Home Mathematics On an inverse problem with high-order overdetermination conditions
Article
Licensed
Unlicensed Requires Authentication

On an inverse problem with high-order overdetermination conditions

  • Vagif Abdullayev ORCID logo EMAIL logo
Published/Copyright: February 10, 2025

Abstract

We study the problem of identifying the constant parameters involved in the right-hand sides of a linear non-autonomous system of differential equations with first-order ordinary derivatives. The specificity of the problem lies in the fact that additional conditions for identifying parameters, firstly, are non-local, and secondly, they include derivatives of an unknown function. We examine the conditions for the existence and uniqueness of a solution to the problem and propose two different approaches to the numerical solution of the problem. The results of computer experiments are presented.

MSC 2020: 65L09; 34A55; 34B10

Acknowledgements

The author is grateful to Professor K. R. Aida-zade, Corresponding Member of the National Academy of Sciences of Azerbaijan, for careful reading of the manuscript and helpful remarks.

References

[1] V. M. Abdullaev, Identification of the functions of response to loading for stationary systems, Cybernet. Systems Anal. 53 (2017), no. 3, 417–425. 10.1007/s10559-017-9942-6Search in Google Scholar

[2] V. M. Abdullaev and K. R. Aida-zade, On the numerical solution of loaded systems of ordinary differential equations, Comput. Math. Math. Phys. 44 (2004), no. 9, 1585–1595. Search in Google Scholar

[3] V. M. Abdullaev and K. R. Aida-zade, Numerical solution of problems of the optimal control of loaded lumped systems, Comput. Math. Math. Phys. 46 (2006), no. 9, 1487–1502. 10.1134/S096554250609003XSearch in Google Scholar

[4] V. M. Abdullaev and K. R. Aida-zade, Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations, Comput. Math. Math. Phys. 54 (2014), no. 7, 1096–1109. 10.1134/S0965542514070021Search in Google Scholar

[5] K. R. Aida-zade, Numerical method of identification of dynamic system parameters, J. Inverse Ill-Posed Probl. 13 (2005), no. 3, 201–211. 10.1515/156939405775199550Search in Google Scholar

[6] K. R. Aida-zade and V. M. Abdullaev, Numerical approach to parametric identification of dynamic systems, J. Automat. Inform. Sci. 46 (2014), no. 3, 30–46. 10.1615/JAutomatInfScien.v46.i3.40Search in Google Scholar

[7] K. R. Aida-zade and V. M. Abdullayev, Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions, J. Inverse Ill-Posed Probl. 24 (2016), no. 5, 543–558. 10.1515/jiip-2015-0011Search in Google Scholar

[8] K. R. Aida-zade and V. M. Abdullaev, Optimization of right-hand sides of nonlocal boundary conditions for a controlled dynamical system, Autom. Remote Control 82 (2021), no. 3, 375–397. 10.1134/S0005117921030012Search in Google Scholar

[9] K. R. Aida-zade and V. M. Abdullayev, To the solution of integro-differential equations with nonlocal conditions, Turkish J. Math. 46 (2022), no. 1, 177–188. Search in Google Scholar

[10] A. T. Assanova, A. E. Imanchiyev and Z. M. Kadirbayeva, Numerical solution of systems of loaded ordinary differential equations with multipoint conditions, Comput. Math. Math. Phys. 58 (2018), no. 4, 508–516. 10.1134/S096554251804005XSearch in Google Scholar

[11] E. A. Bakirova, A. T. Assanova and Z. M. Kadirbayeva, A problem with parameter for the integro-differential equations, Math. Model. Anal. 26 (2021), no. 1, 34–54. 10.3846/mma.2021.11977Search in Google Scholar

[12] O. Baysal, A. Hasanov and S. Kumarasamy, Determination of unknown shear force in transverse dynamic force microscopy from measured final data, J. Inverse Ill-Posed Probl. 32 (2024), no. 2, 243–260. 10.1515/jiip-2023-0021Search in Google Scholar

[13] C. J. de la Vallée-Poussin, Sur l’équation différentielle linéare du second ordre. Détermination d’une integrale par deux valeurs assignées. Extension aux équations d’orde n, J. Math. Pures Appl. 8 (1929), no. 9, 125–144. Search in Google Scholar

[14] A. M. Denisov, Introduction to the Theory of Inverse Problem, Lomonosov Moscow State University, Moscow, 1994. Search in Google Scholar

[15] D. S. Dzhumabaev and A. E. Imanchiev, Correct solvability of a linear multipoint boundary value problem, Math. J. 5 (2005), no. 1(15), 30–38, 134. Search in Google Scholar

[16] D. N. Hào, L. T. Thu Giang, S. Kabanikhin and M. Shishlenin, A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation, J. Inverse Ill-Posed Probl. 26 (2018), no. 6, 835–857. 10.1515/jiip-2018-0060Search in Google Scholar

