Nonlocal problem for a fractional-order mixed-type equation with involution

B. J. Kadirkulov ; G. A. Kayumova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 55-65.

Résumé

In this paper, we examine the unique solvability of a nonlocal problem for a nonlocal analog of a mixed parabolic-hyperbolic equation with a generalized Riemann–Liouville operator and involution with respect to the space variable. A criterion for the uniqueness of the solution is established and sufficient conditions for the unique solvability of the problem are determined. By the method of separation of variables, a solution is constructed in the form of an absolutely and uniformly convergent series with respect to eigenfunctions of the corresponding one-dimensional spectral problem. The stability of the solution of the problem under consideration under a nonlocal condition is established.

Détail

Keywords: mixed-type equation, equation with involution, nonlocal problem, nonlocal differential equation, gluing conditions, Hilfer operator, Mittag-Leffler function, Fourier series.

Bibliographie

[1] Dzhrbashyan M. M., Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966 | MR

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

[3] Sabitov K. B., Guschina V. A., “Zadacha A. A. Dezina dlya neodnorodnogo uravneniya Lavrenteva—Bitsadze”, Izv. vuzov. Mat., 2017, no. 3, 37–50 | Zbl

[4] Sabitov K. B., Martemyanova N. V., “K voprosu o korrektnosti obratnykh zadach dlya neodnorodnogo uravneniya Gelmgoltsa”, Vestn. Samar. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki., 22:2 (2018), 269–292 | Zbl

[5] Islomov B. I., Abdullaev O. Kh., “Zadachi tipa Gellerstedta dlya nagruzhennogo uravneniya parabolo-giperbolicheskogo tipa s operatorami Kaputo i Erdeili—Kobera drobnogo poryadka”, Izv. vuzov. Mat., 2020, no. 10, 33–46 | Zbl

[6] Agarwal P., Abdullaev O. Kh., “A nonlocal problem with integral gluing condition for a third-order loaded equation with parabolic-hyperbolic operator involving fractional derivatives”, Math. Meth. Appl. Sci., 43:6 (2020), 3716–3726 | DOI | MR | Zbl

[7] 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

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

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

[10] Tenreiro Machado J. A. (ed.)., Handbook of Fractional Calculus with Applications, De Gruyter, Berlin–Boston, 2019

[11] Hilfer R. (ed.)., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000 | MR | Zbl

[12] Hilfer R., Luchko Y., Tomovski Z., “Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives”, Fract. Calc. Appl. Anal., 12:3 (2009), 299–318 | MR | Zbl

[13] Karimov E., Mamchuev M., Ruzhansky M., “Non-local initial problem for second order time-fractional and space-singular equation”, Hokkaido Math. J., 49 (2020), 349–361 | DOI | MR | Zbl

[14] Kilbas A. A., Srivastava H. M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006 | MR | Zbl

[15] Kirane M., Sadybekov M. A., Sarsenbi A. A., “On an inverse problem of reconstructing a subdiffusion process from nonlocal data”, Math. Meth. Appl. Sci., 42:6 (2019), 2043–2052 | DOI | MR | Zbl

[16] Kirane M., Turmetov B. Kh., Torebek B. T., “A nonlocal fractional Helmholtz equation”, Fract. Differ. Calc., 7:2 (2017), 225–234 | DOI | MR | Zbl

[17] Kumar D., Baleanu D., “Editorial: Fractional calculus and its applications in physics”, Front. Phys., 7 (2019), 81 | DOI | Zbl

[18] Kim Myong-Ha, Ri Guk-Chol, O Hyong-Chol, “Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives”, Fract. Calc. Appl. Anal., 17:1 (2014), 79–95 | DOI | MR | Zbl

[19] Sabitov K. B., “Nelokalnaya zadacha dlya uravneniya parabolo-giperbolicheskogo tipa v pryamougolnoi oblasti”, Mat. zametki., 89:4 (2011), 596–602 | Zbl

[20] Sandev T., Tomovski Z., Fractional Equations and Models: Theory and Applications, Springer Nature, Switzerland, 2019 | MR | Zbl

[21] Salakhitdinov M. S., Karimov E. T., “Direct and inverse source problems for two-term time-fractional diffusion equation with Hilfer derivative”, Uzbek. Math. J., 4 (2017), 140–149 | MR

[22] Sun H., Chang A., Zhang Y., Chen W., “A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications”, Fract. Calc. Appl. Anal., 22 (2019), 27–59 | DOI | MR | Zbl

[23] 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

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

[25] 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

[26] Yuldashev T. K., Kadirkulov B. J., “Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator”, Ural Math. J., 6:1 (2020), 153–167 | DOI | MR | Zbl

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