Methods of Nonlinear Analysis - Volume 1 (Mathematics in Science & Engineering)

METHODS OF NONLINEAR ANALYSIS Volume I

This is Volume 61 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California A complete list of the books in this series appears at the end of this volume.

METHODS OF NONLINEAR ANALYSIS Richard Bellman Departments of Mathematics, Electrical Engineering, and Medicine University of Southern California Los Angeles, California

VOLUME I

@

1970

ACADEMIC PRESS

New York and London

COPYRIGHT 0 1970,

BY

ACADEMIC PRESS,INC.

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PREFACE

T h e demands of modern science inexorably force the mathematician to explore the nonlinear world. That it is a difficult and often humbling journey with painfully crude maps and rather primitive direction-finders cannot be gainsaid, but in return it can be asserted that it is richly rewarding. The few areas that have been so far examined with any care have been full of surprises and vastly stimulating to the imagination. There is every reason to believe from what so far has been glimpsed that many more surprises lay in store, novel phenomena which will open up undreamt of vistas for mathematics. I t is an exciting prospect in an exciting field in an exciting time. Explicit analytic solutions of nonlinear equations in terms of the familiar, well-tamed functions of analysis are not to be expected, although fortuitous and treasured examples occur here and there. Consequently, if either analytic or computational results are desired, various approximate methods must be applied. By and large, the effective solution of particular problems is an art. However, there d o exist a number of powerful procedures for solving nonlinear problems which have been slowly forged and polished over the last one hundred years. As in all intellectual areas, art combined with method is more effective than untutored art. This book is intended as an introduction to the study of certain systematic and sophisticated techniques. T h e power and versatility of these methods has been tremendously amplified by the digital computer. Even more, this new tool has motivated a careful reexamination of older methods and thus a creation of new techniques specifically designed to exploit the peculiar properties of the electronic calculator. That a good deal of mathematical ingenuity and experience is required to study significant problems with the aid of a computer hardly needs emphasizing. This volunie may also be regarded as a contribution to a new mathematical theory that is slowly emerging, a theory of closure of operations. Abstractly, the general problem may be described in the following terms. We are given the privilege of using a certain limited number of mathematical operations, such as, for example, the solution of finite systems of linear or nonlinear differential equations subject to initial conditions, or the solution of a finite system of linear algebraic equations. The task is then that of solving a particular equation, such as a partial differential equation, a two-point boundary value problem for vii

viii

PREFACE

ordinary differential equations, or an integral equation, to a specified degree of accuracy using only these algorithms. T h e study becomes still more interesting and significant if we impose a constraint on the number of operations of particular types that can be employed, or on the time that may be consumed in the overall calculation. Usually, the computational facilities available automatically impose these constraints. ‘I’he two types of operations mentioned above have been carefully singled out for explicit mention since they correspond to the two major capabilities of the digital computer in the field of analysis. That they are not guaranteed capabilities merely adds to the interest of the zest of using computers to obtain numerical rcsu I ts. We nil1 present a spectrum of methods which can be used for a variety of purposes, ranging from the derivation of a simple exponential or algebraic approximation to a sequence of algorithms of increasing complexity which require a digital computer. At the moment, our orientation, as far as large-scale computing is concerned, is toward a digital computer, which is to say a leaning toward initial value problems. As hybrid computers become more powerful and prevalent, a certain mix of methods involving two-point boundary value problems and initial value problems will occur. In general, the word “solution” must be defined operationally in terms of various technological tools available for obtaining numerical results. T h e arrival of the digital computer has already drastically changed the significance of this term “solution,” and there will be further radical changes over the next twenty-five years. T h e majority of the methods we present here can be applied to the study of partial differential equations and to the still more complex functional equations that the determined engineer and intrepid physicist are forced to face. T h e applications w ithin this broader context are naturally of greater significance than those that can be made using ordinary differential equations. Despite this, we have deliberately refrained from any excursion in force into the area of partial differential equations. In this volume, the first of two, we have discussed only ordinary differential cquations, However, since any complete separation between ordinary and partial differential equations is unnatural, we have broken this self-imposed vow in the second volume. This is particularly the case in the treatment of dynamic programming and invariant imbedding. Since w e are primarily concerned with introducing the reader to a variety of fundamental methods, we feel that there is considerable pedagogical force to keeping the setting as familiar as possible while new ideas are being introduced. Once acquainted with the concepts, the reader can readily apply them to all types of functional equations with a small amount of additional background. References will be found throughout to their utilization in the theory of partial differential equations.

PREFACE

ix

Another strong argument for using ordinary differential equations as a proving ground is that it is relatively easy to provide a number of numerical examples in this area to illustrate different methods. In addition, a large number of interesting analytic results are available as exercises. These have principally been taken from research papers. Having briefly described our overall aims, let us examine the structure of the book. T h e first three chapters contain some of the fundamental results and methods that will serve throughout both as foundation and ancillary tools. Chapter 1 discusses first- and second-order linear differential equations, subject to initial and boundary value problems, with some attention to the Riccati differential equation and a detailed study of the behavior of the physical solutions of nonlinear equations of the form

where p and q are polynomials in their arguments. This last represents a brief account of extensive work by Bore1 and Hardy, very important and useful results which are still not as well known as they should be. Some related results for the Emden-Fowler (or Fermi-Thomas) equation will be presented at the end of Chapter 4 as applications of stability theory. Throughout we have tried to preserve some sort of a careful balance between general methods and particular problems. We have constantly kept in mind the famous dictum of Hurwitz, “It is easier to generalize than particularize.” In Chapter 2, we present a brief account of basic results in algebraic aspects of matrix analysis that will be employed throughout the remainder of the book. The principal contents are the reduction of quadratic forms to canonical forms and associated variational problems and the Perron theorem for positive matrices. Chapter 3 discusses the use of matrices in the study of systems of linear differential equations with both constant and variable coefficients. It is impossible to study multidimensional problems in any meaningful fashion without matrix theory. Chapter 4 contains some basic results concerning stability theory which we will employ in subsequent chapters to validate certain methods of approximation. Following the lines initiated by PoincarC and Lyapunov, we wish to compare the solutions of

T(u) = 0

(2)

T(u) = N ( 4 ,

(3)

with those of where N(u} is “small” in some sense. The most important case is that. where T is a linear operator with the property that T(u) = 0 possesses a convenient

