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Systems of ordinary differential equations: an introduction
Book
Review.
SYSTEMS ox ORDIWBY DIFFEBEX~XAL EQUATIONS : AN INTRODUCTION, W Jack L. Goldberg and Arthur J. Schwartz. 315 pages, 6 x 9 in. New York, Harper & Row, 1972. Price, $18.95 (approx. $7.90). In their Preface, the authors say, “This book is intended for a$& course (author’s italics) in ordinary different&l equations (o.d.e.‘s).” The book is a sharp, even radical, departure from the usual plan for a first course in o.d.e.‘s. Whereas the usual d.e. text will devote the tist several chapters to the one-dimensional case of a single, dependent variable, this book, true to its title, dives-after Chsp. 1 on matrices and vectors-right into the linear, homogeneous, first-order system 5’ = AZ,
q&J = CEO.
(1)
In Chap. 2 (32 pp.) eigenvalues and eigenvectors are introduced, and the initialvalue problem (1) is disposed of for the case of a full complement of linearly independent eigenveotors. The importance of the special case where A in (1) is symmetric is discussed, and a threecompartment diffusion problem is studied. The style of writing is relaxed, nicely informal and readable enough for students beginning a study of o.d.e.‘s. There are many worked, numerical problems to illustrate the principles. In Chap. 3 (41 pp.), the eigenvalueeigenveotor method of solving (1) is completed with the coverage of the ease of repeated eigenvalues with fewer than a full complement of linearly independent eigenvectors. In fact, this coverage is completed in the first half of the chapter, the last half being devoted to a particular approach the authors call “Another View of the Initial-value Problem”, and to a little electrical network theory. Chapter 4 (33 pp.) develops the theory of fundamental matrices, D(t), for the system Z’ = Ab. The inhomogeneous problem, d = AE+j, is taken up and the particular solution
@(t)
c”W’(s)f(s)
J kl
ds
is obtained by a method known ~1s variation of parameters. There is no explanation of this terminology, and the phrase “variation of parameters” does not seem appropriate in the vectormatrix treatment of a? = &+f. Then the suthors mention the phrase “undetermined coefficients” with no explanation of the origin of the phrase. This phrase also is inappropriate in the vector-matrix context. Why not omit these tmditiond phrases until the traditional context is adduced ? On p. 156, beginning Chap. 5, the nth-order single dependent variable equation yen)+ a,_, y(n-l) + . . . =f(t)
+a,y(l)+a,y
(2)
finally appears. This is converted to matrix-vector form, introducing the companion matrix l-
L
0
1
0
...
0
0
0
1
...
0
.
.
.
-al
...
...
-a,
-a,_,
1
J
and the theory of the preceding chapters is brought to bear, treating (2) in eight pages. On p. 164 the authors allow that, “ . . . the nth order equation occurs so often that it is inefficient to involve the entire theory of linear systems whenever we wish its solution”. The traditional “auxiliary equation” of (2) is then introduced and used. A mechanics problem with several degrees of freedom is briefly considered. For a teacher who would prefer that his beginning o.d.e. class learn first-in a more traditional way-the theory and application of o.d.e.‘s in a single dependent variable, this is not the text to use. The solution of the first-order, linear case using an integrating factor does not 194. Separation of appear until p. variables does not appear until p. 232. There are very few applied problems or word problems. In Chap. 6, series solutions are presented first in the veotormatrix form. Later in the chapter the
245
Book Reviews authors allow (p. 209) that, “ . . . it is often more convenient to study specific one-dimensional problems directly”. Then Legendre’s equation is treated as it stands, without first converting it to vector-matrix form. If it is not useful as a beginning text in o.d.e.‘s, can the book be better used as a second text ? Yes, but with a reservation which is that a lot of mathematical power is generated and then hardly used. The power is used merely to obtain solutions: whereas it could be used and interpreted in areas where the necessity for it arose-normal mode behavior and state-space analysis of dynamical systems, uncoupling of dependent variables, Jordan normal form. Returning to Chap. 3, the authors develop the theory of root vectors (or generalized eigenvectors) ((1) p. 88; (2) p. 189; (3) p. 67) and chains of root vectors. This is a rather intricate business, and unnecessary if all one wants is to obtain solutions to 5’ = Ad. In order to write a general solution of 8’ = AZ, one only needs to find bases (any bases) for the subspaces ( (4) p. 283) defined by (‘4 -hj I)? 5 = 0,
j = 1,2,3
I .*., Ic,
