Fractional convolution quadrature based on generalized Adams methods
type: Item
instance of: scholarly article
title: Fractional convolution quadrature based on generalized Adams methods
zbMATH Open document ID: 1314.65090
author: Lidia Aceto
author: Cecilia Magherini
author: P. Novati
published in: Calcolo
publication date: 2015-03-23T00:00:00Z
review text: The authors obtain numerical solutions of fractional differential equations of the form \[ D^{\alpha}_{t_{0}}y(t)=f(t,y(t)),\, t_{0}<t\leq T,\, 0<\alpha<1,\tag{1} \] \(y(t_{0})=y_{0}\), where \( D^{\alpha}_{t_{0}}y(t)\) denotes the Caputo fractional derivative defined by \[ D^{\alpha}_{t_{0}}y(t)=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{t_{0}}\frac{y'(u)}{(t-u)^{\alpha}}du. \tag{2} \] Since the solution of (1) can be written as \[ y(t)=y_{0}+\frac{1}{\Gamma(\alpha)}\int^{t}_{t_{0}}(t-u)^{\alpha-1}f(u,y(u))du,\tag{3} \] each quadrature scheme for (3) leads to a numerical solver for (2). The generalized Adam's method for ordinary differential equations is extended to define product quadrature rules for the solution of (3). Numerical examples are given to illustrate the results obtained.
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 65L20
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A08
zbMATH DE Number: 6418679
zbMATH Keywords: fractional differential equations
zbMATH Keywords: product quadrature rule
zbMATH Keywords: generalized Adams methods
zbMATH Keywords: Caputo fractional derivative
zbMATH Keywords: numerical example
MaRDI profile type: Publication
full work available at URL: s10092-013-0094-4
OpenAlex ID: W1991339940
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DOI: 10.1007/S10092-013-0094-4
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description: scientific article; zbMATH DE number 6418679