Representation and Approximation of the Solution of an Initial Value Problem for a First Order Differential Equation in Banach Spaces

Abstract

An initial value problem $x(0)=x_0$ for the first order differential equation $x(t)+Ax(t)=g(t)$ with an unbounded operator coefficient $A$ in a Banach space is considered. Using the Cayley transform we give explicit formulas for the solution of this problem in case the operator $A$ is strongly positive. On the basis of these formulas we propose numerical algorithms for the approximate solution of the initial value problem and give error estimates. The main property of these algorithms is the following: the accuracy of the approximate solutions depends automatically on the "smoothness" of the initial data (the initial vector ro and the right-hand side $g$).