X

PREFACE

solution. Closely connected with this question is the problem of estimating the difference between the solution of (2) and a function w satisfying the inequality

where I/ ... / / denotes some appropriate norm. With these “back-up” results available, we can turn to our principal goal, the study of certain powerful methods of analytic and computational approximation. I n Chapter 5 , we present the Bubnov-Galerkin method, and in Chapters 7 and 8 that of Rayleigh-Ritz. Although in certain special, but important, cases the methods overlap, they are quite different conceptually and extend in different ways. T h e guiding ideas are simple, as are all fundamental mathematical ideas. As always, effort and ingenuity enter in making these procedures work in particular cases. Let us begin with a description of the Bubnov-Galerkin method. Let T(u) = 0 be the equation whose solution is desired. This is equivalent to minimizing the scalar quantity 11 T(u)ll,for any norm, over the class of admissible functions. We now introduce a closure technique by restricting u to some smaller class of functions, for example, one defined by a finite set of parameters. T h e most important example of this is that where the restricted class is defined by N IZ =

akUk k=l

9

(5)

where the uk are fixed functions and the ak are parameters. T h e infinitedimensional problem of minimizing I/ T(u)lI is then replaced by the approximating finite-dimensional problem of minimizing the function

with respect to the ak . This problem may be attacked by any of a number of techniques developed in optimization theory over the last twenty years: search techniques, gradient methods, Newton-Raphson, nonlinear programming, expansion methods, and so on. Let us note that with such methods in mind, we have deliberately refrained from any automatic use of the usual quadratic norm in the foregoing description. In the text, however, succumbing to the lure of analytic simplicity, we have considered principally quadratic functionals. More general nonlinear functionals give us an opportunity to discuss the Newton-Raphson-Kantorovich method and the use of the Lagrange expansion. Closely associated with the Galerkin method are the methods of mean-square

PREFACE

xi

approximation and differential approximation. The first is discussed at the end of Chapter 5, the second in Chapter 6 . The technique of mean-square approximation may be described in the following terms. Let T ( u ) = 0, as usual, be the original equation, and let S(v, u)

=0

(7)

be another equation, depending on the vector parameter a,which is analytically or computationally more tractable than the original equation. Thus, for example, the equation in (7) may be linear with the original equation nonlinear, or it may be a nonlinear differential equation subject to an initial value condition with the original condition linear and subject to multipoint boundary conditions. Alternatively, the original equation may contain stochastic elements, while (7) is deterministic, or conversely. The existence of analog, digital, and hybrid computers, as well as the availability of many powerful analytic theories has considerably altered the concept of “tractable.” A great deal of flexibility now exists. Many different types of mathematical models are available to treat various kinds of physical processes. We have avoided stochastic processes in this volume since a good deal of effort is required to make various useful methods rigorous. We wish to determine the parameter a so that

is small, where u is the solution of T(u) = 0 and some convenient norm is employed. Presumably, this ensures that v, the solution of (7), is close to u. This is a stability question. Interesting complications arise from the fact that u itself is unknown. There are various “bootstrap” methods that can be employed to circumvent this annoying “little detail.” Here we make brief contact with “self-consistent” methods, of such importance in modern physics. A major analytic problem is that of choosing the operator S(v, u ) in such a way that the function v preserves certain desirable properties of u. Very little is known in this area. The method of differential approximation is the following. Let R(u, b ) be a family of operators depending on a finite-dimensional vector b, and let b be chosen so that

is minimized where u is now given implicitly as the solution of T(u) = 0. We then use the solution of

R(v, 6)

=0

(10)

as an approximation to u. Once again, any discussion of the validity of this approach requires stability considerations.

xii

PREFACE

A case of particular importance is that where

with the Rk(u)differential operators. Observe that the aim of this procedure is once again closure. We want to solve complex functional equations using only the algorithms required to solve the more familiar and placid differential equations. We now turn to an entirely different type of artifice. T h e Rayleigh-Ritz method hinges upon the observation that many equations of the form T ( u ) = 0 may be viewed as the Euler equation of an associated functional J(u), By this we mean that a solution of T(u) = 0 is a stationary point for J(u) as u varies over an appropriate space. Let us suppose that we are looking for a minimum value. The question of determining the minimum of J(u) over the original infinite-dimensional space is then replaced by the finite-dimensional problem of minimizing J(u) over a finite-dimensional subspace, each element of which is characterized by a finite number of parameters, say U =

1

bkUk.

k=l

Here the uk are carefully chosen functions. The new problem, that of minimizing the expression

can now be approached in a number of ways. In many cases of significance, J(u) is a quadratic functional, the minimization of which leads to linear equations for the b, . There are, however, numerous difficulties associated with the solution of large systems of linear algebraic equations, which means that the real difficulties often begin at this point. Observe that in both of the principal methods described above, monotone approximation is obtained immediately upon increasing the dimension of the finite space of functions over which the variation is allowed. Thus, if we set A N =

it is clear that

I! c )I ,

min T {a;}

(k:l

PREFACE

xiii

Similarly, if we set

we have

Several fundamental questions immediately present themselves. The first is that of determining when lim A ,

N-CC

= m$

11 T(u)ll,

lim dN = min J(u).

N+m

U

T h e second is that of determining when the function u ( N ) which yields 4 , converges to the function u which yields min, 11 T(u)ll, with the corresponding problem for the Rayleigh-Ritz functional. Under reasonable conditions on T(u),J(u), and the spaces over which u varies, these are not difficult to answer. Far more difficult and important are the associated stability problems of estimating 11 u - u") (1 in terms of A , - A , , or d , - d, , and in determining 11 u - d N I/) as a function of N . These are essential matters when the effective determination of u ( ~is) of importance. A few numerical examples, together with references to extensive work in this area will be given. In view of the considerable effort required to treat the finite-dimensional variational problems when N is large, there is considerable motivation for finding ways of obtaining useful estimates for small N . In a sense, the major problem is the converse. It is one of determining the smallest value of N which yields an acceptable approximation. Questions of acceleration of convergence and extrapolation arise in this connection, with techniques that go back to Euler and Kronecker. We shall touch briefly on these matters. In Chapter 8, we show how a linear equation containing a parameter can be considered to be the Euler equations associated with the minimization of a functional subject to a global constraint. Once again, Rayleigh-Ritz methods can be employed to obtain approximate results. Many of the problem areas discussed in this volume can be further illuminated, or considered by alternative techniques, using the methods of the second volume. There we consider duality as a technique for providing upper and lower bounds, Caplygin's method, and differential inequalities, quasilinearization, dynamic programming, invariant imbedding, the theory of iteration, and truncation techniques. The work was divided into two volumes to prevent the single,

xiv

PREFACE

massive volume that is so forbidding and discouraging to the newcomer into a field. Let us encourage the reader with the flat statement that very little is known about nonlinear analysis and that it is not obvious that major breakthroughs will be made in the near future, or ever. Hundreds and thousands of fascinating and significant problems abound, each of which may require a new theory for its elucidation. I have been fortunate in having three friends read through the book and help considerably with all aspects of preparation of the manuscript: David Collins, Thomas J. Higgins, and Art Lew. I wish to express my appreciation for their help, and to Jeanette Blood and Rebecca Karush for typing the manuscript. RICHARDBELLMAN

Los Angeles, 1969

CONTENTS

vii

Preface

Chapter 1.

