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A suitable abstract setting of the initial value problem for the first order differential equation with an unbounded operator coefficient in a Banach space where the domain of the operator depends on the dependent variable t is introduced. A new exponentially convergent algorithm for such problems is proposed. This algorithm is based on a generalization of… (More)

The m-point nonlocal problem for the first order differential equation with an operator coefficient in a Banach space X is considered. An exponentially convergent algorithm is proposed and justified provided that the operator coefficient A is strongly positive and some existence and uniqueness conditions are fulfilled. This algorithm is based on… (More)

We propose and justify an exponentially convergent algorithm for a second order differential equation with an operator coefficient in Banach space. A particular example of such equation is the strongly damped wave equation arising in viscoelasticity. Numerical examples are given which confirm theoretical results.

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