Comparison of Nonlocal Operators Utilizing Perturbation Analysis
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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.
KeywordsKernel Function Transition Point Classical Operator Perturbation Expansion Boundary Condition
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.
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