论文标题
具有某些非平滑核的Volterra Integrodiverention方程的时间离散化的全球行为
Global behavior of temporal discretizations for Volterra integrodifferential equations with certain nonsmooth kernels
论文作者
论文摘要
在这项工作中，提出了Z变换以分析希尔伯特领域中非滑动多个多项内核的Volterra Integrodifferentifentix方程（VIDE）的时间二差解决方案，首先考虑了这类连续问题，并由Hannsgen和Wheeler进行了分析（Siam J Math Analy 15（1984）579-579-59-59-594）。这项工作讨论了三个内核$β_Q（t）$的三个案例，用于多学期vides的积分中，我们从中使用相应的数值技术来近似不同情况下的多项VIDE的解决方案。首先，对于$β_1（T），β_2（t）\ in \ Mathrm {l} _1（\ Mathbb {r} _+）$，曲柄 - 尼科尔森（CN）方法和插值Quadration Quadration（IQ）规则用于多段时间的时间 - 二级典范解决方案的多 - 列表vides的时间限制;其次，对于$β_1（t）\ in \ Mathrm {l} _1（\ Mathbb {r} _+）$和$β_2（t）\ in \ in \ Mathrm {l} _ {l} _ {1，{1，\ text}}（loc}}（lot {loc}}（\ mathbb {二阶卷积正交（CQ）用于在时间方向上离散多个期限问题；第三，对于$β_1（t），β_2（t）\ in \ mathrm {l} _ {1，\ text {loc}}（\ m mathbb {r} _+）$，我们利用cn方法和斑点CQ（tcq）统治多个临时时间。然后，对于三种情况的离散解决方案，基于Z变换和某些适当的假设证明了长期的全局稳定性和收敛性。此外，数值测试证实了第三种情况的长期估计值。
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 $β_q(t)$ 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 $β_1(t), β_2(t) \in \mathrm{L}_1(\mathbb{R}_+)$, 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 $β_1(t)\in \mathrm{L}_1(\mathbb{R}_+)$ and $β_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$, 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 $β_1(t), β_2(t)\in \mathrm{L}_{1,\text{loc}}(\mathbb{R}_+)$, 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.
