A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions
type: Item
instance of: scholarly article
title: A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions
zbMATH Open document ID: 1227.65075
author: Xuan Zhao
author: Zhizhong Sun
published in: Journal of Computational Physics
publication date: 2011-07-13T00:00:00Z
review text: Fractional differential equations and their novel numerical schemes are investigated. These equations play an important role in applications in physics, chemistry, engineering, finance and other sciences. The model presented here is of one space dimension with fractional sub-diffusion equation with the Riemann-Liouville operator and with Neumann boundary conditions. For deriving new numerical schemes, the authors choose the methodology of order reduction and transform the original problem into an equivalent system of lower order differential equations. For this system a construction of a so-called box-type numerical scheme is done using classical finite differences. Due to the fact that the fractional derivative operator is nonlocal, the authors call it box-type scheme. First, a new discrete inner product, \(L^2\) norm, \(H^1\) semi-norm and maximum norm are defined. Then the stability estimations for the numerical solution are proved, using the novel technique. Finally, under some smoothing conditions on the continuous solution, and for \( 0<\alpha<1\) , which represents the fractional derivative of the type \(1-\alpha\), the error estimate of the type \(C(\tau^{2-\alpha} +h^2) \) in the maximum norm is proved for \(\tau\) and \(h\) time and space grid mesh size, respectively. Two numerical test examples, which confirm the accuracy and efficiency of the proposed scheme, conclude the paper.
reviewed by: Angela Handlovičová
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 5920327
zbMATH Keywords: Sub-diffusion equation
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: box-type scheme
zbMATH Keywords: stability
zbMATH Keywords: convergence
zbMATH Keywords: energy method
zbMATH Keywords: fractional differential equations
zbMATH Keywords: Riemann-Liouville operator
zbMATH Keywords: order reduction
zbMATH Keywords: finite differences
zbMATH Keywords: error estimate
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: j.jcp.2011.04.013
OpenAlex ID: W2057206818
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DOI: 10.1016/J.JCP.2011.04.013
description: scientific article; zbMATH DE number 5920327