Approximate solutions of operator equations

353 Pages·1997·21.51 MB·English

by  Mingjun Chen

APPROXIMATE
SOLUTIONS OF
OPERATOR
EQUATIONS
APPROXIMATIONS AND DECOMPOSITIONS
Editor-in-Chief: CHARLES K. CHUI
Vol. 1: Wavelets: An Elementary Treatment of Theory and Applications
Tom H. Koornwinder, ed.
Vol. 2: Approximate Kalman Filtering
Guanrong Chen, ed.
Vol. 3: Multivariate Approximation: From CAGD to Wavelets
Kurt Jetter and Florencio I. Utreras, eds.
Vol. 4: Advances in Computational Mathematics: New Delhi, India
H. P. Dikshit and C. A. Micchelli, eds.
Vol. 5: Computational Methods and Function Theory
Proceedings of CMFT '94 Conference, Penang, Malaysia
R. M. AH, St. Ruscheweyh and E. B. Saff, eds.
Vol. 6: Approximation Theory VIII
Approximation and Interpolation - Vol. 1
Wavelets and Multi-level Approximation - Vol. 2
C. K. Chui and L. L. Schumaker, eds.
Vol. 7: Introduction to the Theory of Weighted Polynomial Approximation
H. N. Mhaskar
Vol. 8: Advanced Topics in Multivariate Approximation
F. Fontanella, K. Jetter and P. J. Laurent, eds.
Vol. 9: Approximate Solutions of Operator Equations
M. J. Chen, Z. Y. Chen and G. R. Chen
Series in Approximations and Decompositions - Vol. 9
APPROXIMATE
SOLUTIONS OF
OPERATOR
EQUATIONS
Mingjun Chen
Department of Computer Science
Zhongshan University, P P China
Zhongying Chen
Department of Computer Science
Zhongshan University, P P China
Guanrong Chen
Department of Electrical Engineering
University of Houston, ,SA
World Scientific
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Library of Congress Cataloging-in-Publication Data
Chen, Mingjun, 1934-
Approximate solutions of operator equations / by Mingjun Chen,
Zhongying Chen, Guanrong Chen.
p. cm. — (Series in approximations and decompositions; vol. 9)
Includes bibliographical references and index.
ISBN 9810230648 (alk. paper)
1. Operator equations ~ Numerical solutions. 2. Approximation
theory. I. Chen, Zhongying, 1946- II. Chen, G. (Guanrong)
III. Title. IV. Series.
QA329.C475 1997
515'.724--dc21 97-6029
CIP
British Library Cataloguing-in-Publication Data
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Copyright © 1997 by World Scientific Publishing Co. Pte. Ltd.
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Approximations and Decompositions
During the past decade, Approximation Theory has reached out to encompass
the approximation-theoretic and computational aspects of several exciting
areas in applied mathematics such as wavelets, fractals, neural networks,
and computer-aided-geometric design, as well as the modern mathematical
development in science and technology. The objective of this book series
is to capture this exciting development in the form of monographs, lecture
notes, reprint volumes, text books, edited review volumes, and conference
proceedings.
Approximate Solutions of Operator Equations, the 8th volume of this
series, represents one of the computational aspects of Approximation Theory.
It emphasizes on efficient algorithms for numerical solutions of differential and
integral equations, including various linear and nonlinear as well as evolution
equations. The series editor would like to thank the authors for their very
fine contributions to this series.
World Scientific Series in
APPROXIMATIONS AND DECOMPOSITIONS
Editor-in-Chief: CHARLES K. CHUI
Texas A&M University, College Station, Texas
TThhiiss ppaaggee is intentionally left blank
Preface
The rapid development in mathematical analysis of complex real-world phys­
ical systems has led to many kinds of linear and nonlinear, ordinary and
partial differential, integral or integro-differential, and functional evolution
equations, formulated in elementary or abstract function spaces, with initial-
boundary value conditions and possible additional constraints. The classical
approach to solving such equations for analytic solutions has proved impossi­
ble and, in reality, unnecessary with the great advances of modern computer
facilities and technology. Approximate solutions to these problems by means
of numerical computation have thus become the main stream of research and
applications developed in recent years, in both academia and industry. In
the endeavor of advancing efficient computational theories and methods, one
successful and unified approach is to formulate various apparently different
but intrinsically related equations as a certain type of operator equations in
Banach spaces of real or complex functions. Approximate solutions to these
operator equations can provide deep insights and feasible resolutions to many
