Numerical solution of the Cauchy problem for integro-differential equations with a difference kernel

The problems of numerical solution of the Cauchy problem for the evolutionary equation with memory, when the kernel of the integral term is a difference one, are considered. The computational implementation is associated with the need to work with an approximate solution for all previous moments of time. The non-local problem under consideration is transformed into a local one, and a weakly coupled system of equations with additional ordinary differential equations is solved. This approach is based on the approximation of the difference kernel by the sum of exponentials. Estimates of the stability of the solution with respect to the initial data and the right-hand side for the corresponding Cauchy problem are obtained. Two-layer schemes with weights with convenient computational implementation are constructed and investigated.