Convergence Outside the Initial Layer for a Numerical Method for the Time-Fractional Heat Equation

  • José Luis Gracia
  • Eugene O’Riordan
  • Martin StynesEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)


In this paper a fractional heat equation is considered; it has a Caputo time-fractional derivative of order \(\delta \) where \(0<\delta <1\). It is solved numerically on a uniform mesh using the classical L1 and standard three-point finite difference approximations for the time and spatial derivatives, respectively. In general the true solution exhibits a layer at the initial time \(t=0\); this reduces the global order of convergence of the finite difference method to \(O(h^2+\tau ^\delta )\), where h and \(\tau \) are the mesh widths in space and time, respectively. A new estimate for the L1 approximation shows that its truncation error is smaller away from \(t=0\). This motivates us to investigate if the finite difference method is more accurate away from \(t=0\). Numerical experiments with various non-smooth and incompatible initial conditions show that, away from \(t=0\), one obtains \(O(h^2+\tau )\) convergence.


Time-fractional heat equation Caputo fractional derivative Initial-boundary value problem L1 scheme Layer region Smooth and non-smooth data Compatibility conditions 



The research of José Luis Gracia was partly supported by the Institute of Mathematics and Applications (IUMA), the project MTM2016-75139-R and the Diputación General de Aragón. The research of Martin Stynes was supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • José Luis Gracia
    • 1
  • Eugene O’Riordan
    • 2
  • Martin Stynes
    • 3
    Email author
  1. 1.Department of Applied MathematicsUniversity of ZaragozaZaragozaSpain
  2. 2.School of Mathematical SciencesDublin City UniversityDublinIreland
  3. 3.Applied and Computational Mathematics DivisionBeijing Computational Science Research CenterBeijingChina

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