On Katugampola-Prabhakar fractional integral-differential operators

This work systematically investigates the Katugampola-Prabhakar (K-P) fractional operators, introducing rigorous definitions for K-P integrals and integro-differential operators while establishing their fundamental properties. We develop analytical solutions to Cauchy problems for fractional ordinary differential equations incorporating K-P and Caputo-type K-P derivatives, expressed through bivariate Mittag-Leffler functions E_2(x,y). The study further advances operational calculus for these operators by deriving m_p-Laplace transform relations applicable to: (i) infinite series representations, (ii) K-P integrals, (iii) left-sided K-P derivatives, and (iv) Caputo-type K-P operators. As a key application, we analyze an initial-boundary value problem for a subdiffusion equation governed by Caputo-type K-P derivatives, demonstrating the practical utility of these theoretical developments. The obtained results provide new analytical methods for studying fractional differential equations with K-P operators and their applications to subdiffusion processes. The developed techniques significantly expand the possibilities for solving boundary value problems involving these generalized fractional operators.

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