[17] M. I. Ismailov, F. Kanca and D. Lesnic, Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions, Appl. Math. Comput. 218 (2011), no. 8, 4138–4146. 10.1016/j.amc.2011.09.044Search in Google Scholar

[18] M. I. Ivanchov, Methods for Solving Inverse Problems in Mathematical Physics, VNTL, Lviv, 2003. Search in Google Scholar

[19] S. I. Kabanikhin, Inverse and Ill-Posed Problems: Theory and Applications, Walter de Gruyter, Berlin, 2011. 10.1515/9783110224016Search in Google Scholar

[20] A. L. Karchevsky, Computational algorithms for solving the coefficient inverse problem for parabolic equation, Numer. Anal. Appl. 1 (2008), 114–122. 10.1134/S1995423908020031Search in Google Scholar

[21] H. E. Kunze and E. R. Vrscay, Solving inverse problems for ordinary differential equations using the Picard contraction mapping, Inverse Problems 15 (1999), no. 3, 745–770. 10.1088/0266-5611/15/3/308Search in Google Scholar

[22] L. Ling, M. Yamamoto, Y. C. Hon and T. Takeuchi, Identification of source locations in two-dimensional heat equations, Inverse Problems 22 (2006), no. 4, 1289–1305. 10.1088/0266-5611/22/4/011Search in Google Scholar

[23] M. J. Mardanov, R. S. Mammadov, S. Yu. Gasimov and Y. A. Sharifov, Existence and uniqueness results for the first-order non-linear impulsive integro-differential equations with twopoint boundary conditions, Bull. Karaganda Univ. 102 (2021), no. 2, 74–83. 10.31489/2021M2/74-83Search in Google Scholar

[24] A. N. Naimov, M. V. Bystretskii and A. B. Nazimov, Identification of periodic regimes in a dynamic system, Autom. Remote Control 84 (2023), no. 5, 470–475. 10.1134/S0005117923050077Search in Google Scholar

[25] O. B. Nesterenko, Modified projection-iterative method for weakly nonlinear integrodifferential equations with parameters, J. Math. Sci. (N. Y.) 198 (2014), no. 3, 328–335. 10.1007/s10958-014-1793-3Search in Google Scholar

[26] T.-E. Oussaeif and A. Bouziani, An inverse coefficient problem for a parabolic equation under nonlocal boundary and integral overdetermination conditions, Int. J. Partial Differ. Equ. Appl. 2 (2014), no. 3, 38–43. Search in Google Scholar

[27] I. N. Parasidis and E. Providas, An exact solution method for a class of nonlinear loaded difference equations with multipoint boundary conditions, J. Difference Equ. Appl. 24 (2018), no. 10, 1649–1663. 10.1080/10236198.2018.1515928Search in Google Scholar

[28] I. N. Parasidis and E. Providas, Closed-form solutions for some classes of loaded difference equations with initial and nonlocal multipoint conditions, Modern Discrete Mathematics and Analysis, Springer Optim. Appl. 131, Springer, Cham (2018), 363–387. 10.1007/978-3-319-74325-7_19Search in Google Scholar

[29] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Monogr. Textb. Pure Appl. Math. 231, Marcel Dekker, New York, 2000. Search in Google Scholar

[30] E. Rothe, Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben, Math. Ann. 102 (1930), no. 1, 650–670. 10.1007/BF01782368Search in Google Scholar

[31] A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics, Inverse Ill-posed Probl. Ser. 52, Walter de Gruyter, Berlin, 2007. 10.1515/9783110205794Search in Google Scholar

[32] M. Slodička, A parabolic inverse source problem with a dynamical boundary condition, Appl. Math. Comput. 256 (2015), 529–539. 10.1016/j.amc.2015.01.103Search in Google Scholar

[33] Y. D. Tamarkin, Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions, Math. Z. 27 (1928), 1–54. 10.1007/BF01171084Search in Google Scholar

[34] P. N. Vabishchevich and V. I. Vasil’ev, Computational algorithms for solving the coefficient inverse problem for parabolic equations, Inverse Probl. Sci. Eng. 24 (2016), no. 1, 42–59. 10.1080/17415977.2014.993984Search in Google Scholar

[35] F. Yang, H. Guo and X. Li, The method of central difference for the inverse time-dependent heat source problem, Appl. Math. Comput. 218 (2011), no. 7, 3025–3034. 10.1016/j.amc.2011.09.016Search in Google Scholar

[36] S. P. Zubova and E. V. Raetskaya, Algorithm to solve linear multipoint problems of control by the method of cascade decomposition, Autom. Remote Control 78 (2017), no. 7, 1189–1202. 10.1134/S0005117917070025Search in Google Scholar

Received: 2024-05-01
Revised: 2024-12-26
Accepted: 2024-12-26
Published Online: 2025-02-10
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 10.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jiip-2024-0027/pdf?lang=en
Scroll to top button