Exponentially convergent method for an abstract nonlocal problem with integral nonlinearity
We consider a problem for the first-order differential equation with unbounded operator coefficient in Banach space and a nonlinear integral nonlocal condition. An exponentially convergent method for the numerical solution of this problem is proposed and justified under assumption that the indicated operator coefficient A is strongly positive and certain existence and uniqueness conditions are satisfied. This method is based on the reduction of the posed problem to an abstract Hammerstein equation, discretization of this equation by the collocation method, and its subsequent solution by the fixed-point iteration method. Each iteration of the method involves the Sinc-based numerical evaluation of the exponential operator function represented by the Dunford – Cauchy integral over the hyperbola enveloping the spectrum of A. The integral part of the nonlocal condition is approximated by using the Clenshaw – Curtis quadrature formula.
Citation Example: Makarov V. L., Vasylyk V. B. Exponentially convergent method for an abstract nonlocal problem with integral nonlinearity // Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1587-1597.