The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterra equation that appears in the integral models of developing systems. The Volterra integral equations of the first kind with piecewise smooth kernels are considered. Illustrative examples are presented.
The paper presents a review of the studies that were conducted at Energy Systems Institute (ESI) SB RAS in the field of mathematical modeling of nonlinear input-output dynamic systems with Volterra polynomials. The first part presents an original approach to identification of the Volterra kernels. The approach is based on setting special multi-parameter families of piecewise constant test input signals. It also includes a description of the respective software; presents illustrative calculations on the example of a reference dynamic system as well as results of computer modeling of real heat exchange processes. The second part of the review is devoted to the Volterra polynomial equations of the first kind. Studies of such equations were pioneered and have been carried out in the past decade by the laboratory of ill-posed problems at ESI SB RAS. A special focus in the paper is made on the importance of the Lambert function for the theory of these equations.
In this paper the method of obtaining unimprovable (in certain sense) estimates of solutions of some integral inequalities with the operators of Volterra type is stated. The basis of this method is the theory of monotone operators in partially ordered Banach spaces. This theory allows us to reduce obtaining these estimates to solving corresponding equations. The paper consists of two parts. The first part is devoted to unimprovable estimates of solutions for linear multidimensional inequalities. In the second part the author states nonlinear inequalities which arise while researching multilinear Volterra equations of the first kind connected with modelling nonlinear dynamic systems of black body type by Volterra polynomials.
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