A Singularly Perturbed Boundary Value Problems with Fractional Powers of Elliptic Operators

A boundary value problem for a fractional power 0 < epsilon < 1 of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when epsilon -> 0. It is solved numerically using a time-dependent problem for a pseudo-parabolic equation. For the auxiliary Cauchy problem, the standard two-level schemes with weights are applied. The numerical results are presented for a model two-dimensional boundary value problem with a fractional power of an elliptic operator. Our work focuses on the solution of the boundary value problem with 0 < epsilon << 1.