Please use this identifier to cite or link to this item: https://hdl.handle.net/20.500.11851/7688
Title: THEORETICAL FOUNDATIONS OF INCORPORATING LOCAL BOUNDARY CONDITIONS INTO NONLOCAL PROBLEMS
Authors: Aksoylu, Burak
Reinhard Beyer, Horst
Çeliker, Fatih
Keywords: nonlocal wave equation
nonlocal operator
peridynamics
boundary condition
Hilbert-Schmidt operator
operator theory
Issue Date: 2017
Publisher: Pergamon-Elsevier Science Ltd
Abstract: We study nonlocal equations from the area of peridynamics on bounded domains. We present four main results. In our recent paper, we have discovered that, on R, the governing operator in peridynamics, which involves a convolution, is a bounded function of the classical (local) governing operator. Building on this, as main result 1, we construct an abstract convolution operator on bounded domains which is a generalization of the standard convolution based on integrals. The abstract convolution operator is a function of the classical operator, defined by a Hilbert basis available due to the purely discrete spectrum of the latter. As governing operator of the nonlocal equation we use a function of the classical operator, this allows us to incorporate local boundary conditions into nonlocal theories. As main result 2, we prove that the solution operator can be uniquely decomposed into a Hilbert Schmidt operator and a multiple of the identity operator. As main result 3, we prove that Hilbert Schmidt operators provide a smoothing of the input data in the sense a square integrable function is mapped into a function that is smooth up to boundary of the domain. As main result 4, for the homogeneous nonlocal wave equation, we prove that continuity is preserved by time evolution. Namely, the solution is discontinuous if and only if the initial data is discontinuous. As a consequence, discontinuities remain stationary.
URI: https://doi.org/10.1016/S0034-4877(17)30061-7
https://hdl.handle.net/20.500.11851/7688
ISSN: 0034-4877
Appears in Collections:Matematik Bölümü / Department of Mathematics
Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection

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