Abstract:
In a Hilbert space, we consider the Cauchy problem for a first-order linear differential equation. The coefficient at the unknown function in the equation is an unbounded normal operator. The exact solution of such a problem is expressed in terms of an operator exponential. We suggest a spectral method for constructing an approximate solution based on the calculation of some rational function of the normal operator. Namely, we take a scalar rational function approximating the exponential function on the spectrum of the operator, then we expand the obtained rational function into the sum of elementary fractions and substitute the operator into it. As a result we obtain a linear combination of values of the resolvent of the normal operator at various points of its resolvent set. Theorems on the estimation of the absolute and relative accuracy of the approximation are proved. A variant of the proposed approach for a non-homogeneous equation with a special free term is also discussed. The results of numerical experiments are presented.
Keywords:spectral method, normal operator, operator exponential, rational function of operator, pade approximation, error estimates.
