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A Numerical Study of the Homogeneous Elliptic Equation with Fractional Boundary Conditions

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Abstract

We consider the homogeneous equation Au = 0, where A is a symmetric and coercive elliptic operator in H1(Ω) with Ω bounded domain in ℝd. The boundary conditions involve fractional power α, 0 < α < 1, of the Steklov spectral operator arising in Dirichlet to Neumann map. For such problems we discuss two different numerical methods: (1) a computational algorithm based on an approximation of the integral representation of the fractional power of the operator and (2) numerical technique involving an auxiliary Cauchy problem for an ultra-parabolic equation and its subsequent approximation by a time step** technique. For both methods we present numerical experiment for a model two-dimensional problem that demonstrate the accuracy, efficiency, and stability of the algorithms.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Raytcho Lazarov

  2. Institute of Mathematics and Informatics–Bulg. Acad. Sci., “Acad. G. Bontchev” Str., Block 8, Sofia, 1113, Bulgaria

    Raytcho Lazarov

  3. Nuclear Safety Institute of RAS 52, B. Tulskaya, Moscow, Russia

    Petr Vabishchevich

  4. Peoples’ Friendship University of Russia (PRUDN University) 6, Miklukho-Maklaya Str., Moscow, Russia

    Petr Vabishchevich

Authors
  1. Raytcho Lazarov
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  2. Petr Vabishchevich
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Correspondence to Raytcho Lazarov.

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Lazarov, R., Vabishchevich, P. A Numerical Study of the Homogeneous Elliptic Equation with Fractional Boundary Conditions. FCAA 20, 337–351 (2017). https://doi.org/10.1515/fca-2017-0018

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  • DOI: https://doi.org/10.1515/fca-2017-0018

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