Inversion formulas and their finite-dimensional analogs for multidimensional Volterra equations of the first kind

Abstract

The paper focuses on solving one class of Volterra equations of the first kind, which is characterized by the variability of all integration limits. These equations were introduced in connection with the problem of identifying nonsymmetric kernels for constructing integral models of nonlinear dynamical systems of the “input-output” type in the form of Volterra polynomials. The case when the input perturbation of the system is a vector function of time is considered. To solve the identification problem, previously introduced test signals of duration h (mesh step) are used in the form of linear combinations of Heaviside functions with deviating arguments. The paper demonstrates a method for obtaining the desired solution, developing a method of steps for a one-dimensional case. The matching conditions providing the desired smoothness of the solution are established. The mesh analogs of the studied integral equations based on the formulas of middle rectangles are considered.

Integral Models Based on Volterra Equations with Prehistory and Their Applications in Energy

“…The method for obtaining (21) and ( 22) is described in detail in [49] and is based on the classical method of steps [50], which has proven itself in solving one-dimensional Volterra equations of the first kind with prehistory [26]. For convenience, we rewrite (18) in operator form, using the change of variables (11): 𝑠 0 = 𝑡, 𝑠 1 = 𝑡 − 𝜈, so that, taking into account The method for obtaining (21) and ( 22) is described in detail in [49] and is based on the classical method of steps [50], which has proven itself in solving one-dimensional Volterra equations of the first kind with prehistory [26]. For convenience, we rewrite (18) in operator form, using the change of variables (11):…”

“…Let 𝜑(𝚳) ∈ 𝐶 Ω be a solution to (24), (19), (20). Then By analogy, it is easy to clarify the boundaries between the domains Ω k for other values of k. The proof is not particularly difficult, it can be carried out according to the scheme given in [49], and follows from geometric considerations.…”

“…define a continuous on Ω solution ϕ of the pair Equation ( 24) with prehistory ( 19) and (20). Indeed, the continuity of ϕ on Ω follows from ( 27)-( 29), (32) [49]. Considering that s 0 ∈ [h, T], s 1 ≥ h, then we should consider the situations when s 0 = s 1 and s = s 0 = s 1 separately, since in the second case V 1,2 ϕ = V 2,1 ϕ and so…”