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We consider the homogeneous equation ${\mathcal A} u=0$, where ${\mathcal A}$is a symmetric and coercive elliptic operator in $H^1(\Omega)$ with $\Omega$bounded domain in ${{\mathbb R}}^d$. The boundary conditions involve fractionalpower $\alpha$, $ 0 < \alpha <1$, of the Steklov spectral operator arising inDirichlet to Neumann map. For such problems we discuss two different numericalmethods: (1) a computational algorithm based on an approximation of theintegral representation of the fractional power of the operator and (2)numerical technique involving an auxiliary Cauchy problem for anultra-parabolic equation and its subsequent approximation by a time steppingtechnique. For both methods we present numerical experiment for a modeltwo-dimensional problem that demonstrate the accuracy, efficiency, andstability of the algorithms.

A numerical study of the homogeneous elliptic equation with fractional order boundary conditions