Integro-differential equations and functional analysis

Inversion formulas for the three-dimensional Volterra integral equation of the first kind with prehistory

Ekaterina D. Antipina<sup>ab

a</sup> Irkutsk State University, Irkutsk, Russian Federation
^b Melentiev 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.

Keywords: Volterra polynomial of the first kind, method of steps, variable limits of integration, solvability conditions, inversion formulas.

UDC: 517.968

MSC: 45D05

Received: 28.03.2022
Revised: 12.07.2022
Accepted: 22.08.2022

DOI: 10.26516/1997-7670.2022.41.69

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