Geodesic
Operational Rules for a Mixed Operator of the Erdélyi-Kober Type
Fractional calculus and applied analysis, Tome 7 (2004) no. 3, pp. 339-364.

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In the paper, the machinery of the Mellin integral transform is applied to deduce and prove some operational relations for a general operator of the Erdélyi-Kober type. This integro-differential operator is a composition of a number of left-hand sided and right-hand sided Erdélyi-Kober derivatives and integrals. It is referred to in the paper as a mixed operator of the Erdélyi-Kober type. For special values of parameters, the operator is reduced to some well known differential, integro-differential, or integral operators studied earlier by different authors. The differential operators of hyper-Bessel type, the Riemann-Liouville fractional derivative, the Caputo fractional derivative, and the multiple Erdélyi-Kober fractional derivatives and integrals are examples of its particular cases. In the general case however, the constructions suggested in the paper are new objects not yet well studied in the literature. The initial impulse to consider the operators presented in the paper arose while the author studied a problem to find scale-invariant solutions of some partial differential equations of fractional order: It turned out, that scale-invariant solutions of these partial differential equations of fractional order are described by ordinary differential equations of fractional order containing some particular cases of the mixed operator of Erdélyi-Kober type.
Keywords: Operational Relations, Fractional Derivatives and Integrals, Erdélyi-Kober Fractional Operators, Fractional Differential Equations, 26A33, 44A40, 44A35, 33E30, 45J05, 45D05 
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Luchko, Yury. Operational Rules for a Mixed Operator of the Erdélyi-Kober Type. Fractional calculus and applied analysis, Tome 7 (2004) no. 3, pp. 339-364. http://geodesic.mathdoc.fr/item/FCAA_2004_7_3_a4/