Detailed error analysis for a fractional Adams method with graded meshes
type: Item
instance of: scholarly article
title: Detailed error analysis for a fractional Adams method with graded meshes
zbMATH Open document ID: 1398.65173
author: Yanzhi Liu
author: Y. Yan
author: J. A. Roberts
published in: Numerical Algorithms
publication date: 2018-08-10T00:00:00Z
review text: The paper is concerned with the analysis of the fractional Adams method for the nonlinear fractional differential equation \[ D^{\alpha} y \left( t \right) = f \left( t, y \left( t \right) \right), \quad t \in \left(0,T\right), \quad \alpha>0, \] equipped with initial conditions. The fractional derivative \(D^{\alpha} y\) is understood in the Caputo sense. The authors focus on the case that the function \(D^{\alpha} y\) does not belong to the space \(C^2\left[0,T\right]\), because it is known that in this case the optimal order of convergence for the fractional Adams method can not be guaranteed on the uniform mesh. Therefore, they study the fractional Adams method with graded meshes and they prove that the optimal order of convergence is achieved under the assumption that \(D^{\alpha} y\) is twice continuously differentiable on \(\left( 0,T\right]\) and \(D^{\alpha} y\) behaves as \(t^{\sigma}\), \(0< \sigma <1\). The authors present several numerical experiments to show that the experimental orders of convergence under different assumptions on smoothness of \(D^{\alpha} y\) correspond to theoretical results.
reviewed by: Dana Černá
Mathematics Subject Classification ID: 65L06
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65L20
zbMATH DE Number: 6916369
zbMATH Keywords: fractional Adams method
zbMATH Keywords: fractional differential equation
zbMATH Keywords: initial value problem
zbMATH Keywords: Caputo derivative
Wikidata QID: Q59613416
describes a project that uses: FODE
describes a project that uses: Adaptative L1 TE and Predictive
MaRDI profile type: Publication
full work available at URL: s11075-017-0419-5
OpenAlex ID: W2746911607
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DOI: 10.1007/S11075-017-0419-5
description: scientific article; zbMATH DE number 6916369