Geodesic
On the solvability of some boundary-value problems for the fractional analog of the nonlocal Laplace equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 211 (2022), pp. 14-28.

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

In this paper, we examine methods for solving the Dirichlet boundary-value problem and the periodic boundary-value problem for one class of nonlocal second-order partial differential equations with involutive argument mappings. The concept of a nonlocal analog of the Laplace equation is introduced. A method for constructing eigenfunctions and eigenvalues of the spectral problem based on separation of variables is proposed. The completeness of the system of eigenfunctions is examined. The concept of a fractional analog of the nonlocal Laplace equation is introduced. For this equation, boundary-value problems with the Dirichlet and periodic conditions are considered. The well-posedness of these problems is verified and the existence and uniqueness of the solution of boundary-value problems are proved.
Keywords: Gerasimov–Caputo fractional derivative, nonlocal differential equation, involution, Dirichlet problem, periodic boundary-value problem, eigenfunction, Mittag-Leffler function, Fourier series. 
@article{INTO_2022_211_a1,
     author = {B. Kh. Turmetov and B. J. Kadirkulov},
     title = {On the solvability of some boundary-value problems for the fractional analog of the nonlocal {Laplace} equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {14--28},
     publisher = {mathdoc},
     volume = {211},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_211_a1/}
}
TY  - JOUR
AU  - B. Kh. Turmetov
AU  - B. J. Kadirkulov
TI  - On the solvability of some boundary-value problems for the fractional analog of the nonlocal Laplace equation
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 14
EP  - 28
VL  - 211
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_211_a1/
LA  - ru
ID  - INTO_2022_211_a1
ER  - 
%0 Journal Article
%A B. Kh. Turmetov
%A B. J. Kadirkulov
%T On the solvability of some boundary-value problems for the fractional analog of the nonlocal Laplace equation
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 14-28
%V 211
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_211_a1/
%G ru
%F INTO_2022_211_a1
B. Kh. Turmetov; B. J. Kadirkulov. On the solvability of some boundary-value problems for the fractional analog of the nonlocal Laplace equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 211 (2022), pp. 14-28. http://geodesic.mathdoc.fr/item/INTO_2022_211_a1/

[1] Andreev A. A., “Ob analogakh klassicheskikh kraevykh zadach dlya odnogo differentsialnogo uravneniya vtorogo poryadka s otklonyayuschimsya argumentom”, Differ. uravn., 40:8 (2004), 1126–1128 | MR | Zbl

[2] Burlutskaya M. Sh., Khromov A. P., “Metod Fure v smeshannoi zadache dlya uravneniya s chastnymi proizvodnymi pervogo poryadka s involyutsiei”, Zh. vychisl. mat. mat. fiz., 51:12 (2011), 2233–2246 | MR | Zbl

[3] Ilin V. A., Poznyak E. G., Osnovy matematicheskogo analiza, Fizmatlit, M., 2009 | MR

[4] Linkov A. V., “Obosnovanie metoda Fure dlya kraevykh zadach s involyutivnym otkloneniem”, Vestn. Samar. un-ta., 12:2 (1999), 60–66

[5] Masaeva O. Kh., “Edinstvennost resheniya zadachi Dirikhle dlya mnogomernogo uravneniya v chastnykh proizvodnykh drobnogo poryadka”, Vestn. KRAUNTs. Fiz.-mat. nauki., 2018, no. 4 (24), 50–53 | MR | Zbl

[6] Masaeva O. Kh., “Zadacha Dirikhle dlya obobschennogo uravneniya Laplasa s drobnoi proizvodnoi”, Chelyab. fiz.-mat. zh., 2:3 (2017), 312–322 | MR | Zbl

[7] Masaeva O. Kh., “Zadacha Neimana dlya obobschennogo uravneniya Laplasa”, Vestn. KRAUNTs. Fiz.-mat. nauki., 2018, no. 3 (23), 83–90 | MR | Zbl

