Algorithms without accuracy saturation and exponentially convergent algorithms for evolution equations in Hilbert and Banach spaces

We consider the Cauchy problem for the first and the second order

differential equations in Banach and Hilbert spaces with an

operator coefficient A(t) depending on the parameter t. We

develop discretization methods with high parallelism level and

without accuracy saturation, i.e. the accuracy depends

automatically on the smoothness of the solution. For analytical

solutions the rate of convergence is exponential. These results

can be viewed as a development of parallel approximations of the

operator exponential and of the operator cosine family

with a constant operator A from

earlier papers possessing an exponential

accuracy and based on the Sinc-quadrature approximations

of the corresponding Dunford-Cauchy integral

representations of solutions or the solution operators.