Abstract
Time-dependent fractional-derivative problems \(D_t^\alpha u + Au = f\) are considered, where \(D_t^\alpha\) is a Caputo fractional derivative of order α ∈ (0, 1)∪(1, 2) and A is a classical elliptic operator, and appropriate boundary and initial conditions are applied. The regularity of solutions to this class of problems is discussed, and it is shown that assuming more regularity than is generally true—as many researchers do—places a surprisingly severe restriction on the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Al-Refai, Basic results on nonlinear eigenvalue problems of fractional order. Electron. J. Differ. Equ. 2012 (2012), Article # 191, 1–12.
V.V. Anh, N.N. Leonenko, and M.D. Ruiz-Medina, Fractional-in-time and multifractional-in-space stochastic partial differential equations. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1434–1459; 10.1515/fca-2016–0074 https://www.degruyter.com/view/j/fca.2016.19.issue-6/issue-files/fca.2016.19.issue-6.xml
E. Cuesta, C. Lubich, and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comp. 75 (2006), Article #254, 673–696 (electronic).
K. Diethelm, The Analysis of Fractional Differential Equations. Ser. Lecture Notes in Mathematics, Vol. 2004, Springer-Verlag, Berlin, 2010.
B. Jin, R. Lazarov, J. Pasciak, and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion. IMA J. Numer. Anal. 35, No 2 (2015), 561–582.
B. Jin, R. Lazarov, and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, No 1 (2016), A146–A170.
J. Korbel and Y. Luchko, Modeling of financial processes with a space-time fractional diffusion equation of varying order. Fract. Calc. Appl. Anal. 19, No 6 (2016), 1414–1433; DOI: 10.1515/fca-2016–0073; https://www.degruyter.com/viewZj/fca.2016.19.issue-6/issue-files/fca.2016.19.issue-6.xml
X. Li and C. Xu, Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8, No 5 (2010), 1016–1051.
Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, No 2 (2007), 1533–1552.
Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation. Fract. Calc. Appl. Anal. 15, No 1 (2012), 141–160.
W. McLean, Regularity of solutions to a time-fractional diffusion equation. ANZIAM J. 52, No 2 (2010), 123–138.
R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, No 1 (2000), 1–77.
M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations Springer-Verlag, New York, 1984; Corrected reprint of the 1967 original.
K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, No 1 (2011), 426–447.
M. Stynes, E. O’Riordan, and J.L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. Submitted (2016) for publication to: SIAM J. Numer. Anal.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Stynes, M. Too Much Regularity May Force Too Much Uniqueness. FCAA 19, 1554–1562 (2016). https://doi.org/10.1515/fca-2016-0080
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2016-0080