Nonlinear inverse problems with a stationary unknown element for equations with Dzhrbashyan–Nersesyan derivatives
https://doi.org/10.25587/2411-9326-2024-3-53-72
Abstract
Sufficient conditions for unique solvability in the classical and generalized sense of the inverse problem for a nonlinear equation in a Banach space resolved with respect to the highest fractional derivative of Dzhrbashyan– Nersesyan are obtained. The overdetermination condition of the inverse problem is given by the Stieltjes integral; the lower derivatives are present in the equation non-linearly. The operator by the unknown function in the linear part of the equation is assumed to be bounded or generating an analytical resolving family of the corresponding linear homogeneous equation. Using our previous results for the direct problem for a linear inhomogeneous equation we obtain the main results here by the method of contraction mappings. An example of an inverse problem for a partial differential equation for which the conditions of an abstract theorem are fulfilled is given.
About the Authors
V. E. Fedorov
Chelyabinsk State University, Mathematical Analysis Department
Russian Federation
Russian Federation
Vladimir E. Fedorov
129 Kashirin Brothers Street, Chelyabinsk 454001
M. V. Plekhanova
Chelyabinsk State University, Mathematical Analysis Department
Russian Federation
Russian Federation
Marina V. Plekhanova
129 Kashirin Brothers Street, Chelyabinsk 454001
A. O. Sagimbaeva
Chelyabinsk State University, Mathematical Analysis Department
Russian Federation
Russian Federation
Angelina O. Sagimbaeva
129 Kashirin Brothers Street, Chelyabinsk 454001
References
1. Kozhanov A. I., Composite Type Equations and Inverse Problems, VSP, Utrecht (1999).
2. Prilepko A. I., Orlovsky D. G., and Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York (2000).
3. Belov Yu. Ya., Inverse Problems for Parabolic Equations, VSP, Utrecht (2002).
4. Hasanov Hasanoˇglu A. and Romanov V. G., Introduction to Inverse Problems for Differential Equations, Springer, Cham (2017).
5. Klibanov M. V. and Timonov A. A., Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht; Boston (2004).
6. Ramm A. G., Inverse Problems. Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York (2004).
7. Kabanikhin S. I., Inverse and Ill-Posed Problems: Theory and Applications, Walter de Gruyter, Utrecht (2012).
8. Pyatkov S. G. and Potapkov A. A., “On some classes of coefficient inverse problems of determining thermophysical parameters in layered media [in Russian],” Mat. Zamet. SVFU, 31, No. 2, 31–45 (2024).
9. Samko S. G., Kilbas A. A., and Marichev O. I., Integrals and Derivatives of Fractional Order and Some of Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).
10. Nakhushev A. M., Fractional Calculus and its Application [in Russian], Fizmatlit, Moscow (2003).
11. Pskhu A. V., Partial Differential Equations of Fractional Order [in Russian], Nauka, Moscow (2005).
12. Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).
13. Glushak A. V., “On an inverse problem for an abstract differential equation of fractional order,” Math. Notes, 87, No. 5–6, 654–662 (2010).
14. Orlovskii D. G., “Parameter determination in a differential equation of fractional order with Riemann–Liouville fractional derivative in a Hilbert space,” J. Sib. Fed. Univ., Math. Phys., 8, No. 1, 55–63 (2015).
15. Fedorov V. E. and Ivanova N. D., “Identification problem for degenerate evolution equations of fractional order,” Fract. Calc. Appl. Anal., 20, No. 3, 706–721 (2017).
16. Fedorov V. E. and Nazhimov R. R., “Inverse problems for a class of degenerate evolution equations with Riemann–Liouville derivative,” Fract. Calc. Appl. Anal., 22, No. 2, 271–286 (2019).
17. Orlovsky D. G., “Determination of the parameter of the differential equation of fractional order with the Caputo derivative in Hilbert space,” J. Phys., Conf. Ser., 1205, No. 1, paper ID 012042 (2019).
18. Fedorov V. E. and Kosti´c M., “Identification problem for strongly degenerate evolution equations with the Gerasimov–Caputo derivative,” Differ. Equ., 56, No. 12, 1613–1627 (2020).
19. Fedorov V. E., Nagumanova A. V.,and Avilovich A. S., “A class of inverse problems for evolution equations with the Riemann–Liouville derivative in the sectorial case,” Math. Methods Appl. Sci., 44, No. 15, 11961–11969 (2021).
20. Fedorov V. E., Nagumanova A. V., and Kosti´ c M., “A class of inverse problems for fractional order degenerate evolution equations,” J. Inverse Ill-Posed Probl., 29, No. 2, 173–184 (2021).
