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

## TL;DR

This paper introduces a super-exponentially convergent parallel algorithm for solving fractional Jacobi-type eigenvalue problems, transforming a nonlinear problem into a sequence of linear problems with high convergence speed.

## Contribution

It proposes a novel recursive algorithm using truncated functional discrete schemes that achieves super-exponential convergence for fractional eigenvalue problems.

## Key findings

- Algorithm converges super-exponentially as m increases.
- Eigenpairs can be computed in parallel for all indices.
- Numerical examples validate the theoretical convergence rates.

## Abstract

A new algorithm for eigenvalue problems for the fractional Jacobi type ODE is proposed. The algorithm is based on piecewise approximation of the coefficients of the differential equation with subsequent recursive procedure adapted from some homotopy considerations. As a result, the eigenvalue problem (which is in fact nonlinear) is replaced by a sequence of linear boundary value problems (besides the first one) with a singular linear operator called the exact functional discrete scheme (EFDS). A finite subsequence of $m$ terms, called truncated functional discrete scheme (TFDS), is the basis for our algorithm. The approach provides an super-exponential convergence rate as $m \to \infty$. The eigenpairs can be computed in parallel for all given indexes. The algorithm is based on some recurrence procedures including the basic arithmetical operations with the coefficients of some expansions only. This is an exact symbolic algorithm (ESA) for $m=\infty$ and a truncated symbolic algorithm (TSA) for a finite $m$. Numerical examples are presented to support the theory.