Exponentially convergent method for an abstract nonlocal problem with integral nonlinearity - Vasylyk, V. B.; Makarov, V. L. (Ukrains’kyi Matematychnyi Zhurnal, 2016)
We consider a problem for the first-order differential equation with unbounded operator coefficient in Banach space anda nonlinear integral nonlocal condition. An exponentially convergent method for the numerical solution of this problemis proposed and justified under assumption that the indicated operator coefficient A is strongly positive and certainexistence and uniqueness conditions are satisfied. This method is based on the reduction of the posed problem to anabstract Hammerstein equation, discretization of this equation by the collocation method, and its subsequent solution by thefixed-point iteration method. Each iteration of the method involves the Sinc-based numerical evaluation of the exponentialoperator function represented by the Dunford – Cauchy integral over the hyperbola enveloping the spectrum of A. Theintegral part of the nonlocal condition is approximated by using the Clenshaw – Curtis quadrature formula.