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