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

Abstract

Abstract:A 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.

Super-Exponentially Convergent Parallel Algorithm for a Fractional Eigenvalue Problem of Jacobi-Type

“…We apply to problem (3) the FD-method (see e.g. [7] for details) which is a combination of perturbation of the original differential operator by a parameter dependent operator (embedding) and then a "trip" along this parameter from a "simple" problem to the original one (homotopy). The homotopy idea was exploited in various ways e.g.…”

Symbolic Algorithm of the Functional-Discrete Method for a Sturm–Liouville Problem with a Polynomial Potential

“…Using the formulas (22) and (16) (it follows from (13) or (25)) we obtain the following formula for the corrections of the eigenvalues:…”

“…Returning to the replacements (32), (39) and the notations (36), (37), we obtain the following recursive representation for the coefficients in (22) (see the notations (28), (29)):…”

“…The solution of the system (22) can be determined only up to the constant b (0) n,N,0 , which we calculate from the orthogonality condition (17), which leads to the following equation…”

“…with respect to the unknown b (0) n,N,0 . Thus, the formulas (9)-(12), (22), (23), (26) and (44)-(49) represent the new symbolic algorithmic implementation of the general scheme of the FD-method for the problem (1) with the polynomial potential (2). Our method uses the algebraic operations only and does not need the solutions of any boundary value problems (5)-(8) and computations of any integrals (15)-(18) unlike the previously known traditional implementations of the FDmethod [13,25,26].…”