Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-10T13:31:47.933Z Has data issue: false hasContentIssue false
  • English
  • Français

REGULARITY OF SOLUTIONS TO A TIME-FRACTIONAL DIFFUSION EQUATION

Part of: Partial differential equations Representations of solutions

Published online by Cambridge University Press:  18 July 2011

WILLIAM MCLEAN
WILLIAM MCLEAN*
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia (email: w.mclean@unsw.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We prove estimates for the partial derivatives of the solution to a time-fractional diffusion equation posed over a bounded spatial domain. Such estimates are needed for the analysis of effective numerical methods, particularly since the solution is typically less regular than in the familiar case of classical diffusion.

Keywords

fractional derivativeLaplace transformSobolev spacesingular behaviour

MSC classification

Secondary: 35B65: Smoothness and regularity of solutions 35C15: Integral representations of solutions 35C20: Asymptotic expansions

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 2 , October 2010 , pp. 123 - 138
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Cuesta, E., Lubich, C. and Palencia, C., “Convolution quadrature time discretization of fractional diffusion-wave equations”, Math. Comp. 75 (2006) 673–696, doi:10.1090/S0025-5718-06-01788-1.CrossRefGoogle Scholar
[2]Erdelyi, A., Higher transcendental functions, Volume 3 (McGraw-Hill, New York, 1955).Google Scholar
[3]Golding, I. and Cox, E. C., “Physical nature of bacterial cytoplasm”, Phys. Rev. Lett. 96 (2006) 098102, doi:10.1103/PhysRevLett.96.098102.CrossRefGoogle ScholarPubMed
[4]Gorenflo, R., Loutchko, J. and Luchko, Y., “Computation of the Mittag-Leffler function E α,β(z) and its derivative”, Fract. Calc. Appl. Anal. 5 (2002) 491–518, and correction 6 (2003) 111–112.Google Scholar
[5]Grebenkov, D. S., “Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary”, Phys. Rev. E 81 (2010) 021128, doi:10.1103/PhysRevE.81.021128.CrossRefGoogle Scholar
[6]Henry, B. I. and Wearne, S. L., “Fractional reaction-diffusion”, Phys. A 276 (2000) 448–455.Google Scholar
[7]Mainardi, F., Mura, A. and Pagnini, G., “The M-Wright function in time-fractional diffusion processes: a tutorial survey”, Int. J. Differ. Equ. (2010) Article ID 104505, 29, doi:10.1155/2010/104505.Google Scholar
[8]McLean, W. and Mustapha, K., “A second-order accurate numerical method for a fractional wave equation”, Numer. Math. 105 (2007) 481–510, doi:10.1007/s00211-006-0045-y.CrossRefGoogle Scholar
[9]McLean, W. and Mustapha, K., “Convergence analysis of a discontinuous Galerkin method for a fractional diffusion equation”, Numer. Algorithms 52 (2009) 69–88, doi:10.1007/s11075-008-9258-8.CrossRefGoogle Scholar
[10]McLean, W. and Thomée, V., “Numerical solution of an evolution equation with a positive-type memory term”, J. Aust. Math. Soc. Ser. B 35 (1993) 23–70, doi:10.1017/S0334270000007268.CrossRefGoogle Scholar
[11]McLean, W. and Thomée, V., “Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation”, IMA J. Numer. Anal. 30 (2010) 208–230, doi:10.1093/imanum/drp004.CrossRefGoogle Scholar
[12]Metzler, R. and Klafter, J., “The random walk’s guide to anomalous diffusion: a fractional dynamics approach”, Phys. Rep. 339 (2000) 1–77, doi:10.1016/S0370-1573(00)00070-3.CrossRefGoogle Scholar
[13]Metzler, R. and Klafter, J., “Boundary value problems for fractional diffusion equations”, Phys. A 278 (2000) 107–125, doi:10.1016/S0378-4371(99)00503-8.Google Scholar
[14]Mustapha, K. and McLean, W., “Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation”, Numer. Algorithms 56 (2011) 159–184, doi:10.1007/s11075-010-9379-8.CrossRefGoogle Scholar
[15]Thomée, V., Galerkin finite element methods for parabolic problems, Volume 1054 of Lecture Notes in Mathematics (Springer, Berlin, 1984).Google Scholar
[16]Triebel, H., Interpolation theory, function spaces, differential operators (Johann Ambrosius Barth, Heidelberg, 1995).Google Scholar