A boundary value problem for a fractional power 0 < ε < 1 of the second-order elliptic operator is considered. The boundary value problem is singularly perturbed when ε → 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 < ε 1.
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