Numerical-Analytic Methods in the Theory of Boundary-Value Problems

by Ronto, M.; Samoilenko, A. M.

• ISBN13: 9789810236762
• ISBN10: 981023676X
• Format: Hardcover
• Copyright: 1999-04-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This book contains the main results of the authors' investigations on the development and application of numerical-analytic methods (NAM) for ordinary nonlinear boundary value problems (BVPs). The methods under consideration provide an opportunity to solve the two important problems of the BVP theory -- namely, to establish existence theorems and to build approximation solutions. They can be used to investigate a wide variety of BVPs.

Table of Contents

Preface	v
Introduction	1	(18)
Numerical-Analytic Method of Successive Approximations for Two-Point Boundary-Value Problems	19	(70)
Abstract Scheme of the Method	19	(7)
Choice of the Form of Successive Approximations and Their Uniform Convergence	26	(15)
Sufficient Conditions for the Existence of Solutions	41	(4)
Necessary Conditions for the Solvability of the Boundary-Value Problem	45	(7)
Error of Calculation of the Initial Value of a Solution	52	(12)
Special Types of Successive Approximations and Estimates	64	(8)
Numerical-Analytic Method in the Case of Nonlinear Two-Point Boundary Conditions	72	(9)
Boundary-Value Problems with Small Parameter	81	(8)
Modification of the Numerical-Analytic Method for Two-Point Boundary-Value Problems	89	(60)
Periodic Boundary-Value Problem	89	(9)
Theorems on the Properties of Determining Functions of a Periodic Boundary-Value Problem	98	(7)
Solvability of the Approximate Determining Equation and the Error of the Initial Value of a Periodic Solution	105	(10)
Modification of the Method for Two-Point Problems	115	(8)
Relationship between Exact and Approximate Determining Equations	123	(8)
Determination of Initial Values of Solutions of Two-Point Boundary-Value Problems	131	(11)
Realization of the Method for Systems of Two Equations	142	(7)
Numerical-Analytic Method for Boundary-Value Problems with Parameters in Boundary Conditions	149	(72)
Successive Approximations for Problems with One Parameter in Linear Boundary Conditions	150	(8)
Sufficient Solvability Conditions and Determination of the Initial Value of a Solution of the Boundary-Value Problem with Parameter	158	(10)
Boundary-Value Problems with Nonfixed Right Boundary	168	(9)
Solvability Theorems for Problems with Nonfixed Right Boundary	177	(16)
The Case of Several Parameters in Boundary Conditions	193	(16)
Linear Dependence of Boundary Conditions on Two Parameters	209	(12)
Collocation Method for Boundary-Value Problems with Impulses	221	(96)
Green Function of a Homogeneous Two-Point Boundary-Value Problem	222	(7)
Inhomogeneous Linear Impulsive Boundary-Value Problems	229	(5)
Convergence of the Algebraic Collocation Method for Nonlinear Systems	234	(11)
Method of Trigonometric Collocation for Periodic Systems	245	(5)
Practical Solution of Impulsive Problems	250	(7)
Green Function for a Three-Point Boundary-Value Problem with Single Impulse Influence	257	(14)
Semihomogeneous Linear Two-Point Boundary-Value Problem with m-Impulse Influence	271	(8)
Inhomogeneous Linear m-Impulse Two-Point Boundary-Value Problem	279	(1)
Multipoint m-Impulse Boundary-Value Problem	280	(6)
Equations with Piecewise-Continuous Right-Hand Side	286	(22)
Construction of Solutions of Two-Impulse Systems	308	(9)
Appendix. THE THEORY OF THE NUMERICAL-ANALYTIC METHOD: ACHIEVEMENTS AND NEW TRENDS OF DEVELOPMENT	317	(98)
A1. History of the Method of Periodic Successive Approximations	317	(12)
A2. Relation to Other Investigations	329	(10)
A3. Application of Modifications of the Numerical-Analytic Method to the Investigation of Various Boundary-Value Problems	339	(76)
References	415