论文标题
准线性抽象双曲方程的平行类型分解方案
Parallel Type Decomposition Scheme for Quasi-Linear Abstract Hyperbolic Equation
论文作者
论文摘要
希尔伯特空间中考虑了带有Lipschitz连续操作员的抽象双曲方程的Cauchy问题。对应于方程的椭圆部分的操作员是运算符的总和$ a_ {1}，\，a_ {2}，\，\，\ ldots，\，a_ a_ {m {m {m} $。每个加成都是一个自动伴侣和积极的确定运算符。构建了一种平行类型的分解方案，用于构建所述问题的近似解决方案。该方案的主要思想是，在每个本地间隔上，经典差异问题分别与操作员$ a_ {1}，\，a_ {2}，\，\，\ ldots，\，\，a_ {m} $并行解决。在本地间隔的右端宣布了接收解决方案的加权平均值作为近似解决方案。证明了所提出的方案的收敛性，并估计了近似解决方案误差，以及当初始问题数据满足解决方案存在的自然条件时，第一阶导数的差异类似物的误差。
Cauchy problem for an abstract hyperbolic equation with the Lipschitz continuous operator is considered in the Hilbert space. The operator corresponding to the elliptic part of the equation is a sum of operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. Each addend is a self-adjoint and positive definite operator. A parallel type decomposition scheme for an approximate solution of the stated problem is constructed. The main idea of the scheme is that on each local interval classic difference problems are solved in parallel (independently from each other) respectively with the operators $A_{1},\,A_{2},\,\ldots,\,A_{m}$. The weighted average of the received solutions is announced as an approximate solution at the right end of the local interval. Convergence of the proposed scheme is proved and the approximate solution error is estimated, as well as the error of the difference analogue for the first-order derivative for the case when the initial problem data satisfy the natural sufficient conditions for solution existence.
