A Method for Solving Initial Value Problems for Linear Differential Equations in Hilbert Space Based on The Cayley Transform

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Abstract
A first-order differential equation with a bounded operator coefficient in Hilbert space is considered. In case the coefficient is constant we establish a one-to-one correspondence between the continuous initial value problem and some discrete initial value problem. Using the Cayley transform we give explicit formulas for their solutions as well as formulas connecting corresponding continuous and discrete semigroups. On the basis of these formulas we propose a numerical algorithm for solving initial value problems with a bounded constant operator coefficient which has an exponential rate of convergence. If the operator coefficient is variable we use its piecewise constant approximation and the previously cited algorithm. Error estimates are given.

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