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

Raytcho Lazarov 1  and Petr Vabishchevich 2
  • 1 Department of Mathematics, TX, USA
  • 2 Nuclear Safety Institute of RAS, 52, B. Tulskaya, Moscow, RUSSIA
Raytcho Lazarov
  • Corresponding author
  • Department of Mathematics, Texas A&M University, College Station, TX, USA, Institute of Mathematics and Informatics – Bulg. Acad. Sci., Sofia, BULGARIA
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and Petr Vabishchevich
  • Nuclear Safety Institute of RAS, 52, B. Tulskaya, Moscow, RUSSIA, Peoples’ Friendship University of Russia (PRUDN University), 6, Miklukho-Maklaya Str., Moscow, RUSSIA
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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.

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Fractional Calculus and Applied Analysis (FCAA) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order.