First- and Second-order Differential Equations

1.1. Introduction 1.2. T h e First-order Linear Differential Equation 1.3. Fundamental Inequality 1.4. Second-order Linear Differential Equations 1.5. Inhomogeneous Equation 1.6. Lagrange Variation of Parameters 1.7. Two-point Boundary Value Problem 1.8. Connection with Calculus of Variations 1.9. Green’s Functions 1.10. Riccati Equation 1.1 1. T h e Cauchy-Schwarz Inequality 1.12. Perturbation and Stability Theory 1.13. A Counter-example 1.14. Sm If(t)l dt < co 1.15. Sm If’(t)l dt < co 1.1 6. Asymptotic Behavior 1.17. T h e Equation u” - (1 + f ( t ) ) t ~ = 0 1.18. More Refined Asymptotic Behavior 1.19. J m f z d t < 00 1.20. T h e Second Solution 1.21. T h e Liouville Transformation 1.22. Elimination of Middle Term 1.23. T h e WKB Approximation 1.24. T h e One-dimensional Schrodinger Equation 1.25. u” (1 + f ( t ) ) u = 0; Asymptotic Behavior 1.26. Asymptotic Series 1.27. T h e Equation u’ = p(u, I)/q(u, t ) 1.28. Monotonicity of Rational Functions of u and t 1.29. Asymptotic Behavior of Solutions of u’ = p(u,t ) / q ( u ,t ) Miscellaneous Exercises Bibliography and Comments

+

Chapter 2. 2.1. 2.2.

1 2 3 5

7 8 10 11 12 14 16 18 20 21 22 23 24 26 27 29 30 31

33 33 33 35 37 38 39 42 51

Matrix Theory 54 55

Introduction Determinantal Solution xv

CONTENTS

xvi

2.3. Elimination 2.4. Ill-conditioned Systems 2.5. T h e lmportance of Notation 2.6. Vector Notation 2.7. Norm of a Vector 2.8. Vector Inner Product 2.9. Matrix Notation 2.10. Noncommutativity 2.1 I. ‘l’he Adjoint, or Transpose, Matrix 2.12. ‘l’hc Inverse Matrix 2.13. Matrix Norin 2.14. Relative Invariants 2.15. Constrained Minimization 2.1 6. Symmetric Matrices 2.17. Quadratic Forms 2.18. Multiple Characteristic Roots 2.19. Maximization and Minimization of Quadratic Forms 2.20. illin-Max Characterization of the A, 2.21. Positive Definite Matrices 2.22. Ileterminantal Criteria 2.23. Representation for A-’ 2.24. Canonical Representation for Arbitrary A 2.25. Perturbation of Characteristic Frequencies 2.26. Separation and Reduction of Dimensionality 2.27. Ill-conditioned Matrices and Tychonov Regularization 2.28. Self-consistent Approach 2.29. Positive Matrices 2.30. Variational Characterization of h ( A ) 2.3 I . Proof of Minimum Property 2.32. Equivalent Definition of A(A) Miscellaneous Exercises Bibliography and Comments

Chapter 3.

58 59 60 60 61 61

63

64 65 65 67 68 71 72 14 75 76 77 79 81 82 82 84 85 86 88 88 89 91 92 94 101

Matrices and Linear Differential Equations

3.1. Introduction 3.2. Vector-Matrix Calculus 3.3. Existence and Liniqueness of Solution 3.4. ’rhe Matrix Exponential 3.5. Commutators 3.6. Inhomogeneous Equation 3.7. ‘I’he Euler Solution 3.8. Stability of Solution 3.9. Idinear Differential Equation with Variable Coefficients 3.10. Linear Inhomogeneous Equation 3.1 I. Adjoint Equation XB 3.12. T h e Equation X’ = A X 3.13. Periodic hlatriccs: the Floquet Representation 3.14. Calculus of Variations 3.15. Two-point Boundary Condition

+

104 104

105 107 108 110 111 113 114 116 118 118 1 20 121 122

CONTENTS

3.16. 3.17. 3.18. 3.19. 3.20.

Green’s Functions T h e Matrix Riccati Equation Kronecker Products and Sums

AX

+XB = C

Random Difference Systems Miscellaneous Exercises Bibliography and Comments

Chapter 4.

Introduction Dini-Hukuhara Theorem-I Dini-Hukuhara Theorem-I1 Inverse Theorems of Perron Existence and Uniqueness of Solution Poincark-Lyapunov Stability Theory Proof of Theorem Asymptotic Behavior T h e Function q ( c ) More Refined Asymptotic Behavior Analysis of Method of Successive Approximations Fixed-point Methods Time-dependent Equations over Finite Intervals Alternative Norm Perturbation Techniques Second Method of Lyapunov Solution of Linear Systems Origins of Two-point Boundary Value Problems Stability Theorem for Two-point Boundary Value Problem Asymptotic Behavior Numerical Aspects of Linear Two-point Boundary Value Problems Difference Methods Difference Equations Proof of Stability Analysis of Stability Proof T h e General Concept of Stability Irregular Stability Problems T h e Emden-Fowler-Fermi-Thomas Equation Miscellaneous Exercises Bibliography and Comments

5.6.

134 135 138 140 140 142 143 146 148 149 150

152 152 155 156 157 157 158 159 160 161 163 165 165 166 168 168 170 171 182

The Bubnov-Galerkin Method

5.1. Introduction 5.2. Example of the Bubnov-Galerkin Method 5.3. Validity of Method 5.4. Discussion 5.5.

123 123 124 125 127 127 131

Stability Theory and Related Questions

4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.1 0. 4.1 1 . 4.12. 4.13. 4.14. 4.1 5. 4.16. 4.1 7. 4.18. 4.19. 4.20. 4.21. 4.22. 4.23. 4.24. 4.25. 4.26. 4.27. 4.28.

Chapter 5.

xvii

T h e General Approach Two Nonlinear Differential Equations

187 188 189 190 190 192

xvi ii

CONTENTS

5.7. T h e Nonlinear Spring 5.8. Alternate Average 5.9. Straightforward Perturbation 5.10. A “Tucking-in’’ Technique 5.1 1. l‘he Van der Pol Equation 5.12. Two-point Boundary Value Problems 5.13. T h e Linear Equation L(u) = g 5.14. Method of Moments 5.15. Nonlinear Case 5.16. Newton-Raphson Method 5.17. Multidimensional Newton-Raphson 5.18. Choice of Initial Approximation 5.19. Nonlinear Extrapolation and Acceleration of Convergence 5.20. Alternatives to Newton-Raphson 5.21. Lagrange Expansion 5.22. Method of Moments Applied to Partial Differential Equations Miscellaneous Exercises Bibliography and Comments

Chapter 6.

