Exponentially Convergent Method For An Abstract Nonlocal Problem With Integral Nonlinearity
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We consider a nonlocal problem for the first-order differential equation with unbounded operator coefficient in a Banach space and a nonlinear integral nonlocal condition. We propose an exponentially convergent method for the numerical solution of this problem and justified this method under the assumptions that the indicated operator coefficient A is sectorial and that certain conditions for the existence and uniqueness of the solution are satisfied. This method is based on the reduction of the posed problem to an abstract Hammerstein-type equation, discretization of this equation by the method of collocation, and its subsequent solution by the method of simple iterations. Each iteration of the method is based on the Sinc-quadrature approximation 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.
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