论文标题
关于球的稳定性和三层半混凝土方案的稳定性和收敛性，用于球形差异方程的抽象类似物
On Stability and Convergence of a Three-layer Semi-discrete Scheme for an Abstract Analogue of the Ball Integro-differential Equation
论文作者
论文摘要
我们考虑了希尔伯特空间中二阶非线性进化方程的库奇问题。该方程代表球形差异方程的抽象概括。考虑到包括梯度规范的平方的方程式条款的一般非线性案例。为了找到近似解决方案，提出了三层半差异方案。在此方案中，使用积分平均值来执行取决于梯度的非线性项的近似值。我们表明，非线性离散问题的解及其相应的一阶导数差异类似物是统一界限的。对于相应的线性离散问题的解决方案，通过使用两种可变量的Chebyshev多项式来获得高阶的先验估计。根据这些估计，我们证明了非线性离散问题的稳定性。对于平滑解决方案，我们为近似解决方案提供了错误估计。应用一种迭代方法，以找到每个时间步骤的近似解决方案。证明了迭代过程的收敛性。
We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved.
