Integral Equations Related to Volterra Series and Inverse Problems: Elements of Theory and Applications in Heat Power Engineering

Abstract

The paper considers two types of Volterra integral equations of the first kind, arising in the study of inverse problems of the dynamics of controlled heat power systems. The main focus of the work is aimed at studying the specifics of the classes of Volterra equations of the first kind that arise when describing nonlinear dynamics using the apparatus of Volterra integro-power series. The subject area of the research is represented by a simulation model of a heat exchange unit element, which describes the change in enthalpy with arbitrary changes in fluid flow and heat supply. The numerical results of solving the problem of identification of transient characteristics are presented. They illustrate the fundamental importance of practical recommendations based on sufficient conditions for the solvability of linear multidimensional Volterra equations of the first kind. A new class of nonlinear systems of integro-algebraic equations of the first kind, related to the problem of automatic control of technical objects with vector inputs and outputs, is distinguished. For such systems, sufficient conditions are given for the existence of a unique, sufficiently smooth solution. A review of the literature on these problem types is given.

A Numerical Method for Computing Double Integrals with Variable Upper Limits

“…The change in the number of nodes in Table 4 is described by the decreasing function n(τ) = 10 4 / 3 √ τ, and then 7 demonstrates comparative graphs of absolute and relative errors in numerical integration by Formula (15) of the case of A, ( 16) of the case of B, (17) of the cases of C1 for increasing function n(τ) and C2 for decreasing n(τ). 1 shows the corresponding values for the case when only the size of the imposed grid 2.…”

“…General overviews to the problems of numerical approximation of multiple integrals are contained in [13,16]. Encyclopedic work on multiple numerical integrations was published by Arthur H. Stroud [17]. Cubature formulas have been developed for different regions of integration, namely cubature formulas for the cube [17,18], for triangle [18,19], for the sphere [20], for the space with weight function exp(−r 2 ) [16,21], for the space with weight function exp(−r) [16,22] and for the simplex [23,24].…”

“…Encyclopedic work on multiple numerical integrations was published by Arthur H. Stroud [17]. Cubature formulas have been developed for different regions of integration, namely cubature formulas for the cube [17,18], for triangle [18,19], for the sphere [20], for the space with weight function exp(−r 2 ) [16,21], for the space with weight function exp(−r) [16,22] and for the simplex [23,24]. In addition, we can mention the class of Monte Carlo methods, which has been extended by S. M. Ulam and John von Neumann [25,26].…”

Block-by-block method for solving non-linear Volterra integral equation of the first kind

“…2021 ; Markova et al. 2021 ), engineering Solodusha and Bulatov ( 2021 ), in studies of the evolution of the rapid spread and mutation of Corona virus (Gao et al. 2022 ; Noeiaghdam et al.…”

Optimization of Sliding Mode Control to Save Energy in a SCARA Robot

“…However, one of the major drawbacks in the control of robots with SMC is the use of big forces in the reaching law that is required to move the robots. During industrial operations, the big forces in the electric actuator control of the robots can result in spending large amounts of energy [2].…”