Approximation And Regularisation Methods For Operator Functional Equations
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Approximation And Regularisation Methods For Operator-functional Equations
This book presents an overview of the most recent research and findings in the field of approximation and regularisation methods for operator-functional equations, and explores their applications in electrical and power engineering. It presents the state of the art in building operator theory, regularised numerical methods, and the verification of mathematical models for dynamical models based on integral and differential equations. Special attention is paid to Volterra models, a powerful tool for modelling hereditary dynamics.This book begins by exploring the solvability of singular integral equations and moves on to study approximation methods for linear operator equations and nonlinear integral equations. Following this, it examines loaded equations and bifurcation analysis, before concluding with an investigation of the applications of the contents of the book in electrical engineering and automation. Each chapter provides an overview and analysis of the relevant problem statements, outlines current methods within the field, and identifies future directions for research.With an interdisciplinary approach, this book is essential reading for anyone interested in operator-functional equations. Graduate students and professors in the fields of applied mathematics, physics, materials science, and numerical analysis will find this work insightful and valuable, as will industry professionals in related fields.
Linear Operator Equations: Approximation And Regularization
Many problems in science and engineering have their mathematical formulation as an operator equation Tx=y, where T is a linear or nonlinear operator between certain function spaces. In practice, such equations are solved approximately using numerical methods, as their exact solution may not often be possible or may not be worth looking for due to physical constraints. In such situations, it is desirable to know how the so-called approximate solution approximates the exact solution, and what the error involved in such procedures would be.This book is concerned with the investigation of the above theoretical issues related to approximately solving linear operator equations. The main tools used for this purpose are basic results from functional analysis and some rudimentary ideas from numerical analysis. To make this book more accessible to readers, no in-depth knowledge on these disciplines is assumed for reading this book.
Pseudosolution of Linear Functional Equations
Author: Alexander S. Mechenov
language: en
Publisher: Springer Science & Business Media
Release Date: 2005-03-21
In the book there are introduced models and methods of construction of pseudo-solutions for the well-posed and ill-posed linear functional equations circumscribing models passive, active and complicated experiments. Two types of the functional equations are considered: systems of the linear algebraic equations and linear integral equations. Methods of construction of pseudos6lutions are developed in the presence of passive right-hand side errors for two types of operator errors: passive measurements and active representation errors of the operator, and all their combinations. For the determined and stochastic models of passive experiments the method of the least distances of construction of pseudosolutions is created, the maximum likelihood method of construction of pseudosolutions is applied for active experiments, and then methods for combinations of models of regression, of passive and of active experiments are created. We have constructed regularized variants of these methods for systems of the linear algebraic equations with the degenerated matrices and for linear integral equations of the first kind. In pure mathematics, the solution techniques of the functional equations with exact input data more often are studied. In applied mathematics, problem consists in construction of pseudosolutions, that is, solution of the hctional equations with perturbed input data. Such problem in many cases is incomparably more complicated. The book is devoted to a problem of construction of a pseudosolution (the problem of a parameter estimation) in the following fundamental sections of applied mathematics: confluent models passive, active and the every possible mixed experiments.