A new algebraic approach to the eigenvalue problems of linear differential operators without integrations

1986 ◽  
Vol 29 (2) ◽  
pp. 221-227 ◽  
Author(s):  
Metin Demiralp
Keyword(s):  
Differential Operators ◽  
Eigenvalue Problems ◽  
Algebraic Approach ◽  
Linear Differential Operators ◽  
Linear Differential

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

2016 ◽  
Vol 16 (4) ◽  
pp. 633-652 ◽  
Author(s):  
Ihor Demkiv ◽  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov
Keyword(s):  
Convergence Rate ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Exponential Convergence ◽  
Piecewise Constant ◽  
Numerical Examples ◽  
Linear Differential Operators ◽  
Principal Property ◽  
Linear Differential

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.

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