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Fractional Calculus and Applied Analysis

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

Raytcho Lazarov
  • Department of Mathematics Texas A&M University, College Station, TX, 77843, USA and: Institute of Mathematics and Informatics – Bulg. Acad. Sci. “Acad. G. Bontchev” Str., Block 8, Sofia – 1113, BULGARIA
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/ Petr Vabishchevich
  • Nuclear Safety Institute of RAS 52, B. Tulskaya, Moscow, RUSSIA and: Peoples’ Friendship University of Russia (PRUDN University) 6, Miklukho-Maklaya Str., Moscow, RUSSIA
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Published Online: 2017-04-28 | DOI: https://doi.org/10.1515/fca-2017-0018


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 stepping technique. For both methods we present numerical experiment for a model two-dimensional problem that demonstrate the accuracy, efficiency, and stability of the algorithms.

MSC 2010: Primary 65N30; Secondary 34A08; 65M12; 35R11

Key Words and Phrases: fractional partial differential equations; fractional boundary conditions; numerical methods for fractional powers of elliptic operators; ultra-parabolic equations


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About the article

Received: 2016-09-12

Published Online: 2017-04-28

Published in Print: 2017-04-25

Citation Information: Fractional Calculus and Applied Analysis, Volume 20, Issue 2, Pages 337–351, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2017-0018.

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