where h, is an eigenvector of A of multiplicity nj. This may be done efficiently by reducing (A -h,.l)? to row echelon form, after which a basis for the null space of (A --Xj_l)‘? is easily obtained. The effort to obtain chains of root vectors goes beyond this and finds particular bases which provide initial conditions leading to modal responses, to the modal matrix M such that M-1 AM is in Jordan normal form, and to the uncoupling of the dependent variables. These are very interesting and important concepts and it seems too bad to lift students (with chains of root vectors) to the point where they can take a look at these concepts, and then not mention them. However, every author must stop somewhere, and the teacher now has a chance to write up his own interpretations in his own way. Chapter 7 (22 pp.) is an introduction to nonlinear equations with a development of the Euler numerical method in vectormatrix form. Chapter 8 (43 pp.) treats two-dimensional autonomous systems,
246
phase-planeanalysisandstability.Answers to odd-numbered problems are provided. References (1) A. I. Mal’cev, “Foundations of Linear Algebra”, Freeman, San Francisco, 1963. (2) R. Bronson, “Matrix Methods, An Academic Press, Introduction”, New York, 1969. (3) B. Friedman, “Principles and Techniques of Applied Mathematics”, Wiley, New York, 1956. (4) F. Brauer and J. A. Nohel, “Ordinary Differential Equations, A First Benjamin, New York, Course”, 1967. DOUGLAS H. MOORE College of Environmental Sciences University of Wisconsin Green Bay, Wisconsin
PRINCIPLES OF DIFFERENTIAL AND INTEGRAL EQUATIONS, by C. Corduneanu. 202 pages, diagrams, 6 x 9 in. Boston, Mass., Allyn & Bacon, 1971. Price, $16.95 (approx. di7.05). Written in a concise and rigorous manner, this book presents these basic aspects in the theory of differential and integral equations which are needed for an undergraduate two-semester course in the subject. The major aspects discussed in the book, as offered to students of pure or applied and include : existence mathematics, uniqueness theory for ordinary different,ial equations, linear systems, stability theory, Volterra integral equations, Fredholm theory for linear integral equations, selfadjoint integral equations, and SturmLiouville problems. The excellent presentation of the standard topics in differential and integral equations, complemented by certain advanced topics that do not normally appear in the textbook literature, both provide sufficient background for the study of such areas as control theory and systems theory and stimulate the reader’s interest toward further study in differential equations and (especially) integral equations. (See for instance, the recent excellent book, Linear Integral Equations,
Jcwmal
of The Franklin
Institute
Review.
SYSTEMS ox ORDIWBY DIFFEBEX~XAL EQUATIONS : AN INTRODUCTION, W Jack L. Goldberg and Arthur J. Schwartz. 315 pages, 6 x 9 in. New York, Harper & Row, 1972. Price, $18.95 (approx. $7.90). In their Preface, the authors say, “This book is intended for a$& course (author’s italics) in ordinary different&l equations (o.d.e.‘s).” The book is a sharp, even radical, departure from the usual plan for a first course in o.d.e.‘s. Whereas the usual d.e. text will devote the tist several chapters to the one-dimensional case of a single, dependent variable, this book, true to its title, dives-after Chsp. 1 on matrices and vectors-right into the linear, homogeneous, first-order system 5’ = AZ,
q&J = CEO.
(1)
In Chap. 2 (32 pp.) eigenvalues and eigenvectors are introduced, and the initialvalue problem (1) is disposed of for the case of a full complement of linearly independent eigenveotors. The importance of the special case where A in (1) is symmetric is discussed, and a threecompartment diffusion problem is studied. The style of writing is relaxed, nicely informal and readable enough for students beginning a study of o.d.e.‘s. There are many worked, numerical problems to illustrate the principles. In Chap. 3 (41 pp.), the eigenvalueeigenveotor method of solving (1) is completed with the coverage of the ease of repeated eigenvalues with fewer than a full complement of linearly independent eigenvectors. In fact, this coverage is completed in the first half of the chapter, the last half being devoted to a particular approach the authors call “Another View of the Initial-value Problem”, and to a little electrical network theory. Chapter 4 (33 pp.) develops the theory of fundamental matrices, D(t), for the system Z’ = Ab. The inhomogeneous problem, d = AE+j, is taken up and the particular solution
@(t)
c”W’(s)f(s)
J kl
ds
is obtained by a method known ~1s variation of parameters. There is no explanation of this terminology, and the phrase “variation of parameters” does not seem appropriate in the vectormatrix treatment of a? = &+f. Then the suthors mention the phrase “undetermined coefficients” with no explanation of the origin of the phrase. This phrase also is inappropriate in the vector-matrix context. Why not omit these tmditiond phrases until the traditional context is adduced ? On p. 156, beginning Chap. 5, the nth-order single dependent variable equation yen)+ a,_, y(n-l) + . . . =f(t)
+a,y(l)+a,y
(2)
finally appears. This is converted to matrix-vector form, introducing the companion matrix l-
L
0
1
0
...