193 196 196 198 198 200 200 202 202 204 207 208 210 21 1 212 214 215 222

Differential Approximation

6.1. Introduction 6.2. Differential Approximation 6.3. Linear Differential Operators 6.4. Computational Aspects-I 6.5. Computational Aspects-I1 6.6. Degree of Approximation 6.7. Orthogonal Polynomials 6.8. Improving the ApDroximation 6.9. Extension of Classical Approximation Theory 6.10. liiccati Approximation 6. I I . Transcendentally-transcendent Functions 6.12. Application to Renewal Equation 6.13. An Example 6.14. Differential-Difference Equations 6.15. An Example 6.16. Functional-Differential Equations 6.17. Reduction of Storage in Successive Approximations 6.18. Approximation by Exponentials 6.19. Mean-square Approximation 6.20. Validity of the Method 6.21. A Bootstrap Method 6.22. T h e Nonlinear Spring 6.23. ’I‘he Van der Pol Equation 6.24. Self-consistent Techniques 6.25. T h e Riccati Equation 6.26. Higher-order Approximation 6.27. Mean-square Approximation-Periodic Solutions Miscellaneous Exercises Bibliography and Comments

225 225 226 226 227 228 229 23 1 23 1 232 233 233 236 238 239 240 242 242 242 243 244 244 246 248 24 8 250 25 1 253 255

CONTENTS

Chapter 7.

The Rayleigh-Ritz Method

7.1. Introduction 7.2. T h e Euler Equation 7.3. T h e Euler Equation and the Variational Problem 7.4. Quadratic Functionals: Scalar Case 7.5. Positive Definiteness for Small T 7.6. Discussion 7.7. T h e Rayleigh-Ritz Method 7.8. Validity of the Method 7.9. Monotone Behavior and Convergence 7.10. Estimation of I u z1 I in Terms of J(v) - J(u) 7.11. Convergence of Coefficients 7.12. Alternate Estimate 7.13. Successive Approximations 7.14. Determination of the Cofficients 7.15. Multidimensional Case 7.16. Reduction of Dimension 7,17. Minimization of Inequalities 7.1 8. Extension to Quadratic Functionals 7.19. Linear Integral Equations 7.20. Nonlinear Euler Equation 7.21. Existence and Uniqueness 7.22. Minimizing Property 7.23. Convexity and Uniqueness 7.24. Implied Boundedness 7.25. Lack of Existence of Minimum 7.26. Functional Analysis 7.27. T h e Euler Equation and Haar's Device 7.28. Discussion 7.29. Successive Approximations 7.30. Lagrange Multiplier 7.3 I . A Formal Solution Is a Valid Solution 7.32. Raising the Price Diminishes the Demand 7.33. T h e Courant Parameter 7.34. Control Theory ~

Miscellaneous Exercises Bibliography and Comments

Chapter 8. 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9.

xix

259 259 260 261 263 264 265 265 267 268 269 270 27 1 272 273 274 275 277 279 280 281 282 282 283 284 284 286 287 288 288 289 289 290 291 29 1 301

Sturm-Liouville Theory

Equations Involving Parameters Stationary Values Characteristic Values and Functions Properties of Characteristic Values and Functions Generalized Fourier Expansion Discussion Rigorous Formulation of Variational Problem Rayleigh-Ritz Method Intermediate Problem of Weinstein

304 305 306 307 312 313 314 315 316

xx

CONTENTS

8.10. Transplantation 8.1 1 . Positive Definiteness of Quadratic Functionals 8.12. Finite Difference Approximations 8.13. Monotonicity 8.14. Positive Kernels Miscellaneous Exercises Bibliography and Comment

316 317 318 319 320 322 329

Author Index

331

Subject Index

337

METHODS OF NONLINEAR ANALYSIS Volume I

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Chapter 1 FIRST- AND SECONDORDER DIFFERENTIAL EQUATIONS

1.l.Introduction

I n this chapter we wish to prepare the way for the subsequent chapters by considering in some detail a number of results pertaining to firstand second-order ordinary differential equations. T h e pathbreaking consists of two parts. I n the first place, we want to observe the kinds of results that can be obtained in these relatively simple situations, and in the second place we want to note the methods that are employed. I n many cases the same methods can be employed to treat higher order equations. We will consider first the linear first-order differential equation

+p ( t ) u = q(t),

u(0)

u’

=

c,

(1.1.1)

and then the second-order linear differential equation U”

+ p ( t ) u’+ q ( t )

U = f(t),

( 1.1.2)

subject to both initial and two-point boundary conditions. T h e twopoint boundary conditions introduce Green’s functions. Following this, we will analyze the asymptotic behavior of solutions of u”f.(1 +f(t)) u

=

0

(1 .I -3)

as t -+ 00 wheref(t) is “small” in some sense. This will lay the groundwork for the study of the solutions of the more general equation 24’’

fR ( t ) U

=

0,

(1.1.4)

relying upon an ingenious change of dependent variable due to Liouville. T h e Riccati equation plays an important role in the study of (1.1.3) and (1.1.4). Although it appears here only as an artifice, its true fundamental role in analysis will be made apparent in Volume I1 in connection with dynamic programming and invariant imbedding. T h e results obtained in connection with (1.1.3) will permit us to discuss the validity of the WKB approximation. 1

2

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

Finally, we will indicate how elementary, albeit occasionally tedious, analysis can be used to obtain precise asymptotic behavior of the equation (1.1.5)

where p and q are polynomials in their arguments, in the important case where we are considering only solutions which remain finite for all t > t o , the so-called physical solutions. This work was initiated by Bore1 and Hardy. T h e explicit results obtained here can be used occasionally in the later chapters to illustrate the fancier analytic and computational methods designed to handle the more complex functional equations thrust upon u s by biology, economics, engineering, and physics. Although specific examples are absolutely essential, there is no harm done in occasionally minimizing the algebraic and arithmetic labor involved in this exposition. When an equation is displayed in this chapter, it will be tacitly assumed that a solution exists and, furthermore, that this solution is unique. I n Chapters 2 and 3 we will provide the necessary existence and uniqueness theorems validating both the analysis that follows and various statements that are made about the properties of these solutions. 1.2. The First-order Linear Differential Equation

T h e equation u' + p ( t ) u

=

dt),

4 0 ) = c,

(1.2.1)

plays an important role in the theory of differential equations, due equally to its simplicity and to the fact that it is one of the few equations which can be solved explicitly in terms of elementary functions and elementary operations. T o obtain this explicit analytic solution, we multiply by the integrating factor

and integrate both sides between 0 and t. We thus obtain

1.3. FUNDAMENTAL INEQUALITY

or u = cexp

3

1-

Let us denote the expression on the right by T ( p , q, c ) . This is an operation on the functions p and q and a simple linear function of c. Observe that it is also linear in q. Observe further that the positivity of the exponential function permits us to assert that the operation T is monotone in both c and q, that is to say, if c, 3 c2 , T(P, 9, c1> 3 T(P,9, cz), (1.2.4) 91 3 92 * T(P, 91, c) 3 T(p,qz ,4, if These properties will play an important role in the chapter on quasilinearization in Volume 11. Exercises

1. Use the preceding results to show that if u satisfies the differential 0, where v is inequality u' + p(t)u < q(t), u(0) = c, then u < v , t the solution of the corresponding equation.

