Numerical-analytic methods in the theory of boundary-value problems

462 Pages·2000·11.627 MB·English

by  N I Ronto, A M Samoĭlenko

Numerical-Analytic
Methods in
the Theory of
Boundary-Value
Problems
This page is intentionally left blank
Numerical-Analytic
Methods in
the Theory of
Boundary-Value
Problems
M. Ronto
University of Miskolc, Hungary
A. M. Samoilenko
National Academy of Sciences, Ukraine
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NUMERICAL-ANALYTIC METHODS IN THE THEORY OF
BOUNDARY-VALUE PROBLEMS
Copyright m 2000 by WorldScientific PublishingCo. Pte. Ltd.
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PREFACE
Formulation and development of constructive methods is one of new directions in the
contemporary mathematical analysis and simulation. In spite of the fact that investiga
tions in this field were carried out for only several decades, the class of constructive
mathematical methods draws more and more attention. It is likely that there is no ac
cepted definition which would strictly restrict the class of constructive methods. Never
theless, this term becomes more and more common. Apparently, the most natural way is
to regard constructive methods as certain methods for the construction of solutions of dif
ferent classes of equations and investigation of the existence and properties of exact and
approximate solutions. Furthermore, the main characteristic of constructive methods is
the fact that they enable one to completely solve the problem (up to the numerical values)
and practically verify the theoretical background and conditions that guarantee the appli
cability of these methods to specific classes of problems.
Apparently, for the first time, the constructive side of methods attracted attention in
the fields of mathematics that are now known as the theory of nonlinear oscillations and
nonlinear mechanics. It is widely recognized that asymptotic methods using different
averaging schemes became, in fact, one of the main tools of constructive investigation
and construction of solutions of various problems of nonlinear mechanics. The mono
graph [GrR3], for example, convincingly points to this fact. In the monograph men
tioned, the constructivity of methods is investigated with the use of asymptotic decompo
sitions, the averaging principle, various iterative versions of the method of Lyapunov-
Poincare" series in powers of a small parameter, and the iteration method with accelerated
convergence. These investigations are mainly aimed at studying periodic and quasiperi-
odic solutions.
It is clear that there is an urgent need in developing constructive methods for other
branches of the theory of differential equations, in particular, for the theory of boundary-
value problems for ordinary differential equations. The analysis of the contemporary
state of methods for the investigation of boundary-value problems convincingly shows
that the classes of analytic, functional-analytic, numerical, and numerical-analytic meth
ods are most often used in this field. Obviously, each group of these methods has ad
vantages and disadvantages. However, it should be noted that, in the theory of boundary-
value problems, the numerical-analytic methods compare favorably with other methods
by their constructivity both at the stage of construction of solutions and in the course of
v
vi Preface
investigation of the principal qualitative problems such as establishing the existence of
solutions, the verification of convergence of approximate solutions to exact solutions, and
obtaining error estimates for approximate solutions that can be practically verified. All
problems indicated above demonstrate that the numerical-analytic methods open good
prospects of further development of constructive methods for the investigation of solu-
tions of boundary-value problems for ordinary differential equations.
I am pleased to introduce this book to readers. It is written by the well-known experts
in the field of numerical-analytic methods and is a natural continuation of their works
[SaRI], [SaR2], and [SaR4].
The present monograph is characterized by the perfect combination of the most estab-
lished elements of the theory of the numerical-analytic method of successive approxima-
tions, which has proved to be very efficient; new directions of research are also described.
