# Operational Rules for a Mixed Operator of the Erdélyi-Kober Type

Fractional Calculus and Applied Analysis (2004)

- Volume: 7, Issue: 3, page 339-364
- ISSN: 1311-0454

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topLuchko, Yury. "Operational Rules for a Mixed Operator of the Erdélyi-Kober Type." Fractional Calculus and Applied Analysis 7.3 (2004): 339-364. <http://eudml.org/doc/11251>.

@article{Luchko2004,

abstract = {2000 Mathematics Subject Classification: 26A33 (main), 44A40, 44A35, 33E30, 45J05, 45D05In 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.},

author = {Luchko, Yury},

journal = {Fractional Calculus and Applied Analysis},

keywords = {26A33; 44A40; 44A35; 33E30; 45J05; 45D05},

language = {eng},

number = {3},

pages = {339-364},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Operational Rules for a Mixed Operator of the Erdélyi-Kober Type},

url = {http://eudml.org/doc/11251},

volume = {7},

year = {2004},

}

TY - JOUR

AU - Luchko, Yury

TI - Operational Rules for a Mixed Operator of the Erdélyi-Kober Type

JO - Fractional Calculus and Applied Analysis

PY - 2004

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 7

IS - 3

SP - 339

EP - 364

AB - 2000 Mathematics Subject Classification: 26A33 (main), 44A40, 44A35, 33E30, 45J05, 45D05In 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.

LA - eng

KW - 26A33; 44A40; 44A35; 33E30; 45J05; 45D05

UR - http://eudml.org/doc/11251

ER -

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