2. Show that if

c

>, 0, q >, 0, then T ( p , , q, c)

3. How many solutions of

+

< T ( p , , q, c) for p , 3 p , .

u' au = 1 can satisfy the condition that 1imt+- u ( t ) exists ? Consider separately the cases a > 0, a < 0, a = 0.

+

4. Consider the same question for u' a(t)u = f ( t ) , under the assumption that the limits of a(t) andf(t) exist as t -+ 00.

5 . Obtain an explicit representation for the solution of u'

+ p,(t) u = pz(t)U"+1,

u(0) = c

1.3. Fundamental Inequality

From the monotonicity noted above we can readily establish the following basic inequality:

If u(t) < c 3 0, then

Lemma.

for t

+ JAu(tl)v ( t l ) dt, , where c > 0 and u, v 3 0 (1.3.1)

4

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

T o obtain this result from the result of the foregoing section, let rt

Then the integral inequality yields, after multiplication by v(t), the differential inequality ~

dw dt

< cv + vw,

w ( 0 ) = 0,

(1.3.3)

whence, taking account of Exercise 1 following the last section, w

< exp [,: v dt,]

1:

/cu(tl)exp

[-

f'

1

v(t2)dt2] dt,

.

(1.3.4)

Since the integration can be readily carried out, we have w

< c [exp

(1:

v dt,)

-

13.

(1.3.5)

Using this estimate in the original integral inequality, we have

11

=

c exp

(1:.

dt,),

(1.3.6)

the desired result. Exercises

1. Establish the foregoing result starting with the relation

and integrating between 0 and t. (This proof is much shorter, but it masks the reason for the validity of the result.)

2. What can we deduce if u(t) < f ( t ) v >O? 3 . Is the restriction

z1

+ $, u(tl)u(tl)dt, for t 3 0 with

3 0 essential ?

4. Carry through a proof of (1.3.1) by iteration of the inequality.

+

5. Consider the inequality u < h(t) aecbt $, ebL1w(t,)dt, J,,1 eb%u(tl) dt, \< v(t) where v' = heb1 av, u(0) = 0.

+

. Show

that

1.4. SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

5

6. Hence, show that if 0 < a < b and h(t) + O

as t-+ co, then f = o(g) to signify that f / g + 0. (Th e notation f = O ( g ) signifies that 1 f 111g 1 is uniformly bounded as t + co.)

Jot e b t w ( t l )dt,

=

o(ebl) as t + 00. We use the notation

+

(1 - a ) J: u(tl) dt, + b as t + co, where 0 can we say about the limiting behavior of u(t) ?

7. If au(t)

+

< a < 1, what

8. If au(t) (1 - a ) So u(tl) dt,/t -+ b as t -+ co, where 0 what can we say about the limiting behavior of u(t) ? 1

+

(1 - a ) C;=, uk/n+ b as n -+ co, where 0 < a can we say about the limiting behavior of u, as n + co ?

9. If au,

< a < 1,

< 1, what

1.4. Second-order Linear Differential Equations

Consider the linear equation U"

+ p ( t ) u' + q ( t ) u = 0,

(1.4.1)

where p and q are assumed continuous in [0, TI. This is a fundamental equation of mathematical physics, arising in analytical mechanics, quantum mechanics, wave propagation, and many other areas. It remains challenging despite all of the work that has been devoted to its study since we cannot read off the properties of the solution with the aid of a general explicit analytic solution in terms of the elementary functions and a finite number of integrations and differentiations. Every solution of this equation is a linear combination of two particular solutions u1 , u2 , the principal solutions, determined by the initial conditions =

1,

u1'(0) = 0,

Uz(O) =

0,

Uz'(O) =

Ul(0)

(1.4.2)

1.

T h e Wronskian of two solutions, u, v, of (1.4.1) is a useful function defined by the determinantal expression W(u,v) =

I

u

v

u)

v'

=

(1.4.3)

uv' - u'v.

It is easily seen that u'

21'

I

=

-p(t)

w.

(1.4.4)

6

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

I n particular, W ( u , , u2) = I in the case wherep principal solutions.

= 0, and

u and

are

Exercises

1. Use the Wronskian relation to obtain the general solution of (1.4.1) given one particular solution.

2. Show that W(u,v) # 0 for any t > 0 if W(u(O),v(0)) # 0.

+

3 . If zil and u2 are principal solutions, show that a,u,(t,) a,u2(tl)= b, , a,u,’(t,) + a,u,’(t,) = b, has a unique solution for the coefficients a, and a 2 .

4. If p ( t ) 3 0, show that W(u, , u 2 ) is a constant for t 3 0. 5. Show that W ( u , , u2) is never zero unless it is identically zero.

6. Show that the general solution of the linear difference equation

where a , , a2 are constants, is un = clrln

+ cZrzn,

where rl and r2 are the roots of the quadratic equation r2

provided that r, #

Y,

+ alr + a2

=

0,

. What is the form of the solution if rl

= r,

?

7. Show that a necessary sufficient condition that all solutions of the foregoing equation approach zero is that the roots of r2 a,r + a2 = 0 be less than one in absolute value.

+

8. Determine the analytic form of r, where

Y , + ~ = (ar, Hence determine the asymptotic form of rn as n + Y, u,&, and pick u, and v ,judiciously. 1

+ b)/(cr, + d ) . 00.

Hint: Set

1.5. INHOMOGENEOUS EQUATION

7

1.5. Inhomogeneous Equation

Consider first the case where the coefficients and u satisfies the inhomogeneous equation u"

p and q are constant

+ pu' + qu = f ( t ) .

