Fractional differential equations: alpha-entire solutions, regular and irregular singularities

2008/06/11 by Anatoly N. Kochubei, Kochubei, Anatoly N.
Mathematics · Physics and Astronomy · #26A33 #34M99 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Meromorphic and Entire Functions #math-ph #math.CA #math.MP #msc:26A33 #msc:34M99

paper · pdf · doi:10.48550/arxiv.0806.1826

20 pages; to appear in Fractional Calculus and Applied Analysis

openalex publication_date 2008/06/11 · arxiv created 2008/11/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider fractional differential equations of order α∈ (0,1) for functions of one independent variable t∈ (0,∞) with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order of growth of α-entire solutions is given. An analog of the Frobenius method for systems with regular singularity is developed. For a model example of an equation with a kind of an irregular singularity, a series for a formal solution is shown to be convergent for t>0 (if α is an irrational number poorly approximated by rational ones) but divergent in the distribution sense.

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