Uniform \(l^1\) behavior of the first-order interpolant quadrature scheme for some partial integro-differential equations
type: Item
instance of: scholarly article
title: Uniform \(l^1\) behavior of the first-order interpolant quadrature scheme for some partial integro-differential equations
zbMATH Open document ID: 1472.65104
author: Da Xu
published in: Applied Mathematics Letters
publication date: 2021-04-28T00:00:00Z
review text: In this paper, the time discrete scheme based on the first-order backward difference method for the following integro-differential problem is studied \[ u_t(t)+\int\limits_0^t\beta(t-s)Au(s)ds=0, \quad t>0, \quad u(0)=u_0. \] Here \(A\) is a positive self-adjoint linear operator defined on a dense subspace \(D(A)\) of the real Hilbert space \(H\), \(u_0\in H\), and \(\beta(t)\) is weakly singular kernel at \(t=0\) such that \(\beta(t)=t^{\alpha-1}/\Gamma(\alpha)\), \(0<\alpha<1\). The paper is organized as follows. Section 1 is an introduction. In this section abovementioned problem is stated and the main theorem of this work is formulated. A review of the suitable papers is also given in this section. The memory term is approximated by the interpolating quadrature in this paper. In Section 2, using the Laplace transform technique is shown that this interpolating quadrature scheme has a convergence rate of \(O(\Delta t)\), where \(\Delta t\) denotes the time step. Special attention is that the uniform \(\ell^1\) convergence property of the discretization in time is given.
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 65D30
Mathematics Subject Classification ID: 44A10
zbMATH DE Number: 7340773
zbMATH Keywords: evolution equation
zbMATH Keywords: weakly singular kernel
zbMATH Keywords: non-smooth data
zbMATH Keywords: interpolating quadrature
zbMATH Keywords: error estimates
Wikidata QID: Q115360725
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
full work available at URL: j.aml.2021.107097
OpenAlex ID: W3131890698
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DOI: 10.1016/J.AML.2021.107097
description: scientific article; zbMATH DE number 7340773