We consider the Cauchy problem for a first-order evolution equation with memory in a finite-dimensional Hilbert space when the integral term is related to the time derivative of the solution. The main problems of the approximate solution of such nonlocal problems are due to the necessity to work with the approximate solution for all previous time moments. We propose a transformation of the first-order integrodifferential equation to a system of local evolutionary equations. We use the approach known in the theory of Voltaire integral equations with an approximation of the difference kernel by the sum of exponents. We formulate a local problem for a weakly coupled system of equations with additional ordinary differential equations. We have given estimates of the stability of the solution by initial data and the right-hand side for the solution of the corresponding Cauchy problem. The primary attention is paid to constructing and investigating the stability of two-level difference schemes, which are convenient for computational implementation. The numerical solution of a two-dimensional model problem for the evolution equation of the first order, when the Laplace operator conditions the dependence on spatial variables, is presented.
翻译:当一个整体术语与解决办法的时间衍生物相关时,我们考虑在有限维度的Hilbert空间内以内存为一阶进化方程式的毛细问题。这些非本地问题近似的解决办法的主要问题是,必须先与以前所有时间的近似解决办法合作。我们建议将一阶内分化方程式转换为本地进化方程式系统。我们使用伏尔泰尔整体方程式理论中已知的方程式方法,将差异内核近似于Expents之和。我们形成了一个本地问题,用一种较弱结合的方程式系统,加上额外的普通差异方程式。我们用初始数据和相应的Cauchy问题右侧对解决办法的稳定性作了估计。我们主要注意的是构建和调查两级差异方程式的稳定性,这对计算实施是方便的。当Laplace操作者对空间变量的依赖时,提出了第一级进方程式进化方程式的二维模型问题的数字解决办法。