Abstract
We present a comparative study of integral operators used in nonlocal problems. The size of nonlocality is determined by the parameter δ. The authors recently discovered a way to incorporate local boundary conditions into nonlocal problems. We construct two nonlocal operators which satisfy local homogeneous Neumann boundary conditions. We compare the bulk and boundary behaviors of these two to the operator that enforces nonlocal boundary conditions. We construct approximations to each operator using perturbation expansions in the form of Taylor polynomials by consistently keeping the size of expansion neighborhood equal to δ. In the bulk, we show that one of these two operators exhibits similar behavior with the operator that enforces nonlocal boundary conditions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
B. Aksoylu, M.L. Parks, Variational theory and domain decomposition for nonlocal problems. Appl. Math. Comput. 217, 6498–6515 (2011)
B. Aksoylu, Z. Unlu, Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J. Numer. Anal. 52 (2), 653–677 (2014)
B. Aksoylu, H.R. Beyer, F. Celiker, Application and implementation of incorporating local boundary conditions into nonlocal problems (Submitted)
B. Aksoylu, H.R. Beyer, F. Celiker, Theoretical foundations of incorporating local boundary conditions into nonlocal problems (Submitted)
F. Andreu-Vaillo, J.M. Mazon, J.D. Rossi, J. Toledo-Melero, Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165 (American Mathematical Society, Providence/Real Socied Matematica Espanola, Madrid, 2010)
M. Arndt, M. Griebel, Derivation of higher order gradient continuum models from atomistic models for crystalline solids. Multiscale Model. Simul. 4 (2), 531–562 (2005)
H.R. Beyer, B. Aksoylu, F. Celiker, On a class of nonlocal wave equations from applications. J. Math. Phys. 57 (6) (2016). doi:org/10.1063/1.4953252, http://scitation.aip.org/content/aip/journal/jmp/57/6/10.1063/1.4953252
Q. Du, M. Gunzburger, R.B. Lehoucq, K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54, 667–696 (2012)
E. Emmrich, O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5 (4), 851–864 (2007)
P. Seleson, M.L. Parks, M. Gunzburger, R.B. Lehoucq, Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul. 8, 204–227 (2009)
S. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
Acknowledgements
Burak Aksoylu was supported in part by National Science Foundation DMS 1016190 grant, European Commission Marie Curie Career Integration Grant 293978, and Scientific and Technological Research Council of Turkey (TÜBİTAK) TBAG 112T240 and MAG 112M891 grants. Fatih Celiker’s sabbatical visit was supported in part by the TÜBİTAK 2221 Fellowship for Scientist on Sabbatical Leave Program. He also would like to thank Orsan Kilicer of Middle East Technical University for his careful reading of the paper.
Fatih Celiker was supported in part by the National Science Foundation DMS 1115280 grant.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Aksoylu, B., Celiker, F. (2016). Comparison of Nonlocal Operators Utilizing Perturbation Analysis. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_57
Download citation
DOI: https://doi.org/10.1007/978-3-319-39929-4_57
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-39927-0
Online ISBN: 978-3-319-39929-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)