(1.5.1)

We can take u(0) = u'(0) = 0 without loss of generality since it is sufficient to obtain any particular solution in order to obtain the general solution. Let us use a method which is particularly well suited to equations with constant coefficients, namely the Laplace transform. I t possesses the merit of extending immediately to many other types of functional equations such as linear differential-difference equations and linear partial differential equations and even to certain types of nonlinear equations containing convolution terms. Write L(u) =

I

m

(1.5.2)

e-%(t) dt

0

assumed convergent for Re(s) > 0. Actually, convergence is of no particular significance here since we use the Laplace transform solely to obtain the desired result. I t is then verified using a direct method. From (1.5.1) we derive by repeated integration by parts L(u) =

Let r1 , r2 be the roots of moment. Then

+ 4)

L ( f1

+ps

(IZ

(1.5.3)

*

+ p r + q = 0, assumed

P . ~

Hence, the inverse transform of (s2

+ PS + q)-'

distinct for the

is (1.5.5)

Let us now employ the fundamental relation for the Laplace transform of the convolution of two functions, L

(St

0

u(t

-

t l ) v(2,) dll)

= L(u)L(v).

(1.5.6)

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

8

1

This means that Jflu(t - tl) v(tl) dt, is the Laplace inverse of L(u)L(v) and thus, referring to (1.5.3), that u

1'

k(t

=

-

t l ) f ( t J dt,

0

.

(1.5.7)

Once the result has been obtained, we can readily verify by direct differentiation that it is indeed a solution. Hence, there is no need to impose any conditions on f ( t ) at t = co or worry about the convergence of the integrals involved. I n particular, if the equation is U"

we have r,

+ w2u = f ( t ) ,

(1.5.8)

and

= mi, r p = - w i

- e-wit

k(t) =

2wi

-

sin w t w

,

(1.5.9)

and thus the frequently useful result u =

-I I

t

sin w ( t -

(1.5.10)

w f l

Exercises

1. Determine the form of K(t) in the case that the roots r , and r2 are equal.

+

+

2. Obtain the general solution of u , + ~ a,u,+, 00 a2u, =f, using the generating function approach, i.e., set u(s) = Cn=ou,sn, etc. 3. Obtain an explicit expression for the Laplace transform of the solution of the differential-difference equation u'(t) = alu(t) a,u(t - l), t 3 1, u(t) = f ( t ) , 0 t 1, and thus a contour integral representation for u(t). For further results, see

<


*l(tl)l/W(tl

> t2)-

(1.9.3)

Without loss of generality, we can take u1 and u2 to be principal solutions so that W = 1 . Setting t = 0, we see that c1 = 0. T o obtain c 2 , we set t = T, 0 = C2U2(T)

+J

T

q(T9 tdf(t1) 4 .

(1.9.4)

Using this value of c2 in (1.9.2), we have

T h e dependence of the kernel k on T is not usually explicitly indicated. We ordinarily write

*=J

T

4 4 tl)f(tl) dtl ,

(I .9.6)

where

(1.9.7)

14

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

I t follows from this (cf. 1.9.7) that

a most important reciprocity relation. T h e function k(t, t l ) is called the Green’s function associated with the equation and the boundary conditions. Different conditions produce different Green’s functions. Exercises

<
(b) U” = f ( t ) , (c) 24’’ w2u = f ( t ) , (4 U” w2u = f ( t ) , (el u” wzu = f ( t ) ,

+ + +

the equations u(0) = u ( T ) = 0, u(0) = 0, U ‘ ( T ) = 0, u(0) = u ( T ) = 0 u(0) = 0, u‘(T) = 0, u(0) = 0, U ’ ( T ) bu(T) = 0.

+

3. Consider the Green’s function for Exercise 2(e) as a function of b. Is it continuous as b 4 0 and b -+ co ?

+ u = f ( t ) , u(0) = u ( T ) = 0.

4. Consider the Green’s function for EU” Discuss the limiting behavior as E + 0.

5. Similarly discuss the behavior of the solutions of €24’’ - u = 0, €U“

as

+ tu‘

E 4

--

u =

€24‘‘

0,

+ u‘ = 0, EU“

EU” - u‘ =

+ tu’ + u = 0,

0, €24”

EU”

+ tu‘

+ tu’ = 0, -

u/2 = 0

0. See,

C. E. Pearson, “On a Differential Equation of Boundary Layer Type,” J . Math. Phys., Vol. 47, 1968, pp. 134-154. 1.10. Riccati Equation

T h e change of dependent variable, u

=

exp

(1

z1

dt),

(1.10.1)

1.10. RlCCATl EQUATION

15

replaces the linear second-order equation u” +pu’

+ gu = 0

(1.10.2)

with the nonlinear first-order equation 0’

+

v2

+ g = 0.

+pv

(1.10.3)

This is called a Riccati equation and plays a basic role in the theory of the linear second-order equation. * T h e transformation which appears so formal here is actually well motivated by both analytic and physical considerations, as we will see in the chapters on dynamic programming and invariant imbedding in Volume 11. Exercises

1. If v satisfies a Riccati equation, show that v-l also satisfies a Riccati equation.

+

2. If v satisfies a Riccati equation, show that w

+

= (av b)(cv d) also satisfies a Riccati equation for any functions of t , a , b, c , and d. m

3. If v(t) possesses a power series expansion v ( t ) = En=, antn, and p

and q are analytic in t , show how to obtain a continued fraction expansion for v,

where c1 , c2 are positive integers. Hint: Set v

=

a,

+ a l t c l / v l , etc.

4. If p and q are analytic in a parameter E and v possesses a power series expansion in E, v,(t) q ( t ) ..., show how to obtain a continued fraction expansion for v,

+

+

v=vo+-

+

En1

1

b

E lv,

’

+

where b, is a positive integer. Hint: Set v, = vo E W , etc. (The last two results are connected with the subject of Pad6 approximations, a topic considered in the exercises at the end of the chapter.)

* Also in the study of nonlinear equations. See H. P. F. Swinnerton-Dyer, “On a Problem of Littlewood Concerning Riccati’s Equation,” Proc. Cambridge Phil. SOC., Vol. 65, 1969, pp. 651-662.

16

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

1.11. The Cauchy-Schwarz Inequality

I n the course of various estimations, we will make frequent use of the following basic inequality: (1.11.1) valid for any two real functions for which the right-hand side exists. Let us give two different proofs; each can be extended in different ways. Start with the inequality 0

< (U

b)'

~-

=

+

( 1 .1 1.2)

a2 - 2 ~ b b2,

valid for any two real numbers a and b, equivalent to 2ab

< a2 + b2.

(1.1 1.3)

Now set b

= g/(jTg2dt)lil, 0

(1.11.4)

obtaining (1.1 1.5) Integrating between 0 and T , we obtain (1.1 1.1). I n the second proof, we start with the fact that we have 0


1,

12. Let d(x) = (xlP - x2P - ... - x,P)l/P for xi in the region defined x,P)l/P, p 3 1. by xi 3 0, i = 1, 2,..., n, and x1 > (x2P + Show that if x = (xl,x2 ,..., xn), y = ( y l , y 2 ,...,y,), then x y belongs to the foregoing region when x and y do and that 0 . -

4(. Hint: Write +(x)

=

+

+

+ Y >2 4(4 + 4(Y>* n

min,(xi=l xixi) for z in a suitable region.