0
0
0
1
...
0
.
.
.
-al
...
...
-a,
-a,_,
1
J
and the theory of the preceding chapters is brought to bear, treating (2) in eight pages. On p. 164 the authors allow that, “ . . . the nth order equation occurs so often that it is inefficient to involve the entire theory of linear systems whenever we wish its solution”. The traditional “auxiliary equation” of (2) is then introduced and used. A mechanics problem with several degrees of freedom is briefly considered. For a teacher who would prefer that his beginning o.d.e. class learn first-in a more traditional way-the theory and application of o.d.e.‘s in a single dependent variable, this is not the text to use. The solution of the first-order, linear case using an integrating factor does not 194. Separation of appear until p. variables does not appear until p. 232. There are very few applied problems or word problems. In Chap. 6, series solutions are presented first in the veotormatrix form. Later in the chapter the
245
Book Reviews authors allow (p. 209) that, “ . . . it is often more convenient to study specific one-dimensional problems directly”. Then Legendre’s equation is treated as it stands, without first converting it to vector-matrix form. If it is not useful as a beginning text in o.d.e.‘s, can the book be better used as a second text ? Yes, but with a reservation which is that a lot of mathematical power is generated and then hardly used. The power is used merely to obtain solutions: whereas it could be used and interpreted in areas where the necessity for it arose-normal mode behavior and state-space analysis of dynamical systems, uncoupling of dependent variables, Jordan normal form. Returning to Chap. 3, the authors develop the theory of root vectors (or generalized eigenvectors) ((1) p. 88; (2) p. 189; (3) p. 67) and chains of root vectors. This is a rather intricate business, and unnecessary if all one wants is to obtain solutions to 5’ = Ad. In order to write a general solution of 8’ = AZ, one only needs to find bases (any bases) for the subspaces ( (4) p. 283) defined by (‘4 -hj I)? 5 = 0,
j = 1,2,3
I .*., Ic,
where h, is an eigenvector of A of multiplicity nj. This may be done efficiently by reducing (A -h,.l)? to row echelon form, after which a basis for the null space of (A --Xj_l)‘? is easily obtained. The effort to obtain chains of root vectors goes beyond this and finds particular bases which provide initial conditions leading to modal responses, to the modal matrix M such that M-1 AM is in Jordan normal form, and to the uncoupling of the dependent variables. These are very interesting and important concepts and it seems too bad to lift students (with chains of root vectors) to the point where they can take a look at these concepts, and then not mention them. However, every author must stop somewhere, and the teacher now has a chance to write up his own interpretations in his own way. Chapter 7 (22 pp.) is an introduction to nonlinear equations with a development of the Euler numerical method in vectormatrix form. Chapter 8 (43 pp.) treats two-dimensional autonomous systems,
246
phase-planeanalysisandstability.Answers to odd-numbered problems are provided. References (1) A. I. Mal’cev, “Foundations of Linear Algebra”, Freeman, San Francisco, 1963. (2) R. Bronson, “Matrix Methods, An Academic Press, Introduction”, New York, 1969. (3) B. Friedman, “Principles and Techniques of Applied Mathematics”, Wiley, New York, 1956. (4) F. Brauer and J. A. Nohel, “Ordinary Differential Equations, A First Benjamin, New York, Course”, 1967. DOUGLAS H. MOORE College of Environmental Sciences University of Wisconsin Green Bay, Wisconsin
PRINCIPLES OF DIFFERENTIAL AND INTEGRAL EQUATIONS, by C. Corduneanu. 202 pages, diagrams, 6 x 9 in. Boston, Mass., Allyn & Bacon, 1971. Price, $16.95 (approx. di7.05). Written in a concise and rigorous manner, this book presents these basic aspects in the theory of differential and integral equations which are needed for an undergraduate two-semester course in the subject. The major aspects discussed in the book, as offered to students of pure or applied and include : existence mathematics, uniqueness theory for ordinary different,ial equations, linear systems, stability theory, Volterra integral equations, Fredholm theory for linear integral equations, selfadjoint integral equations, and SturmLiouville problems. The excellent presentation of the standard topics in differential and integral equations, complemented by certain advanced topics that do not normally appear in the textbook literature, both provide sufficient background for the study of such areas as control theory and systems theory and stimulate the reader’s interest toward further study in differential equations and (especially) integral equations. (See for instance, the recent excellent book, Linear Integral Equations,
Jcwmal
of The Franklin
Institute