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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)

Abstract

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.

Keywords

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

Notes

Acknowledgments

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.

References

  1. 1.
    Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Applied Mathematics. Chapman  & Hall/CRC, Boca Raton (2000)zbMATHGoogle Scholar
  3. 3.
    Gracia, J.L., O’Riordan, E., Stynes, M.: Convergence in positive time for a finite difference method applied to a fractional convection-diffusion equation (submitted for publication)Google Scholar
  4. 4.
    Jin, B., Lazarov, R., Zhou, Z.: An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36, 197–221 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Luchko, Y.: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15(1), 141–160 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order. With an Annotated Chronological Bibliography by Bertram Ross. Mathematics in Science and Engineering, vol. 111. Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers), New York-London (1974)Google Scholar
  7. 7.
    Stynes, M.: Too much regularity may force too much uniqueness. Fract. Calc. Appl. Anal. 19(6), 1554–1562 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM. J. Numer. Anal. (to appear)Google Scholar

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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