Super-Exponential Rate of Convergence of the Cayley Transform Method for an Abstract Differential Equation

A boundary-value problem (BVP) for a second-order abstract differential equation with an operator coefficient in a Hilbert space is investigated. The exact solution is presented as an infinite series by means of the Cayley transform of the operator coefficient A and the polynomials of Meixner type in the independent variable x. The approximate solution is given by the truncated sum of the series with N addends. The error estimates (with the weighted function) depending not only on the discretization parameter N but also on the distance of the argument x to the boundary points of the segment are proved. The algorithm has a super-exponential rate of convergence.