Super-Exponentially Convergent Parallel Algorithm for Eigenvalue Problems with Fractional Derivatives

Author:

Demkiv Ihor1,Gavrilyuk Ivan P.2,Makarov Volodymyr L.1

Affiliation:

1. 1Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., 01601 Kyiv-4, Ukraine

2. 2University of Cooperative Education Eisenach, Am Wartenberg 2, 99817 Eisenach, Germany

Abstract

AbstractA new algorithm for eigenvalue problems for linear differential operators with fractional derivatives is proposed and justified. The algorithm is based on the approximation (perturbation) of the coefficients of a part of the differential operator by piecewise constant functions where the eigenvalue problem for the last one is supposed to be simpler than the original one. Another milestone of the algorithm is the homotopy idea which results at the possibility for a given eigenpair number to compute recursively a sequence of the approximate eigenpairs. This sequence converges to the exact eigenpair with a super-exponential convergence rate. The eigenpairs can be computed in parallel for all prescribed indexes. The proposed method possesses the following principal property: its convergence rate increases together with the index of the eigenpair. Numerical examples confirm the theory.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics,Numerical Analysis

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