Geodesic
A problem with generalized fractional integro-differentiation operator of an arbitrary order
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2012), pp. 59-71.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a degenerate hyperbolic equation we study a problem with fractional integro-differentiation operators in the boundary condition on the characteristic part of the boundary. We determine intervals of variation of parameters of generalized operators of an arbitrary order with the Gauss hypergeometric function with which the problem is either uniquely solvable or has more than one solution.
Keywords: Riemann–Liouville integral and derivative of a fractional order, Volterra and Abel integral equations, Gauss hypergeometric function, resolvent of the kernel. 
@article{IVM_2012_12_a5,
     author = {O. A. Repin and S. K. Kumykova},
     title = {A problem with generalized fractional integro-differentiation operator of an arbitrary order},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {59--71},
     publisher = {mathdoc},
     number = {12},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2012_12_a5/}
}
TY  - JOUR
AU  - O. A. Repin
AU  - S. K. Kumykova
TI  - A problem with generalized fractional integro-differentiation operator of an arbitrary order
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2012
SP  - 59
EP  - 71
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2012_12_a5/
LA  - ru
ID  - IVM_2012_12_a5
ER  - 
%0 Journal Article
%A O. A. Repin
%A S. K. Kumykova
%T A problem with generalized fractional integro-differentiation operator of an arbitrary order
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2012
%P 59-71
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2012_12_a5/
%G ru
%F IVM_2012_12_a5
O. A. Repin; S. K. Kumykova. A problem with generalized fractional integro-differentiation operator of an arbitrary order. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2012), pp. 59-71. http://geodesic.mathdoc.fr/item/IVM_2012_12_a5/

[1] Saigo M., “A remark on integral operators involving the Gauss hypergeometric function”, Math. Rep. Kyushu Univ., 11:2 (1978), 135–143 | MR | Zbl

[2] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987 | MR | Zbl

[3] Nakhushev A. M., Drobnoe ischislenie i ego primenenie, Fizmatlit, M., 2003 | Zbl

[4] Nakhushev A. M., Zadachi so smescheniem dlya uravnenii v chastnykh proizvodnykh, Nauka, M., 2006 | Zbl

[5] Zhegalov V. I., “Kraevaya zadacha dlya uravneniya smeshannogo tipa s granichnym usloviem na obeikh kharakteristikakh i s razryvami na perekhodnoi linii”, Uchen. zap. Kazansk. un-ta, 122, no. 3, 1962, 3–16 | MR | Zbl

[6] Orazov I. O., “Ob odnoi kraevoi zadache so smescheniem dlya obobschennogo uravneniya Trikomi”, Differents. uravneniya, 17:2 (1981), 339–344 | MR

[7] Kumykova S. K., “Nelokalnaya kraevaya zadacha dlya uravneniya Lykova”, Nelokalnye kraevye zadachi matem. fiziki i ikh prilozh., Izd-vo Instituta matematiki AN USSR, Kiev, 1990

[8] Repin O. A., Kumykova S. K., “Nelokalnaya zadacha dlya uravneniya Bitsadze–Lykova”, Izv. vuzov. Matem., 2010, no. 3, 28–35 | MR | Zbl

[9] Bitsadze A. V., Nekotorye klassy uravnenii v chastnykh proizvodnykh, Nauka, M., 1981 | MR | Zbl

[10] Smirnov M. M., Vyrozhdayuschiesya giperbolicheskie uravneniya, Izd-vo Vyssh. shk., Minsk, 1977 | MR

[11] Saigo M., Kilbas A. A., “Generalized fractional integrals and derivatives in Hölder space”, Transforms methods special functions, Proc. Intern. Workshop (Bankya, Bulgaria, August 12–17, 1994), Sofia, 1995, 282–293 | Zbl

[12] Trikomi F., Integralnye uravneniya, In. lit., M., 1960 | MR

[13] Lebedev N. N., Spetsialnye funktsii i ikh prilozheniya, Fizmatgiz, M., 1963 | MR