A numerical study of the homogeneous elliptic equation with fractional boundary conditions
The authors consider the numerical approximation of homogeneous second order elliptic equations with boundary conditions involving fractional power \(\alpha\), with \(0<\alpha <1\), of the Steklov spectral operator arising in Dirichlet to Neumann map. Two numerical methods are presented and discussed. The first one is a computational algorithm based on an approximation of the integral representation of the fractional power of the operator, whereas the second one is based on the introduction of an auxiliary Cauchy problem for an ultra-parabolic equation and its subsequent approximation by a time stepping technique. Some numerical experiments are presented to justify the accuracy and stability of the algorithms for both methods.
fractional partial differential equations
numerical methods for fractional powers of elliptic operators
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