Exponential quadrature rules for linear fractional differential equations
type: Item
instance of: scholarly article
title: Exponential quadrature rules for linear fractional differential equations
zbMATH Open document ID: 1314.65093
author: Roberto Garrappa
author: Marina Popolizio
published in: Mediterranean Journal of Mathematics
publication date: 2015-03-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1311.1388
review text: The authors investigate numerical solutions of linear fractional differential equations of the following form \[ D^{\alpha}_{t_{0}}y(t)+\lambda y(t)=f(t), \] where \( \alpha >0\) is a real number, \( \lambda \in \mathbb R \), \( y:[t_{0},T]\rightarrow \mathbb R \) and the forcing term \( f:[t_{0},T]\rightarrow \mathbb R \) is sufficiently smooth. The symbol \( D^{\alpha}_{t_{0}} \) denotes the derivative operator of non-integer order \( \alpha \), with respect to the origin \( t_{0}\), according to the Caputo definition \[ D^{\alpha}_{t_{0}}y(t)= \frac{1}{\Gamma(m-\alpha)}\int^{t}_{t_{0}}\frac{y^{(m)}(u)}{(t-u)^{\alpha+1-m}}du,\tag{\(*\)} \] with the gamma function \( \Gamma(.) \), \( m=\lceil\alpha\rceil \) the smallest integer such that \( \alpha < m \), \( y^{(m)} \) is the derivative of \( y \) of integer order \( m \) and the initial conditions \( y^{(k)}(t_{0})=y_{0,k}\), \(k=0,1,\dots,m-1 \). In order to obtain high accuracy under suitable and reasonable assumptions of smoothness on \( f(t)\), Eq.(*) is reformulated in terms of some special weighted integrals involving \( f(t) \) alone as the integrand and then specific quadrature rules are formulated for their numerical approximation. Numerical examples are given to illustrate the results obtained. The effectiveness of the present approach is demonstrated by means of comparisons with other standard methods.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34A08
zbMATH DE Number: 6419349
zbMATH Keywords: fractional differential equation
zbMATH Keywords: linear problem
zbMATH Keywords: quadrature rules
zbMATH Keywords: Mittag-Leffler function
zbMATH Keywords: accuracy
zbMATH Keywords: numerical example
describes a project that uses: FracPECE
describes a project that uses: Mittag-Leffler
describes a project that uses: mlf
MaRDI profile type: Publication
OpenAlex ID: W3102950679
arXiv ID: 1311.1388
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DOI: 10.1007/S00009-014-0396-Z
description: scientific article