specific initial-boundary value problems of differential and integral equations.
This book is designed as an elementary and self-contained introduc­
tion to some important notions such as the solvability issue, computational
schemes, convergence analysis, stability conditions, and error estimates of
approximate solutions for several types of operator equations in abstract Ba­
nach spaces. The operator equations studied in this treatise include various
linear and nonlinear, ordinary and partial differential, integral and evolu­
tion equations that are frequently encountered in applied mathematics and
engineering applications.
The book serves also as a textbook for graduate students and as a refer­
ence for researchers and professionals in the fields of approximation theory,
numerical analysis, scientific computation, applied mathematics, and engi­
neering. Among the many important and elegant results in the literature of
approximate solutions of operator equations, we only include certain elemen­
tary and fundamental topics in this introductory text of modest size. Other
results and more advanced topics can be found from the references provided
vn
Preface
Vlll
at the end of the book. The presentation is organized as follows.
Chapter 1 is an overview of the projection approximation technique,
which is of fundamental importance for operator equations throughout the
entire text. Projection operators, projective approximation schemes, and
their properties are discussed in this chapter.
In Chapter 2, compact linear operator equations and their approximate
solutions in a Banach space setting are investigated. Projection approxima­
tion of eigenvalues of self-adjoint compact linear operators in Hilbert spaces
is also studied. The Fredholm integral equation is used as a case study of the
theory and computational methods.
General linear operator equations and their perturbation problems are
the main topics in Chapter 3. Some sufficient and necessary conditions for
unique approximate solvability and stability are derived for bounded linear
operator equations in reflexive Banach spaces. Several general computational
frameworks for finding generalized solutions of densely defined linear opera­
tor equations are established, in Banach space and/or Hilbert space settings.
Applications of these schemes to numerical solutions of boundary value prob­
lems of linear ordinary and partial differential equations are discussed.
The basic theory of topological degrees is presented in Chapter 4. It
includes the topological degree of continuous operators in Euclidean spaces
(the Brouwer degree), topological degree of compact fields in Banach spaces
(the Leray-Schauder degree), and the generalized topological degree of the
so-called A-proper operators. Some important fixed-point theorems are given
in this chapter, with applications to projective approximate solutions of non­
linear integral equations.
Chapter 5 provides a review of some fundamental concepts of Calculus
in Banach space, and studies the projective approximate solvability problem
for monotone and K-monotone nonlinear operator equations, including their
perturbation problems. Numerical solutions of boundary value problems of
some typical nonlinear elliptic differential equations are discussed along with
several examples.
Finally, in Chapter 6, both continuous-time and discrete-time (semi-
discrete as well as fully discrete) projection approximation methods are de­
veloped for approximate solutions of first and second order abstract evolution
equations. Their applications to initial-boundary values problems of differ­
ential evolution equations are illustrated via concrete examples.
The present book is a significant expansion and revision of the text
Operator Equations and Their Projection Approximate Solutions by the first
two authors, published in Chinese by the Guangdong Scientific Publishing
Company in 1992. The revision involves complete reorganization, filling in
many details, updating with certain new techniques, and inclusion of many
illustrative examples and exercises.
Preface ix
The authors wish to acknowledge several individuals, Professors Ronghua
Li, Yuesheng Li, Guoshen Feng, and Jingan Lei, for their interest and en­
couragement during the preparation of this monograph. They would also
like to express their gratitude to Professor Charles Chui, the series editor,
for his continued support, and to Mrs. Margaret Chui for her assistance in
the editorial work.
Mingjun Chen
Guangzhou Zhongying Chen
Houston Guanrong Chen
Summer, 1996
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