In view of the small volume of this monograph, the authors consider a narrow range of
problems concerning only the generalization of nonlinear, boundary-value problems for
ordinary differential equations to new classes.
One of the advantages of this work is the fact that the authors demonstrate the possi-
bility of application of the numerical-analytic schemes under consideration not only to
classic boundary-value problems, but also to nonstandard boundary-value problems, e.g.,
boundary-value problems with parameters in boundary conditions or pulse influence.
Another distinctive feature of this book is the development of the idea of possibility and
expediency of combining various numerical-analytic methods for the investigation of pe-
riodic solutions and solutions of nonlinear boundary-value problems of the general form.
Clearly, this monograph contains an interesting and necessary material, which will be
interesting for experts in the theory of boundary-value problems and nonlinear oscilla-
tions and will contribute to the further development of constructive numerical-analytic
methods for the investigation of periodic boundary-value problems and boundary-value
problems of general form.
Academician Yu. A. Mitropolsky
comm
Preface v
Introduction 1
Chapter 1. NUMERICAL-ANALYTIC METHOD OF SUCCESSIVE
APPROXIMATIONS FOR TWO-POINT
BOUNDARY-VALUEPROBLEMS 19
1. Abstract Scheme of the Method 19
2. Choice of the Form of Successive Approximations and Their
Uniform Convergence 26
3. Sufficient Conditions for the Existence of Solutions 41
4. Necessary Conditions for the Solvability of the Boundary-Value
Problem 45
5. Error of Calculation of the Initial Value of a Solution 52
6. Special Types of Successive Approximations and Estimates 64
7. Numerical-Analytic Method in the Case of Nonlinear Two-Point
Boundary Conditions 72
8. Boundary-Value Problems with Small Parameter 81
Chapter2. MODIFICATION OF THE NUMERICAL-ANALYTIC
METHODFOR TWO-POINT BOUNDARY-VALUE
PROBLEMS 89
9. Periodic Boundary-Value Problem 89
10. Theorems on the Properties of Determining Functions of
a Periodic Boundary-Value Problem 98
11. Solvability of the Approximate Determining Equation and
the Error of the Initial Value of a Periodic Solution 105
vu
viii Contents
12. Modification of the Method for Two-Point Problems 115
13. Relationship between Exact and Approximate Determining
Equations 123
14. Determination of Initial Values of Solutions of Two-Point
Boundary-Value Problems 131
15. Realization of the Method for Systems of Two Equations 142
Chapter 3. NUMERICAL-ANALYTIC METHOD FOR
BOUNDARY-VALUE PROBLEMS WITH PARAMETERS
IN BOUNDARY CONDITIONS 149
16. Successive Approximations for Problems with One Parameter
in Linear Boundary Conditions 150
17. Sufficient Solvability Conditions and Determination of the Initial
Value of a Solution of the Boundary-Value Problem with Parameter 158
18. Boundary-Value Problems with Nonfixed Right Boundary 168
19. Solvability Theorems for Problems with Nonfixed Right Boundary 177
20. The Case of Several Parameters in Boundary Conditions 193
21. Linear Dependence of Boundary Conditions on Two Parameters 209
Chapter4. COLLOCATIONMETHOD FOR BOUNDARY-VALUE
PROBLEMS WITH IMPULSES 221
22. Green Function of a Homogeneous Two-Point Boundary-Value
Problem 222
23. Inhomogeneous Linear Impulsive Boundary-Value Problems 229
24. Convergence of the Algebraic Collocation Method for Nonlinear
Systems 234
25. Method of Trigonometric Collocation for Periodic Systems 245
26. Practical Solution of Impulsive Problems 250
27. Green Function for a Three-Point Boundary-Value Problem
with Single Impulse Influence 257
28. Semihomogeneous Linear Two-Point Boundary-Value Problem
with m-Impulse Influence 271
Contents ix
29. Inhomogeneous Linear m-Impulse Two-Point Boundary-Value
Problem 279
30. Multipoint m-Impulse Boundary-Value Problem 280
31. Equations with Piecewise-Continuous Right-Hand Side 286
32. Construction of Solutions of Two-Impulse Systems 308
Appendix. THE THEORY OF THE NUMERICAL-ANALYTIC
METHOD: ACHIEVEMENTS AND NEW TRENDS
OF DEVELOPMENT 317
Al. History of the Method of Periodic Successive Approximations 317
A2. Relation to Other Investigations 329
A3. Application of Modifications of the Numerical-Analytic Method
to the Investigation of Various Boundary-Value Problems 339
REFERENCES 415