On solvability of nonlocal problems with Ionkin conditions for partial differential equations. II
https://doi.org/10.25587/2411-9326-2024-1-48-55
Abstract
Considering the differential equations of any order with variable coefficients, we study the solvability of nonlocal boundary value problems with the Ionkin classical condition in Sobolev spaces. We prove the unique existence of regular solutions, i.e., those that enter the equations with all weak derivatives.
About the Author
A. I. Kozhanov
Sobolev Institute of Mathematics; Banzarov Buryat State University
Russian Federation
Russian Federation
Aleksandr I. Kozhanov
4 Koptyug Avenue, 630090 Novosibirsk
24a Smolin Street, 670000 Ulan-Ude
References
1. Ionkin N. I., “Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition [in Russian],” Differents. Uravn., 13, 204–211 (1977).
2. Ionkin N. I., “The stability of a problem in the theory of heat equations with nonclassical boundary conditions [in Russian],” Differents. Uravn., 15, No. 7, 1279–1283 (1979).
3. Ionkin N. I. and Moiseev E. I., “On the problem for the heat equation with two-point boundary conditions [in Russian],” Differents. Uravn., 15, No. 7, 1284–1295 (1979).
4. Beilin S. A., “Existence of solutions for one-dimentional wave equations with nonlocal conditions,” Electron J. Differ. Equ., 76, 1–8 (2001).
5. Berdyshev A. S., Cabada A., and Kadirkulov B. J., “The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional differential operator,” Comput. Math. Appl., 62, No. 10, 3884–3893 (2011).
6. Kaliev I. A. and Sabitova M. M., “The task of determining the temperature and density of heat sources from the initial and final temperatures (in Russian),” Sib. J. Ind. Math., 12, No. 1, 89–97 (2009).
7. Ismailov M. I. and Kanka F., “An inverse coefficient problems for a parabolic equation in the case of nonlocal boundary and overdetermination conditions,” Math. Methods Appl. Sci., 34, No. 6, 692–702 (2011).
8. Kerimov N. B. and Ismailov M. I., “An inverse coefficient problems for the heat equation in the case of nonlocal boundary conditions,” J. Math. Anal. Appl., 396, 546–554 (2012).
9. Moiseev E. I., “On the solution of a nonlocal boundary value problem by the spectral method,” Differents. Uravn., 35, No. 8, 1105–1112 (1999).
10. Moiseev E. I., “Solvability of a nonlocal boundary value problem,” Differents. Uravn., 37, No. 11, 1643–1646 (2001).
11. Sabitov K. B. and Martemyanova N. V., “Nonlocal inverse problem for a mixed type equation [in Russian],” Izv. Vuzov, Mat., 2, 71–85 (2011).
12. Sabitov K. B. and Martemyanova N. V., “Inverse problem for an elliptic-parabolic type equation with a nonlocal boundary condition [in Russian],” Sib. Math. J., 53, No. 3, 633–647 (2012).
13. Nakhushev A. M., Problems with Shift for Partial Differential Equation [in Russian], Nauka, Moscow (2006).
14. Nakhusheva Z. A., “Samarsky problem for fractal diffusion equations [in Russian],” Mat. Zamet., 95, No. 6, 878–883 (2014).
15. Sadybekov M. A., “Initial-boundary value problem for a heat equation with not strongly regular boundary conditions,” in: Functional Analysis in Interdisciplinary Applications, pp. 330– 348, Springer, Cham (2017) (Springer Proc. Math. Stat.; vol. 216).
16. Kozhanov A. I., “To the question of the solvability of the Ionkin problem for partial differential equations,” Math., 12, 487 (2024).
17. Kozhanov A. I., “Nonlocal problems with the generalized Samarsky–Ionkin condition for some classes of nonstationary differential equations [in Russian],” Dokl. Akad. Nauk, Mat., Inform., Protsessy Upravl. 509, 50–53 (2023).
18. Gupta C. P., “Existence and uniqueness theorems for the bending of an elastic beam equation,” Appl. Anal., 26, 289–304 (1988).
19. Ma T. F., “Existence results for a model of nonlinear beam on elastic bearings,” Appl. Math. Lett., 13, 11–15 (2000).
20. Sorrentino S., Marchesiello S., and Piombo B. A. D., “A new analytical technique for vibration analysis of non-proportionally damped beams,” J. Sound Vib., 265, 765–782 (2003).
21. Kozhanov A. I. and Pinigina N. R., “Boundary-value problems for some higher-order nonclassical differential equations [in Russian],” Mat. Zamet., 101, No. 3–4, 467–474 (2017).
22. Kozhanov A. I. and Abdrakhmanov A. M., “Spatially-nonlocal boundary value problems with the generalized Samarskii–Ionkin condition for quasi-parabolic equations [in Russian],” Sib. Electron. Math. Rep., 20, No. 1, 110–123 (2023).
23. Kozhanov A. I., “Initial-boundary value problems with generalized Samarskii–Ionkin condition for parabolic equations with arbitrary evolution direction,” J. Math. Sci., 274, No. 2, 228–240 (2023).
24. Kozhanov A. I. and Dyuzheva A. V., “Well-posedness of the generalized Samarskii–Ionkin problem for elliptic equations in a cylindrical domain [in Russian],” Differents. Uravn., 59, 230–242 (2023).
Review
For citations:
Kozhanov A.I. On solvability of nonlocal problems with Ionkin conditions for partial differential equations. II. Mathematical notes of NEFU. 2024;31(1):48-55. (In Russ.) https://doi.org/10.25587/2411-9326-2024-1-48-55
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