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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs
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by David Bolin, Mihály Kovács, Vivek Kumar and Alexandre B. Simas;
Math. Comp. 93 (2024), 2439-2472
DOI: https://doi.org/10.1090/mcom/3929
Published electronically: December 27, 2023

Abstract:

The fractional differential equation $L^\beta u = f$ posed on a compact metric graph is considered, where $\beta >0$ and $L = \kappa ^2 - \nabla (a\nabla )$ is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients $\kappa ,a$. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when $f$ is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power $L^{-\beta }$. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the $L_2(\Gamma \times \Gamma )$-error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for ${L = \kappa ^2 - \Delta , \kappa >0}$ are performed to illustrate the results.
References
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Bibliographic Information
  • David Bolin
  • Affiliation: Statistics Program, Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
  • MR Author ID: 918715
  • ORCID: 0000-0003-2361-5465
  • Email: david.bolin@kaust.edu.sa
  • Mihály Kovács
  • Affiliation: Department of Mathematical Sciences,Chalmers University of Technology and University of Gothenburg, SE-41296 Gothenburg, Sweden; Department of Differential Equations, Budapest University of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary; and Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, Práter utca 50/a., H-1083 Budapest, Hungary
  • ORCID: 0000-0001-7977-9114
  • Email: mkovacs@math.bme.hu
  • Vivek Kumar
  • Affiliation: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, 8th Mile Mysore Road, Bangalore, 560059, Karnataka, India
  • MR Author ID: 1319257
  • Email: vivekkumar\!_ra@isibang.ac.in
  • Alexandre B. Simas
  • Affiliation: Statistics Program, Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
  • MR Author ID: 864018
  • ORCID: 0000-0003-2562-2829
  • Email: alexandre.simas@kaust.edu.sa
  • Received by editor(s): February 8, 2023
  • Received by editor(s) in revised form: September 19, 2023
  • Published electronically: December 27, 2023
  • Additional Notes: The second author was supported by the Marsden Fund of the Royal Society of New Zealand (grant no. 18-UOO-143), the Swedish Research Council (VR) (grant no. 2017-04274) and the National Research, Development, and Innovation Fund of Hungary (grant no. TKP2021-NVA-02 and K-131545). The third author was supported in part by a NBHM post-doctoral fellowship from the Department of Atomic Energy (DAE), Government of India (file no. 0204/6/2022/R&D-II/5635).
  • © Copyright 2023 American Mathematical Society
  • Journal: Math. Comp. 93 (2024), 2439-2472
  • MSC (2020): Primary 35R02, 35A01, 35A02, 60H15, 60H40
  • DOI: https://doi.org/10.1090/mcom/3929
  • MathSciNet review: 4759380