13. Show that

This is an example of the interesting class of problems which arise in an attempt to show explicitly that an expression is nonnegative. 1.12. Perturbation and Stability Theory

T h e equation u”

+ a(t)u = 0

(1.12.1)

19

1.12. PERTURBATION AND STABILITY THEORY

#

plays an important role in mathematical analysis and in mathematical physics, particularly in connection with the Schrodinger equation. Since the solution cannot, in general, be obtained in terms of the elementary functions of analysis and a finite number of operations of integration and differentiation, we must have recourse to a number of approximate techniques. We shall present only a few of the many ingenious methods that exist in the pages that follow. An important case to begin with is that where a ( t ) is close to a constant, which is to say, the equation has the form u”

+ (1 +f(t)).

= 0,

(1.12.2)

where f is small in some sense. Once we have analyzed equations of this quite special type in detail, we will be in an excellent position to study the behavior of solutions of (1.12.1) with the aid of a simple basic transformation due to Liouville. *

Exercises

+ +

1. Consider the equation u” (1 Ef(t))u = 0, where E is a parameter. Show that we can obtain a formal solution of the form u = uo E U ~ e2u2 + , where

+

+

u;

+ u, = 0,

u;

+ u1 +fu,

= 0,

and so on. We will examine the validity of perturbation expansions of this nature in Chapter 3.

2. Consider the associated Riccati equation v‘

+ v2 + (1 + 4)

Show that we have a formal solution of the form 0

where 0,’

and so on.

= 0,

+ + E’U1

+ + 1 = 0, Oo2

E2V2

Vl’

= 0.

+ .*.,

+ 2v,v1 +f

=

0,

+

3. Consider the equation u“ u = g with the formal solution u = g - U” = g - g(2) + g(4)- .*., Under what conditions does the series represent a solution of the equation ?

* See also K. Stach, “Die allgemeine Eigenschaften der Kummerschen Transformation zweidimensional Raume von stetigen Funktionen,” Math. Rev., Vol. 36, August 1968,

No. 1720.

FIRST- A N D SECOND-ORDER DIFFERENTIAL EQUATIONS

20

1.13. A Counter-example

I t is reasonable to expect that the condition f ( t ) +0 would ensure that all solutions of (1.13.1) u“ ( I +f(t)) u = 0

+

are bounded as t -+ GO * and, furthermore, are well approximated to by the solutions of the tame equation V’’

+

z, =

0.

( 1.13.2)

As the following example shows, the situation is far more complex. Consider the function u = exp

(j‘g(s) cos s ds1 cos t ,

( 1.13.3)

0

where g will be chosen malevolently in a moment. We have 21 21‘

upon setting w u”

=

=

Hence u”

-zu

=

w cos t ,

=

-w sin t

( 1.1 3.4)

+ (g cos2 t ) w,

1

exp(Jog cos s ds), and cos t

-

(g cos t sin t ) w - (2g cos t sin t ) w

+ (g‘cos2 t ) zu

+u

:

-

=

+ (g2

C O S ~t ) w.

+

w cos t[g’ cos t 3g sin t g2 cos2 t ] u[g’ cos t - 3g sin t g2 cos2 t ] , -

+

(1.13.5)

(1.1 3.6)

which means that u satisfies the equation u”

4-(1

+ 4)u

=

( 1.1 3.7)

0,

where (f, = 3s sin t

-

g’ cos t

- g2

cos2 t .

(1.13.8)

Choosing g = cos t i t ,

( 1.13.9)

we see that g, g‘, and g2 all approach zero as t + a.Nonetheless, the integral J” g(s) cos s ds diverges, which means that u increases indefinitely

* As

a matter of fact, a well-known mathematician published a “proof” of this.

1.14.

Jm j

f(r)i dt < co

21

in amplitude as t + co. Hence, the study of (1.13.1) has its interesting aspects in the case whereftt) + 0 as t + CO.

T h e example presented in the preceding section shows that it is not sufficient to require that f ( t ) -+ 0 as t + co if we want all solutions of U" (1 + f ( t ) ) u = 0 bounded as t+co. Let us now show that the condition

+

is sufficient. To do this, we convert the linear differential equation into the linear integral equation u = c1 cos t

t c2 sin t

-

st

sin(t

-

t l ) f ( t l ) u(tl)dt,

0

.

(1.14.2)

Hence IUI


u ( t l ) dt,

0

* Recall

and the associated

our previous use of the o-notation.

,

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

24

where v u =

= c1

v

-

-t c2 sin t . Iterating, show that

cos t

J‘ sin(t

-

t , ) y(tl>v(tl>dt,

0

2. Hence, show that all solutions are bounded if (a)

Jlg)(tl)dt, ,

1

F(tl) sin 2t, dt, , J“,g)(tl) cos 2t, dt, are uniformly bounded for t >, t o . Jto

J“l,

I y ( t l ) sin(t - t,) sin(t, - t2)g)(tl)dt, 1 dt, for some t o . What conditions on y ensure (a) and (b) ? (b)

3 . Show that all solutions of a f 2, b > 1/2.

u”

+ (1 + sin at/t”)u

=

5. Obtain the asymptotic behavior of solutions of u” t -+ cc under the hypothesis that J“ I g I dt < CO. g+o,

J“

/ g ’ l dt

u”

0 are bounded if

+ (1 + sin t2a)u = 0 are bounded

4. Show that all solutions of u” a > 1.

6. Are all solutions of

< k < 1 f o r t 3 to

+ gu‘ + u = 0 bounded

< oo?

1.17. The Equation u”-

(1

if

+ gu’ + u = 0 as if we assume that

+ f(t))u = 0

Let us now consider the equation u”

-

(1 + f ( t ) ) u

=

0

(1.17.1)

under the assumption that f ( t ) + 0 as t + CO. T h e behavior of the solutions of this equation is very much easier to discuss than that of z i ‘ - 1 ( I + f ( t ) ) u = 0 since we can employ the associated Riccati equation, v’

+

212 -

(I

+f(t)) =

0.

( 1.17.2)

We shall carry through the analysis in detail since it will give us an opportunity to introduce techniques that will be employed in more general form in Chapter 3. I n ( 1 . I7.2), set v = 1 4.-w. T h e n w satisfies the equation w‘

+ 2w

1-

w2 - f ( t )

=

0.

