Abstract
The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55. Courier Corporation, Chelmsford (1965)
Antil, H., Otárola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control Optim. 53(6), 3432–3456 (2015)
Arendt, W., Elst, A.F.M.T., Warma, M.: Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator. Commun. Partial Differ. Equ. 43(1), 1–24 (2018). https://doi.org/10.1080/03605302.2017.1363229
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, Berlin (2010)
Bonito, A., Lei, W., Pasciak, J.E.: Numerical approximation of the integral fractional Laplacian. Numer. Math. 142(2), 235–278 (2019)
Budd, C.J., Iserles, A.: Geometric integration: numerical solution of differential equations on manifolds. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 357(1754), 945–956 (1999)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(8), 1245–1260 (2007). https://doi.org/10.1080/03605300600987306
Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75(2), 305–332 (2002)
Chen, L., Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion: a posteriori error analysis. J. Comput. Phys. 293, 339–358 (2015)
Cushman, J.H., Ginn, T.R.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Transp. Porous Media 13(1), 123–138 (1993)
Deimling, K.: Nonlinear Functional Analysis. Courier Corporation, Chelmsford (2010)
D‘Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66(7), 1245–1260 (2013)
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, Berlin (2002)
Galé, J.E., Miana, P.J., Stinga, P.R.: Extension problem and fractional operators: semigroups and wave equations. J. Evol. Equ. 13(2), 343–368 (2013)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer, Berlin (2006)
Hansen, E., Henningsson, E.: A convergence analysis of the Peaceman–Rachford scheme for semilinear evolution equations. SIAM J. Numer. Anal. 51(4), 1900–1910 (2013)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations, vol. 840. Springer, Berlin (2006)
Huang, Y., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52(6), 3056–3084 (2014)
Iserles, A.: A First Course in the Numerical Analysis of Differential Equations, 2nd edn. Cambridge University Press, New York (2008)
Jones, T., Gonzalez, L.P., Guha, S., Sheng, Q.: A continuing exploration of a decomposed compact method for highly oscillatory wave problems. J. Comput. Appl. Math. 299, 207–220 (2016). https://doi.org/10.1016/j.cam.2015.11.044. Recent Advances in Numerical Methods for Systems of Partial Differential Equations
Lumer, G., Phillips, R.S.: Dissipative operators in a Banach space. Pac. J. Math. 11(2), 679–698 (1961)
Martinez, C., Sanz, M.: The Theory of Fractional Powers of Operators, vol. 187. Elsevier, Amsterdam (2001)
Meichsner, J., Seifert, C.: Fractional powers of non-negative operators in Banach spaces via the Dirichlet-to-Neumann operator. arXiv preprint arXiv:1704.01876 (2017)
Munthe-Kaas, H.Z., Føllesdal, K.K.: Lie–Butcher series, geometry, algebra and computation. In: Discrete Mechanics, Geometric Integration and Lie–Butcher Series, pp. 71–113. Springer (2018)
Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733–791 (2015)
Nochetto, R.H., Otarola, E., Salgado, A.J.: A PDE approach to space-time fractional parabolic problems. SIAM J. Numer. Anal. 54(2), 848–873 (2016)
Padgett, J.L., Sheng, Q.: On the positivity, monotonicity, and stability of a semi-adaptive LOD method for solving three-dimensional degenerate Kawarada equations. J. Math. Anal. Appl. 439, 465–480 (2016)
Padgett, J.L., Sheng, Q.: Numerical solution of degenerate stochastic kawarada equations via a semi-discretized approach. Appl. Math. Comput. 325, 210–226 (2018). https://doi.org/10.1016/j.amc.2017.12.034
Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48(1), 175–209 (2000)
Söderlind, G.: The logarithmic norm. history and modern theory. BIT Numer. Math. 46(3), 631–652 (2006)
Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35(11), 2092–2122 (2010). https://doi.org/10.1080/03605301003735680
Acknowledgements
This work was supported by the NSF Grant Number 1903450. The author would like to thank Akif Ibraguimov, of Texas Tech University, for introducing this problem to them. The author is also thankful for Akif’s time and insightful suggestions that greatly improved the quality of the work herein. Finally, the author is thankful to the reviewers who provided numerous useful suggestions that greatly improved the presentation of the numerical experiments in Sect. 5.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Padgett, J.L. Analysis of an approximation to a fractional extension problem. Bit Numer Math 60, 715–739 (2020). https://doi.org/10.1007/s10543-019-00787-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-019-00787-y
Keywords
- Fractional diffusion
- Nonlocal operators
- Singular equations
- Degenerate equations
- Bessel equations
- Semigroup methods