21. Kostin A. B. and Piskarev S. I., “Inverse source problem for the abstract fractional differential equation,” J. Inverse Ill-Posed Probl., 29, No. 2, 267–281 (2021).
22. Plekhanova M. V. and Izhberdeeva E. M., “On the well-posedness of an inverse problem for a degenerate evolutionary equation with the Dzhrbashyan–Nersesyan fractional derivative [in Russian],” Itogi Nauki Tekhniki, Sovremen. Mat. Prilozh., Tem. Obzory, 213, 80–88 (2022).
23. Orlovsky D. and Piskarev S., “Inverse problem with final overdetermination for time-fractional differential equation in a Banach space,” J. Inverse Ill-Posed Probl., 30, No. 2, 221–237 (2022).
24. Ashurov R. R. and Fayziev Yu. E., “Inverse problem of determining the order of fractional derivative in the wave equation,” Math. Notes, 110, No. 6, 824–836 (2021).
25. Plekhanova M., Melekhina D., and Fedorov V., “On local unique solvability for a class of nonlinear identification problems,” J. Math. Sci., 281, No. 6, 882–897 2024.
26. Fedorov V. E., Borel L. V., and Ivanova N. D., “Nonlinear inverse problems for one class of equations with Riemann–Liouville derivatives [in Russian],” Zap. Nauchn. Semin. POMI., 519, 264–288 (2022).
27. Fedorov V. E., Ivanova N. D., Borel L. V., and Avilovich A. S., “Nonlinear inverse problems for fractional differential equations with sectorial operators,” Lobachevskii J. Math., 43, No. 11, 3125–3141 (2022).
28. Fedorov V. E., Plekhanova M. V., Ivanova N. D., Shuklina A. F., and Filin N. V., “Nonlinear inverse problems for some equations with fractional derivatives,” Chelyab. Fiz. Mat. Zh., 8, No. 2, 190–202 (2023).
29. Fedorov V. E., Plekhanova M. V., and Melekhina D. V., ‘Nonlinear inverse problems for equations with Dzhrbashyan–Nersesyan derivatives,” Fractal Fract., 7, No. 6, 464 (2023).
30. Fedorov V. E., Plekhanova M. V., and Melekhina D. V., “On local unique solvability for a class of nonlinear identification problems,” Axioms, 12, No. 11, 1013 (2023).
31. Dzhrbashyan M. M. and Nersesyan A. B., “Fractional derivatives and the Cauchy problem for differential equations of fractional order [in Russian],” Izv. Akad. Nauk Armyan. SSR, Mat., 3, No. 1, 3–28 (1968).
32. Fedorov V. E., Plekhanova M. V., and Izhberdeeva E. M., “Initial value problem for linear equations with the Dzhrbashyan–Nersesyan derivative in Banach spaces,” Symmetry, 13, 1058 (2013).
33. Fedorov V. E., Plekhanova M. V., and Izhberdeeva E. M., “Analytic resolving families for equations with the Dzhrbashyan–Nersesyan derivative,” Fractal Fract., 6, 541 (2022).
34. Izhberdeeva E. M., “Compositions of fractional derivatives as the Dzhrbashyan–Nersesyan derivative,” Chelyab. Fiz. Mat. Zh., 9, No. 1, 35–49 (2024).
35. Bajlekova E. G., “Fractional Evolution Equations in Banach Spaces,” PhD Thes., Eindhoven Univ. Technol., Eindhoven (2001).
36. Fedorov V. E. and Avilovich A. S., “A Cauchy type problem for a degenerate equation with the Riemann–Liouville derivative in the sectorial case,” Sib. Math. J., 60, No. 2, 359–372 (2019).
37. Fedorov V. E., Romanova E. A., and Debbouche A., “Analytic in a sector resolving families of operators for degenerate evolution fractional equations,” J. Math. Sci., 228, No. 4, 380–394 (2018).
38. Hessard B., Kazarinov N., and Wen Y., Theory and Applications of the Cycle Birth Bifurcation [in Russian], Mir, Moscow (1985).
Review
For citations:
Fedorov V.E., Plekhanova M.V., Sagimbaeva A.O. Nonlinear inverse problems with a stationary unknown element for equations with Dzhrbashyan–Nersesyan derivatives. Mathematical notes of NEFU. 2024;31(3):53-72. (In Russ.) https://doi.org/10.25587/2411-9326-2024-3-53-72
Views:
7
JATS XML
Email this article 