Exponentially converging method for a differential equation of the first order in the Banach space with an unlimited operator in a non-local condition

Abstract

Problem for the first order differential equation with an unbounded op-
erator coefficient in Banach space and two-pointed nonlocal condition is
considered. It is assumed that the nonlocal condition possesses an un-
bounded operator coefficient. An exponentially convergent algorithm is
proposed and justified for the numerical solution of this problem under
assumption that the operator coefficient A is sectorial and some existence
and uniqueness conditions are fulfilled. The proposed algorithm is based
on the application of Sinc-quadrature formulae to the Dunford-Cauchy in-
tegral representation of the solution operator and, as a result, requires only
a small number of resolvent evaluations.The efficiency of the proposed al-
gorithm is demonstrated by several numerical examples.