[8] Masaeva O. Kh., “Zadacha Dirikhle dlya obobschennogo uravneniya Laplasa s proizvodnoi Kaputo”, Differ. uravn., 48:3 (2012), 442–446 | MR | Zbl

[9] Mikhlin S. G., Lineinye uravneniya v chastnykh proizvodnykh, Vysshaya shkola, M., 1977

[10] Nakhushev A. M., Uravneniya matematicheskoi biologii, Vysshaya shkola, M., 1995

[11] Al-Salti N., Kerbal S., Kirane M., “Initial-boundary-value problems for a time-fractional differential equation with involution perturbation”, Math. Model. Nat. Phenom., 14:3 (2019), 1–15 | DOI | MR

[12] Ashyralyev A., Sarsenbi A., “Well-posedness of a parabolic equation with involution”, Num. Funct. Anal. Optim., 38:10 (2017), 1295–1304 | DOI | MR | Zbl

[13] Ashyralyev A., Sarsenbi A. M., “Well-posedness of an elliptic equation with involution”, Electron. J. Differ. Equations., 2015, 284 | MR | Zbl

[14] Cabada A., Tojo F. A. F., “On linear differential equations and systems with reflection”, Appl. Math. Comput., 305 (2017), 84–102 | DOI | MR | Zbl

[15] Cascaval R. C., Eckstein E. C., Frota C. L., Goldstein J. A., “Fractional telegraph equations”, J. Math. Anal. Appl., 276:1 (2002), 145–159 | DOI | MR | Zbl

[16] Dulce M., Getmanenko A., “On the relationship between the inhomogeneous wave and Helmholtz equations in a fractional setting”, Abstr. Appl. Anal., 2019, 1483764 | MR | Zbl

[17] Gorenflo R., Kilbas A. A., Mainardi F., Rogosin S. V., Mittag-Leffler Functions, Related Topics, and Applications, Springer-Verlag, New York–Dordrecht–London, 2014 | MR | Zbl

[18] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, North-Holland, Amsterdam, 2006 | MR | Zbl

[19] Kirane M., Al-Salti N., “Inverse problems for a nonlocal wave equation with an involution perturbation”, J. Nonlin. Sci. Appl., 9 (2016), 1243–1251 | DOI | MR | Zbl

[20] Kirane M., Malik S. A., Al-Gwaiz M., “An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions”, Math. Meth. Appl. Sci., 36:9 (2012), 1056–1069 | DOI | MR

[21] Kirane M., Samet B., Torebek B. T., “Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data”, Electron. J. Differ. Equations., 2017, 257 | MR | Zbl

[22] Kopzhassarova A., Sarsenbi A., “Basis properties of eigenfunctions of second-order differential operators with involution”, Abstr. Appl. Anal., 2012, 576843 | MR | Zbl

[23] Malik S. A., Aziz S., “An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions”, Comput. Math. Appl., 73:12 (2017), 2548–2560 | DOI | MR | Zbl

[24] Przeworska-Rolewicz D., “Some boundary-value problems with transformed argument”, Comment. Math. Helvet., 17 (1974), 451–457 | MR | Zbl

[25] Tokmagambetov N., Torebek B. T., “Well-posed problems for the fractional Laplace equation with integral boundary conditions”, Electron. J. Differ. Equations., 2018, 90 | MR | Zbl

[26] Torebek B. T., Tapdigoglu R., “Some inverse problems for the nonlocal heat equation with Caputo fractional derivative”, Math. Meth. Appl. Sci., 40 (2017), 6468–6479 | DOI | MR | Zbl

[27] Turmetov B. Kh., Torebek B. T., “On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation”, New York J. Math., 20 (2014), 1237–1251 | MR | Zbl

[28] Turmetov B. Kh., Torebek B. T., “On a class of fractional elliptic problems with an involution perturbation”, AIP Conf. Proc., 1759 (2016), 020070 | DOI

[29] Turmetov B. Kh., Torebek B. T., Ontuganova Sh., “Some problems for fractional analogue of Laplace equation”, Int. J. Pure Appl. Math., 94:4 (2014), 525–532 | DOI