Representation and approximation of the solution of an initial value problem for a first order differential equation in Banach spaces
type: Item
instance of: scholarly article
title: Representation and approximation of the solution of an initial value problem for a first order differential equation in Banach spaces
zbMATH Open document ID: 0849.65040
DOI: 10.4171/ZAA/712
author: Iwan Gawriljuk
author: Vladimir L. Makarov
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1996-11-11T00:00:00Z
review text: 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 \(g\).
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 889009
zbMATH Keywords: initial value problem
zbMATH Keywords: first-order differential equation
zbMATH Keywords: Banach space
zbMATH Keywords: Cayley transformation
zbMATH Keywords: numerical algorithms
zbMATH Keywords: error estimates
zbMATH Keywords: Laguerre polynomials
reviewed by: Werner H. Schmidt
MaRDI profile type: Publication
Wikidata QID: Q115211685
cites work: A Method for Solving Initial Value Problems for Linear Differential Equations in Hilbert Space Based on The Cayley Transform
cites work: Q4843299
cites work: On well-posedness of the nonlocal boundary value problem for parabolic difference equations
cites work: Q3329838
cites work: Q5325667
cites work: The cayley transform and the solution of an initial value problem for a first order differential equation with an unbounded operator coefficient in hilbert space
cites work: Q3263045
cites work: An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow
cites work: Q3742104
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q3896420
cites work: Spectral Methods in Time for Hyperbolic Equations
cites work: Spectral Methods in Time for Parabolic Problems
description: scientific article; zbMATH DE number 889009