Volterra积分微分方程高效谱配置方法研究

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 propose and analyze the (Legendre, Chebyshev, Jacobi) collocation spectral methods to solve the second kind Volterra integral (integro-differential) equations and delay Volterra integral (integro-differential) equations. We also study spectral collocation methods for the Volterra integral equations of the second kind, with weakly singular kernels. The numerical treatment of these singular equations is not simple, mainly due to the fact that the solutions of them usually have a weak singularity at initial time, even when the inhomogeneous term is regular. Specific methods have been proposed by several authors for equations with smooth solution and for equations with nonsmooth solution, such as Fractional linear multistep methods, ordinary collocation methods, and standard product integration methods. Recently, it has been demonstrated by some researchers that Jacobi weighted Besov and Sobolev spaces are the most appropriate tools for obtaining optimal upper and lower bounds when dealing with weakly singular problems, particularly for those with certain singularity at the end-points. This approach may be useful for analyzing the direct Jacobi-collocation spectral approach outlined in our present work. 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 method is established in infinity norm and weighted L^2 norm. Numerical experiments are also carried out, which will be used to verify the theoretical results. It is found that spectral type collocation methods have high order accuracy and are high efficient comparable with that of ordinary collocation methods.

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。

积分微分方程比一般的微分方程更能精确地反映系统的运动规律，特别是对一些复杂的带有记忆性质的系统的数学建模往往导致延迟微分方程或积分方程，所以对延迟微分积分方程的研究有广泛实际应用背景。积分微分方程的主要特征是所谓的全局依赖性，即方程的解在一个点上的值依赖于积分区域内任何点的值，当离散方法不论是低阶的差分方法和有限元方法还是高阶的谱方法，这个特性导致离散方程的非局部性。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。. 我们重点研究线性与非线性第二类带弱奇异核的Volterra积分方程、带中立项的弱奇异Volterra型微分积分方程Jacobi、常数延迟、变延迟及比例延迟等多种类型Volterra延迟型积分微分方程、高维区域的 Volterra积分微分方程及分数阶微分方程谱配置方法及分片谱配置方法，根据奇异指标数的不同选取合适的正交多项式空间和不同的谱配置方法，针对具有光滑解和具有非光滑解的两种情形分别进行了深入地研究，寻找、探索并选取合适的数学理论工具解决非光滑解奇异问题，通过数学上的严格证明，得到谱配置方法的数值解在最大模范数和加权积分平均范数意义下的收敛性。我们进一步研究Volterra积分微分方程的带光滑核以及带弱奇异核的Legendre谱配置方法、Chebyshev谱配置方法和Jacobi谱配置方法，探讨了如何处理初值条件，将初值条件转化为等价的积分方程。我们不仅证明谱配置方法的数值解本身的收敛性，还要设法得到数值解的导数的误差估计结果。针对各类Volterra型方程，我们都从Volterra型方程的积分项具有光滑核函数和带弱奇异核函数的情形、具有光滑解和具有非光滑解的情形等多方面进行了细致地研究。我们结合有限元方法的思想和我们多年来在有限元方面的研究成果，研究了Volterra型方程的谱Galerkin方法和谱元方法，利用Sobolev空间的算子理论研究数值解的稳定性、收敛性以及超收敛性和后验误差估计等现代计算数学理论热点问题。在数值试验方面，我们构造了并研究了非线性问题谱配置方法的的数值实例，进行了大量的数值试验证实了我们的理论分析结果。