Nonclassical Linear Volterra Equations of the First Kind

By A.S. Apartsyn

November 2003

VSP

ISBN: 90-6764-375-0

172 pages, Illustrated, 6 1/2" x 9 3/4"

$111.00 Hardcover

This monograph in the Inverse and Ill-Posed Problems Series deals with linear integral Volterra equations of the first kind with variable upper and lower limits of integration. Volterra operators of this type are the basic operators for integral models of dynamic systems. The corresponding conventionally well-posed problems were subject of systematic study: existence and uniqueness theorems were proved, estimates of the inverse operator norms for suitable pairs of functional spaces were obtained, etc. Consideration was also given to quadrature methods for the numerical solution of such equations with emphasis on their specific features, as opposed to classical algorithms. Results of computations for the modelled examples are presented. This book is intended for specialists in computer mathematics, inverse and ill-posed problems, and mathematical modelling.

Contents: CLASSICAL VOLTERRA EQUATIONS OF THE FIRST KIND Classification of integral Volterra equations of the first kind The Gronwall-Bellman lemma A difference analog of the Gronwall-Bellman lemma Self-regularization Two-parametric -regularization Inequalities with isotone operators Inequalities with interchangeable isotone operators Unimprovable estimates of solutions of multidimensional integral inequalities The well-posedness of a two-dimensional Volterra Equation of the first kind Unimprovable estimates of solutions for two-dimensional difference inequalities VOLTERRA EQUATIONS OF THE FIRST KIND WITH TWO VARIABLE INTEGRATION LIMITS. THE CASE A(T0) < T0 Problem statement The method of steps Illustrative examples The existence and uniqueness theorem An estimate of the solution stability The study of a special problem of mathematical programming A numerical solution of the test example A geometrical illustration of the reduction by unity in the order of convergence A theorem on the convergence of the quadrature method (the general case) Some numerical results On self-regularization VOLTERRA EQUATIONS OF THE FIRST KIND WITH TWO VARIABLE LIMITS OF INTEGRATION. THE CASE A(T0) = T0 Problem statement Solution of the simplest test equation Existence and uniqueness theorem (the general case) Estimation of the solution stability Some generalizations of the Gronwall-Bellman inequality Numerical solution of a test example The proof of convergence for the quadrature method (the general case) Some numerical results Self-regularization (the case of a disturbance in the right-hand side) Stability of a numerical solution with resepect to disturbances of Multidimensional Volterra equations of the first kind related to the modelling of nonlinear dynamic systems using the Volterra series Bibliography Index

Mathematics

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