Geodesic
Boundary-value problem with an integral conjugation condition for a partial differential equation with the fractional Riemann--Liouville derivative that describes gas flows in a channel surrounded by a porous medium
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 66-76.

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

A boundary-value problem with an integral conjugation condition for a mixed equation with a fractional integro-differential operator was examined. The main result of the work is the proof of the unique solvability of the boundary-value problem with an integral conjugation condition for the equation consisting of two partial differential equations with the fractional Riemann–Liouville derivative in a rectangular domain. The problem is reduced to a Volterra integral equation of the second kind. The special role of the conjugation condition in the solvability of the problem is shown.
Keywords: boundary-value problem, integral conjugation condition, mixed fractional-order equation, gas flow in a channel. 
@article{INTO_2022_210_a7,
     author = {A. K. Urinov and E. T. Karimov and S. Kerbal},
     title = {Boundary-value problem with an integral conjugation condition for a partial differential equation with the fractional {Riemann--Liouville} derivative that describes gas flows in a channel surrounded by a porous medium},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {66--76},
     publisher = {mathdoc},
     volume = {210},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_210_a7/}
}
TY  - JOUR
AU  - A. K. Urinov
AU  - E. T. Karimov
AU  - S. Kerbal
TI  - Boundary-value problem with an integral conjugation condition for a partial differential equation with the fractional Riemann--Liouville derivative that describes gas flows in a channel surrounded by a porous medium
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 66
EP  - 76
VL  - 210
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_210_a7/
LA  - ru
ID  - INTO_2022_210_a7
ER  - 
%0 Journal Article
%A A. K. Urinov
%A E. T. Karimov
%A S. Kerbal
%T Boundary-value problem with an integral conjugation condition for a partial differential equation with the fractional Riemann--Liouville derivative that describes gas flows in a channel surrounded by a porous medium
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 66-76
%V 210
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_210_a7/
%G ru
%F INTO_2022_210_a7
A. K. Urinov; E. T. Karimov; S. Kerbal. Boundary-value problem with an integral conjugation condition for a partial differential equation with the fractional Riemann--Liouville derivative that describes gas flows in a channel surrounded by a porous medium. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 66-76. http://geodesic.mathdoc.fr/item/INTO_2022_210_a7/

[1] Abdullaev O. Kh., “Nelokalnaya zadacha dlya nagruzhennogo uravneniya smeshannogo tipa s integralnym operatorom”, Vestn. Samar. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki., 2016, no. 2, 220–240 | Zbl

[2] Balkizov Zh. A., Kraevye zadachi dlya uravnenii parabolo-giperbolicheskogo tipa vtorogo i tretego poryadkov, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Nalchik, 2014

[3] Gelfand I. M., “Nekotorye voprosy analiza i differentsialnykh uravnenii”, Usp. mat. nauk., 14:3 (1959), 3–19 | Zbl

[4] Kadirkulov B. Zh., Zhalilov M. A., “Ob odnoi nelokalnoi zadache dlya uravneniya smeshannogo tipa chetvertogo poryadka c operatorom Khilfera”, Byull. In-ta mat. im. V. I. Romanovskogo., 2020, no. 1, 59–67 | MR

[5] Kapustin N. Yu., Zadachi dlya parabolo-giperbolicheskikh uravnenii i sootvetstvuyuschie spektralnye voprosy s parametrom v granichnykh tochkakh, Diss. na soisk. uch. step. dokt. fiz.-mat. nauk, M., 2012

[6] Pskhu A. V., Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005

[7] Rakhmanova L. Kh., Kraevye zadachi dlya uravnenii smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Kazan, 2009

[8] Sabitov K. B., Melisheva E. P., “Zadacha Dirikhle dlya nagruzhennogo uravneniya smeshannogo tipa v pryamougolnoi oblasti”, Izv. vuzov. Mat., 2013, no. 7, 62–76 | Zbl

[9] Sabitov K. B., Sidorov S. N., “Obratnaya zadacha dlya vyrozhdayuschegosya parabolo-giperbolicheskogo uravneniya s nelokalnym granichnym usloviem”, Izv. vuzov. Mat., 2015, no. 1, 46–59 | Zbl

[10] Sabitov K. B., “Nachalno-granichnaya i obratnye zadachi dlya neodnorodnogo uravneniya smeshannogo parabolo-giperbolicheskogo uravneniya”, Mat. zametki., 2017, no. 3, 415–435 | Zbl

[11] Sabitov K. B., Sidorov S. N., “Nachalno-granichnaya zadacha dlya neodnorodnykh vyrozhdayuschikhsya uravnenii smeshannogo parabolo-giperbolicheskogo tipa”, Itogi nauki i tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 137 (2017), 26–60

[12] Sabitov K. B., “Nachalno-granichnaya i obratnye zadachi dlya neodnorodnogo uravneniya smeshannogo parabolo-giperbolicheskogo uravneniya”, Mat. zametki., 102:3 (2017), 415–435 | MR | Zbl

[13] Tarasenko A. V., “Kraevaya zadacha dlya nagruzhennogo uravneniya smeshannogo parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Vestn. Samar. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki., 2010, no. 5, 263–267 | Zbl

[14] Yuldasheva A. Yu., “Obratnaya zadacha dlya parabolo-giperbolicheskogo uravneniya”, Voprosy vychislitelnoi i prikladnoi matematiki, Tashkent, 2009, 85–88

[15] Abdullaev O. Kh., Sadarangani K. B., “Non-local problems with integral gluing condition for loaded mixed-type equations involving the Caputo fractional derivative”, Electron. J. Differ. Equations., 2016, 164 | MR | Zbl

[16] Berdyshev A. S., Eshmatov B. E., Kadirkulov B. J., “Boundary-value problems for fourth-order mixed-type equation with fractional derivative”, Electron. J. Differ. Equations., 2016, 36 | MR | Zbl

[17] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999 | MR | Zbl

[18] Ruzhansky M., Tokmagambetov N., Torebek B., “On a non–local problem for a multi–term fractional diffusion-wave equation”, Fract. Calc. Appl. Anal., 2020, no. 2, 324–355 | DOI | MR | Zbl

[19] Ruzhansky M., Tokmagambetov N., Torebek B., “Inverse source problems for positive operators. I. Hypoelliptic diffusion and subdiffusion equations”, J. Inv. Ill-posed Probl., 6 (2019), 891–911 | DOI | MR | Zbl

[20] Yuldashev T. K., Kadirkulov B. J., “Boundary-value problem for weak nonlinear partial differential equations of mixed type with fractional Hilfer operator”, Axioms., 9:2 (2020), 68 | DOI

[21] Yuldashev T. K., Kadirkulov B. J., “Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator”, Ural Math. J., 2020, no. 1, 153–167 | DOI | MR | Zbl

[22] Yuldashev T. K., Karimov E. T., “Inverse problem for a mixed type integro-differential equation with fractional order Caputo operators and spectral parameters”, Axioms., 9:4 (2020), 121 | DOI | MR