We consider the problems of the numerical solution of the Cauchy problem for an evolutionary equation with memory when the kernel of the integral term is a difference one. The computational implementation is associated with the need to work with an approximate solution for all previous points in time. In this paper, the considered nonlocal problem is transformed into a local one; a loosely coupled equation system with additional ordinary differential equations is solved. This approach is based on the approximation of the difference kernel by the sum of exponentials. Estimates for the stability of the solution concerning the initial data and the right-hand side for the corresponding Cauchy problem are obtained. Two-level schemes with weights with convenient computational implementation are constructed and investigated. The theoretical consideration is supplemented by the results of the numerical solution of the integrodifferential equation when the kernel is the stretching exponential function.

翻译：在整体术语的内核存在差异时,我们考虑如何用数字方法解决突尼西岛问题的演变式方程式,在整体术语的内核存在差异时,用记忆来记忆进化式方程式。计算性实施与之前所有时间点的近似解决办法有关。在本文件中,考虑的非局部问题被转化成局部问题;一个松散、互不相干的方程式系统,加上另外的普通差异方程式得到解决。这一方法的基础是以指数总和接近差异内核;获得关于初始数据解决方案的稳定性和相应的Cauchi问题右侧的估计数。构建和调查了具有方便计算实施权重的两级方案。理论考虑还辅之以当内核为伸展指数函数时,对异族方方方方方方方程式的数字解决方案的结果进行补充。