Efficient high order algorithms for fractional integrals and fractional differential equations

Lehel Banjai, M. Lopez-Fernandez

Research output: Contribution to journalArticle

Abstract

We propose an efficient algorithm for the approximation of fractional integrals by using Runge–Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The resulting method is easy to implement, allows for high order, relies on rigorous error estimates and its performance in terms of memory and computational cost is among the best to date. Several numerical results illustrate the method and we describe how to apply the new algorithm to solve fractional diffusion equations. For a class of fractional diffusion equations we give the error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge–Kutta based convolution quadrature in time.

Original languageEnglish
Pages (from-to)1-29
Number of pages29
JournalNumerische Mathematik
Early online date24 Oct 2018
DOIs
Publication statusE-pub ahead of print - 24 Oct 2018

Fingerprint

Integral-differential Equation
Fractional Integral
Fractional Differential Equation
Convolution
Quadrature
Fractional Diffusion Equation
Differential equations
Runge-Kutta
Higher Order
Error Analysis
Integral Representation
Error analysis
Computational Cost
Error Estimates
Efficient Algorithms
Discretization
Space-time
Finite element method
Data storage equipment
Numerical Results

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

@article{97d6985209874321b3eef99526959232,
title = "Efficient high order algorithms for fractional integrals and fractional differential equations",
abstract = "We propose an efficient algorithm for the approximation of fractional integrals by using Runge–Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The resulting method is easy to implement, allows for high order, relies on rigorous error estimates and its performance in terms of memory and computational cost is among the best to date. Several numerical results illustrate the method and we describe how to apply the new algorithm to solve fractional diffusion equations. For a class of fractional diffusion equations we give the error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge–Kutta based convolution quadrature in time.",
author = "Lehel Banjai and M. Lopez-Fernandez",
year = "2018",
month = "10",
day = "24",
doi = "10.1007/s00211-018-1004-0",
language = "English",
pages = "1--29",
journal = "Numerische Mathematik",
issn = "0029-599X",
publisher = "Springer",

}

Efficient high order algorithms for fractional integrals and fractional differential equations. / Banjai, Lehel; Lopez-Fernandez, M.

In: Numerische Mathematik, 24.10.2018, p. 1-29.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Efficient high order algorithms for fractional integrals and fractional differential equations

AU - Banjai, Lehel

AU - Lopez-Fernandez, M.

PY - 2018/10/24

Y1 - 2018/10/24

N2 - We propose an efficient algorithm for the approximation of fractional integrals by using Runge–Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The resulting method is easy to implement, allows for high order, relies on rigorous error estimates and its performance in terms of memory and computational cost is among the best to date. Several numerical results illustrate the method and we describe how to apply the new algorithm to solve fractional diffusion equations. For a class of fractional diffusion equations we give the error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge–Kutta based convolution quadrature in time.

AB - We propose an efficient algorithm for the approximation of fractional integrals by using Runge–Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special quadrature for it. The resulting method is easy to implement, allows for high order, relies on rigorous error estimates and its performance in terms of memory and computational cost is among the best to date. Several numerical results illustrate the method and we describe how to apply the new algorithm to solve fractional diffusion equations. For a class of fractional diffusion equations we give the error analysis of the full space-time discretization obtained by coupling the FEM method in space with Runge–Kutta based convolution quadrature in time.

UR - http://www.scopus.com/inward/record.url?scp=85055725280&partnerID=8YFLogxK

U2 - 10.1007/s00211-018-1004-0

DO - 10.1007/s00211-018-1004-0

M3 - Article

SP - 1

EP - 29

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

ER -