List of issues > Series «Mathematics». 2022. Vol 41
Inversion Formulas for the Three-Dimensional Volterra
Integral Equation of the First Kind with Prehistory
Author(s)
Ekaterina D. Antipina1,2
1Irkutsk State University, Irkutsk, Russian Federation
2Melentiev Energy Systems Institute SB RAS, Irkutsk, Russian Federation
Abstract
The article is devoted to solving one class of Volterra equations of the first
kind with variable upper and lower limits. These equations were introduced in connection
with the problem of identifying asymmetric kernels for constructing integral models of
nonlinear dynamical systems of ”input-output” type in the form of Volterra polynomials.
To solve the identification problem, previously introduced test signals with duration h
(grid sampling step) are used in the form of a linear combination of Heaviside functions.
The article demonstrates a method for obtaining the desired solution, which develops
the step method for the one-dimensional case. Matching conditions are established that
ensure the desired smoothness of the solution.
About the Authors
Ekaterina D. Antipina, Irkutsk
State University, Irkutsk, 664003,
Russian Federation; Melentiev Energy
Systems Institute SB RAS, Irkutsk,
664033, Russian Federation,
kate19961231@gmail.com
For citation
Antipina E. D. Inversion Formulas for the Three-Dimensional Volterra
Integral Equation of the First Kind with Prehistory. The Bulletin of Irkutsk State
University. Series Mathematics, 2022, vol. 41, pp. 69–84. (in Russian)
https://doi.org/10.26516/1997-7670.2022.41.69
Keywords
Volterra polynomial of the first kind, method of steps, variable limits of
integration, solvability conditions, inversion formulas
UDC
517.968
MSC
45D05
DOI
https://doi.org/10.26516/1997-7670.2022.41.69
References
- Abas W.M.A., Harutyunyan R.V. Analysis and optimization of nonlinear systems with memory based on Volterra integro-functional series and Monte-Carlo methods. Izvestiya Vuzov. Severo-Kavkazskiy Region. Tekhnicheskie Nauki, 2021, no. 3, pp. 30–34. https://doi.org/10.17213/1560-3644-2021-3-30-34 (in Russian)
- Apartsyn A.S. Nonclassical Volterra equations of the first kind: theory and numerical methods. Boston, Utrecht, VSP, 2003, 168 p.
- Apartsyn A.S. Existences and uniqueness’ theorems of the solutions of the Volterra equations of the I kind, non-linear dynamic systems connected to identification (scalar case), Preprint no. 9. Irkutsk, Publ. of SEI SB RAS, 1995, 30 p. (in Russian)
- Volkodavov V.F., Rodionova I.N. Inversion formulas for some two-dimensional Volterra integral equations of the first kind. Russian Math., 1998, vol. 42, no. 9, pp. 28–30.
- El’sgol’ts L.E., Norkin S.B. Introduction into the theory of differential equations with a deviating argument. Moscow, Nauka Publ., 1971, 296 p.(in Russian)
- Cheng C. M., Peng Z. K., Zhang W. M., Meng G. Volterra-series-based nonlin-ear system modeling and its engineering applications: A state-of-the-art review. Mechanical Systems and Signal Processing, 2017, vol. 87, pp. 340–364. https://doi.org/10.1016/j.ymssp.2016.10.029
- Markova E., Sidler I., Solodusha S. Integral models based on volterra equations with prehistory and their applications in energy. Mathematics, 2021, vol. 9, no. 10, pp. 1127. https://doi.org/10.1007/978-3-030-87966-2_16
- Solodusha S.V. Identification of Integral Models of Nonlinear Multi-Input Dynamic Systems Using the Product Integration Method. Stability and Control Processes. SCP 2020. LNCIS - Proceedings. Springer, Cham, 2022, pp. 137–147.https://doi.org/10.1007/978-3-030-87966-2_16
- Volterra V. A theory of functionals, integral and integro-differential equations. New York, Dover Publ., 1959, 288 p