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Nonlocal operators with local boundary conditions in higher dimensions

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Abstract

We present novel nonlocal governing operators in 2D/3D for wave propagation and diffusion. The operators are inspired by peridynamics. They agree with the original peridynamics operator in the bulk of the domain and simultaneously enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of separable kernel functions together with even and odd parts of bivariate functions on rectangular/box domains. The operators are bounded and self-adjoint. We present all possible 36 different types of BC in 2D which include pure and mixed combinations of Neumann, Dirichlet, periodic, and antiperiodic BC. Our construction is systematic and easy to follow. We provide numerical experiments that verify our theoretical findings. We also compare the solutions of the classical wave and heat equations to their nonlocal counterparts.

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Funding

Burak Aksoylu was supported in part by the European Commission Marie Curie Career Integration 293978 grant, and Scientific and Technological Research Council of Turkey (TÜ BİTAK) MFAG 115F473 grant. Portion of his work was also supported in part by the Oak Ridge Institute for Science and Engineering (ORISE) contract 1120-1120-99 at the US Army Research Laboratory.

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Author notes

  1. Orsan Kilicer

    Present address: Department of Mathematics, Texas A&M University, College Station, TX, USA

Authors and Affiliations

  1. US Army Research Laboratory, Attn:RDRL-WMM-B, Aberdeen Proving Ground, Aberdeen, MD, 21005, USA

    Burak Aksoylu

  2. Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI, 48202, USA

    Burak Aksoylu & Fatih Celiker

  3. Department of Mathematics, Middle East Technical University, Ankara, Turkey

    Orsan Kilicer

Authors
  1. Burak Aksoylu
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  2. Fatih Celiker
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  3. Orsan Kilicer
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Corresponding author

Correspondence to Burak Aksoylu.

Additional information

Communicated by: Ilaria Perugia

The original version of this article was revised: In the original publication, Figure 4 image should be Figure 5 and Figure 5 image was a repetition of Figure 6. The original article was updated by correcting the images of figures 4 and 5.

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Aksoylu, B., Celiker, F. & Kilicer, O. Nonlocal operators with local boundary conditions in higher dimensions. Adv Comput Math 45, 453–492 (2019). https://doi.org/10.1007/s10444-018-9624-6

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  • DOI: https://doi.org/10.1007/s10444-018-9624-6

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