Abstract
In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.
Highlights
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in Special functions of mathematical physics are usually defined in the form of a power series, or as solutions to some differential equations, or via integral representations
Convolution series generated by Sonine kernels κ ∈ L1 were introduced in [13] for analytical treatment of fractional differential equations with n-fold GFDs of the Caputo type by means of an operational calculus developed for these GFDs
In [7], some of the results presented in [13] were extended to convolution series in the form (44) generated by any function κ ∈ C−1 (0, +∞)
Summary
Special functions of mathematical physics are usually defined in the form of a power series, or as solutions to some differential equations, or via integral representations. The Mittag-Leffler function can be introduced in terms of solutions to the fractional differential equations with the Riemann-Liouville or Caputo fractional derivatives. We employ the convolution series for derivation of analytical solutions to the single- and multi-terms fractional differential equations with the general fractional derivatives in the Riemann-Liouville sense.
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