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

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.

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A convergence theorem for singular integral equations

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s033427000000285x ◽

1981 ◽

Vol 22 (4) ◽

pp. 539-552 ◽

Cited By ~ 10

Author(s):

David Elliott

Keyword(s):

Integral Equation ◽

Singular Integral Equation ◽

Integral Equations ◽

Singular Integral ◽

Sufficient Conditions ◽

Singular Integral Equations ◽

Convergence Theory ◽

Cauchy Kernel ◽

Algebraic Equations ◽

Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

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Markov Jump Linear System Analysis of a Microgrid Operating in Islanded and Grid Tied Modes

10.36227/techrxiv.13490496 ◽

2020 ◽

Author(s):

Gilles Mpembele ◽

Jonathan Kimball

Keyword(s):

Monte Carlo ◽

Power Systems ◽

Power System ◽

System Analysis ◽

Differential Algebraic Equations ◽

Algebraic Equations ◽

Markov Jump ◽

Numerical Application ◽

Conditional Moments ◽

Markov Jump Linear Systems

<div>The analysis of power system dynamics is usually conducted using traditional models based on the standard nonlinear differential algebraic equations (DAEs). In general, solutions to these equations can be obtained using numerical methods such as the Monte Carlo simulations. The use of methods based on the Stochastic Hybrid System (SHS) framework for power systems subject to stochastic behavior is relatively new. These methods have been successfully applied to power systems subjected to</div><div>stochastic inputs. This study discusses a class of SHSs referred to as Markov Jump Linear Systems (MJLSs), in which the entire dynamic system is jumping between distinct operating points, with different local small-signal dynamics. The numerical application is based on the analysis of the IEEE 37-bus power system switching between grid-tied and standalone operating modes. The Ordinary Differential Equations (ODEs) representing the evolution of the conditional moments are derived and a matrix representation of the system is developed. Results are compared to the averaged Monte Carlo simulation. The MJLS approach was found to have a key advantage of being far less computational expensive.</div>

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Existence of Solutions of Nonlinear Stochastic Volterra Fredholm Integral Equations of Mixed Type

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2010/603819 ◽

2010 ◽

Vol 2010 ◽

pp. 1-16 ◽

Cited By ~ 1

Author(s):

K. Balachandran ◽

J.-H. Kim

Keyword(s):

Integral Equations ◽

Existence And Uniqueness ◽

Mixed Type ◽

Fixed Point Theorems ◽

Existence Of Solutions ◽

Stochastic Integral ◽

Sufficient Conditions ◽

Fredholm Integral Equations ◽

Equations Of Mixed Type ◽

Stochastic Integral Equations

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

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Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets

Mathematical Problems in Engineering ◽

10.1155/mpe.2005.113 ◽

2005 ◽

Vol 2005 (1) ◽

pp. 113-121 ◽

Cited By ~ 21

Author(s):

M. Lakestani ◽

M. Razzaghi ◽

M. Dehghan

Keyword(s):

Integral Equations ◽

Algebraic Equations ◽

B Spline ◽

Compactly Supported ◽

Hammerstein Integral Equations ◽

Spline Wavelets

Compactly supported linear semiorthogonal B-spline wavelets together with their dual wavelets are developed to approximate the solutions of nonlinear Fredholm-Hammerstein integral equations. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example.

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Sufficient Conditions for Robust Frequency Stability of AC Power Systems

IEEE Transactions on Power Systems ◽

10.1109/tpwrs.2020.3039832 ◽

2020 ◽

pp. 1-1

Author(s):

Erick Fernando Alves ◽

Gilbert Bergna ◽

Danilo Iglesias Brandao ◽

Elisabetta Tedeschi

Keyword(s):

Power Systems ◽

Sufficient Conditions ◽

Frequency Stability ◽

Ac Power

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On the Almost Sure Stability of Linear Stochastic Distributed-Parameter Dynamical Systems

Journal of Applied Mechanics ◽

10.1115/1.3624976 ◽

1966 ◽

Vol 33 (1) ◽

pp. 182-186 ◽

Cited By ~ 17

Author(s):

P. K. C. Wang

Keyword(s):

Dynamical Systems ◽

Asymptotic Stability ◽

Integral Equations ◽

Sufficient Conditions ◽

Distributed Parameter ◽

Almost Sure Stability ◽

Partial Differential ◽

Stochastic Parameters

In this paper, sufficient conditions for almost sure stability and asymptotic stability of certain classes of linear stochastic distributed-parameter dynamical systems are derived. These systems are described by a set of linear partial differential or differential-integral equations with stochastic parameters. Various examples are given to illustrate the application of the main results.

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Dynamic near-field microwave diagnostics of lungs.

Journal of Radio Electronics ◽

10.30898/1684-1719.2021.8.16 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

L.A. Bokeria ◽

◽

T.T. Kakuchaya ◽

A.M. Kuular ◽

Ye.S. Maksimovitch ◽

...

Keyword(s):

Numerical Simulation ◽

Inverse Problems ◽

Integral Equations ◽

Near Field ◽

Experimental Studies ◽

Heart Activity ◽

Blood Content ◽

Air Content ◽

Microwave Diagnostics ◽

Biomedical Diagnostics

Results of theoretical and experimental studies of the method of the near-field microwave tomography of the thorax are presented. Integral equations of inverse tomography problem of 3D blood- and air content inhomogeneities by data of multisensory measurements are obtained. Methods of air and blood content profiling in processes of breathing and heart activity by data of bistatic measurements of the scattered signal are proposed and solving algorithms of inverse problems are studied in the numerical simulation. Multifrequency and pulse measurements of scattered signals are carried out in processes of cardiorespiratory activity. By data of bistatic measurements of scattered signals parameters from the thorax, profiling relative air- and blood content profiles has been realized. Application possibilities of the method in the biomedical diagnostics are considered.

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On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽

10.3390/axioms10030234 ◽

2021 ◽

Vol 10 (3) ◽

pp. 234

Author(s):

Vladimir Vasilyev ◽

Nikolai Eberlein

Keyword(s):

General Solution ◽

Differential Equations ◽

Integral Equations ◽

Mellin Transform ◽

Algebraic Equations ◽

Conjugation Problem ◽

Solvability Conditions ◽

Linear Algebraic Equations ◽

Linear Integral ◽

Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

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