Representation and approximation of the solution of an initial value problem for a first order differential equation in Banach spaces
An initial value problem for a first-order differential equation \(\dot x+ Ax= g\), \(x(0)= x_0\) with an unbounded operator \(A\) in a Banach space is considered. If \(A\) is a strongly positive explicit operator then formulas for the solution of the initial value problem are given. The Cayley transformation of the operator \(A\) is used to derive such formulas. On the basis of these formulas numerical algorithms are proposed to construct approximate solutions and to give error estimates. Important tools are the Laguerre polynomials here. The accuracy of the approximate solutions depends automatically on the ``smoothness'' of the initial vector \(x_0\) and the right-hand side \(g\).
scientific article; zbMATH DE number 889009
scientific article