In this work, the z-transform is presented to analyze time-discrete solutions for Volterra integrodifferential equations (VIDEs) with nonsmooth multi-term kernels in the Hilbert space, and this class of continuous problem was first considered and analyzed by Hannsgen and Wheeler (SIAM J Math Anal 15 (1984) 579-594). This work discusses three cases of kernels $\beta_q(t)$ included in the integrals for the multi-term VIDEs, from which we use corresponding numerical techniques to approximate the solution of multi-term VIDEs in different cases. Firstly, for the case of $\beta_1(t), \beta_2(t) \in \mathrm{L}_1(\mathbb{R}_+)$, the Crank-Nicolson (CN) method and interpolation quadrature (IQ) rule are applied to time-discrete solutions of the multi-term VIDEs; secondly, for the case of $\beta_1(t)\in \mathrm{L}_1(\mathbb{R}_+)$ and $\beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$, second-order backward differentiation formula (BDF2) and second-order convolution quadrature (CQ) are employed to discretize the multi-term problem in the time direction; thirdly, for the case of $\beta_1(t), \beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$, we utilize the CN method and trapezoidal CQ (TCQ) rule to approximate temporally the multi-term problem. Then for the discrete solution of three cases, the long-time global stability and convergence are proved based on the z-transform and certain appropriate assumptions. Furthermore, the long-time estimate of the third case is confirmed by the numerical tests.
翻译:在本文中,引入z变换分析在希尔伯特空间中具有非光滑多项式核的沃尔特拉奇异积分微分方程(VIDE)的时离散解,这一连续的问题是由Hannsgen和Wheeler(SIAM J Math Anal 15 (1984) 579-594)首次考虑和分析的。本文讨论了三种核 $\beta_q(t)$ 的情况,这些核包含了多项式VIDEs的积分项,我们使用相应的数值技术,在不同的情况下逼近多项式VIDEs的解。首先,对于 $\beta_1(t),\beta_2(t) \in \mathrm{L}_1(\mathbb{R}_+)$ 的情况,将Crank-Nicolson(CN)方法和插值积分(IQ)规则应用于多项式VIDEs的时离散解;其次,对于 $\beta_1(t)\in \mathrm{L}_1(\mathbb{R}_+)$ 和 $\beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$ 的情况,采用二阶后向差分公式(BDF2)和二阶卷积积分(CQ)来离散化时向的多项式问题;第三,对于$\beta_1(t),\beta_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$的情况,我们利用Crank-Nicolson方法和梯形卷积积分(TCQ)规则来逼近多项式问题的时间。然后,根据z变化和适当的假设,证明三种情况的离散解的长时间全局稳定性和收敛性。此外,通过数值测试验证了第三种情况的长时间估计。