Geodesic
Inversion formulas for the three-dimensional Volterra integral equation of the first kind with prehistory
The Bulletin of Irkutsk State University. Series Mathematics, Tome 41 (2022), pp. 69-84 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Volterra polynomial of the first kind, method of steps, variable limits of integration, solvability conditions
Mots-clés : inversion formulas. 
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Ekaterina D. Antipina. Inversion formulas for the three-dimensional Volterra integral equation of the first kind with prehistory. The Bulletin of Irkutsk State University. Series Mathematics, Tome 41 (2022), pp. 69-84. http://geodesic.mathdoc.fr/item/IIGUM_2022_41_a4/

[1] 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, 30–34 (in Russian) | DOI

[2] Apartsyn A. S., Nonclassical Volterra equations of the first kind: theory and numerical methods, VSP, Boston–Utrecht, 2003, 168 pp. | MR

[3] 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, Publ. of SEI SB RAS, Irkutsk, 1995, 30 pp. (in Russian)

[4] Volkodavov V. F., Rodionova I. N., “Inversion formulas for some two-dimensional Volterra integral equations of the first kind”, Russian Math., 42:9 (1998), 28–30 | MR | Zbl

[5] El'sgol'ts L.E., Norkin S. B., Introduction into the theory of differential equations with a deviating argument, Nauka Publ, M., 1971, 296 pp. (in Russian) | MR

[6] Cheng C. M., Peng Z. K., Zhang W. M., Meng G., “Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review”, Mechanical Systems and Signal Processing, 87 (2017), 340–364 | DOI

[7] Markova E., Sidler I., Solodusha S., “Integral models based on volterra equations with prehistory and their applications in energy”, Mathematics, 9:10 (2021), 1127 | DOI

[8] 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, 137–147 | DOI | MR

[9] Volterra V., A theory of functionals, integral and integro-differential equations, Dover Publ, New York, 1959, 288 pp. | MR | Zbl