值计算程序，建立任意d维空间上的Volterra型积分方程谱配置法离散格式，对未知的解函数用高维Lagrange插值多项式离散，从理论和数值模拟两方面证明了原方程的精确解与用谱配置法求得的近似解之间的误差具有指数衰减性。本项目的研究为解决其它非线性及高维空间上的初值问题提供了理论依据，也是对谱方法理论体系的发展和创新。