(1.17.3)

1.17. THE EQUATION U" - (1

+ f(t))u

25

=0

Regardingf(t) - w2 as a forcing term, we see that w satisfies the integral equation w = ce-2'

+

e-2'

j'

n

e2t1[f(tl)

-

w21 dt,

,

(1.17.4)

where c = w(0). Let us now show that if I c I is sufficiently small (a bound will be given below), the solution of (1.17.4) exists for all t > 0 and w -+ 0 as t -+ co. For our present purposes, we could choose c =0, but it is instructive to give the full result. To do this, we use the method of successive approximations. Set wo = ce-2'

w,+,

=

ce-2'

+ +

e-2'

J' e2t1f(tl) dt, , 0

e-2'

J'

e2t1[f(tl)

- w:] dt,

,

n

0

> 0.

(1.17.5)




J't e - z t l [ f ( t l )

Since we want w to approach 0 as t -+ convergent. Choose c so that c

+J

m 0

-

w2] dt,

0

GO,

- to21

dt,

.

(1.20.4)

the infinite integral should be

e-z'l[f(tl) - to21 dt, = 0.

(I .20.5)

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

30

Alternatively, we can begin with the integral equation w

OD

= eZt t

e-2t1[zo2 - f ( t l ) ] dt,

(1.20.6)

and employ the method of successive approximations to establish the existence of the desired solution,

[

'x

w o = -eZt

eUzt'f(t,)dt, ,

(1.20.7)

We leave it to the reader to carry out the details. We shall employ this device again in Chapter 3. 1.21. The Liouville Transformation

Let us now engage in some analytic chicanery which will reduce the equation of (1.12.1) to an equation whose coefficients are close to constants in many cases of significance. T o simplify the algebra, let us take the equation in the form u"

*

d ( t )u

===

0.

(1.21.l)

Perform the change of variable ( 1.21.2)

an idea due to Liouville. T h e n a simple calculation shows that (1.21.1) becomes d2u

-+-ds2

d ( t ) du &u-0. & ( t ) ds

(1.21.3)

We then see that there are two cases of importance where (1.21.3) is close in form to an equation with constant coefficients. T h e first is I in this interval," i.e., where the t-interval is finite and I a'(t)i a(t) is slowly varying. T h e second is where a(t)+ co as t + 00 and 1 a'(t)/a2(t)l 0 as t + co. As we know from what has preceded, a sufficient condition for


0, we see that m

f

$#o(t)dt

=

loga(t)]

co

= 03,

(1.21.5)

if a(t).+03 as t + 03. Hence, we require a further transformation of the equation in this case of frequent occurrence. Let us consider first the case u” + a2(t)u = 0. T h e equation with the minus sign is easier to consider. Exercises

1. Use the Liouville transformation to establish the following result of Gusarov: If u” ( I + f ( t ) u = 0 with 1 + f ( t ) >, a2 > 0, J“ lf”(t)\ dt < 00, then u is bounded as t + co.

+

2. Hence, discuss the boundedness of the solutions of U“

for 0

+ (1 + (cos

t”)/tb)

u =

0

< a, b < 1. See

R. Bellman, “Boundedness of Solutions of Second Order Linear Differential Equations,” Duke Math. J., Vol. 22, 1955, pp. 51 1514.

3. Show that if we set u = g exp[i J ( d t ) / ( p g 2 ) ]then , -(pu’)’ + qu = 0 is converted into ((pg’)’/g)- (I/pg4)= q. For applications of this result, see

J. Walter, “Bemerkungen zu dem Grenzpunktfallkriterium von N. Levinson,” Math. Zeit., Vol. 105, 1968, pp. 345-350. J. F. deSpautz and R. A. Lerman, “Equations Equivalent to NonVol. 18, linear Differential Equations,” Proc. Amer. Math. SOC., 1967, pp. 441-444. 1.22. Elimination of Middle Term

Consider the general linear equation u”

+ p(s) u‘ + q(s) u = 0.

(1.22.1)

32

FIRST- A N D SECOND-ORDER DIFFERENTIAL EQUATIONS

T h e change of variable

u = exp ( - s j1, d s ) v

(1.22.2)

converts this into the equation v"

+ (q

-

p'/2 - p74) v

=

(1.22.3)

0,

as a direct calculation shows. Thus, if we set

(1.22.4) we see that v satisfies the equation

(1.22.5) If the integrals

(1.22.6) converge, we can apply the results of Sec. 1.16 to conclude that the solutions of (1.22.5) are asymptotically given by the solutions of wn

+w

==

(1.22.7)

0

as s + co. From this we can determine the asymptotic behavior of the solution of (1.21.1), as we do below. Prior to this, let us see what the conditions of (1.22.6) are in the 1 t-variable. Since s = fo a(tl) dt, , ds/dt = a(t), we see that they are

< 00.

(1.22.8)

Exercises

+

1. Determine the asymptotic behavior of the solutions of U" tau = 0, a > 0; u'' e"'u : 0, a > 0; U" (log t)"u = 0, a > 0. T h e last equation illustrates the fact that it may be necessary to apply the Liouville transformation twice.

+

+

2. Determinc the asymptotic behavior in the cases where a

3. Determine the asymptotic behavior of the solution of ZL"

f(I

+ 1 j t " ) u -= 0,

I > a > 0.

< 0.

1.25.

U”

+ (1 + f ( t ) ) u = 0 ; ASYMPTOTIC BEHAVIOR

33

1.23. The W K B Approximation

Retracing our steps to the original equation, (1.21.1), we see that under appropriate conditions on a(t), e.g., those of (1.22.8), a useful approximation to the solution of u”

+ &(t) u = 0

as t -+ 00 is given by u

exp [i

1’

a(tl)dt,] /a(t)l/z.

(1.23.1)

(1.23.2)

1.24. The One-dimensional Schrodinger Equation

I n quantum mechanics, it is often a question of solving an equation of the form (1.24.1)

where t is constrained to a finite interval. Carrying out the foregoing changes of independent and dependent variable, we obtain the equation (1.24.2)

Hence, if the functions (1.24.3)

are well behaved over the corresponding finite s-interval, we can write m

v

=

v,,

+ C hkvk,

(1.24.4)

k=l

a convergent series, and thus, retracing our steps, obtain an approximation for u. 1.25. u”

+ (1 + f ( t ) ) u = 0; Asymptotic Behavior

Using the Liouville transformation, we can readily determine the asymptotic behavior of the solutions of u”

+ (1 + f ( t ) ) u = 0

(1.25.1)

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

34

in an important case where f

that

+

0 but J" I f I dt

=

co. Let us suppose

I f ' I dt < co.

Jm

(1.25.2)

Performing the change of variable s =

1'(1 0

(1.25.3)

+ f ) l I 2 dt, ,

we obtain the new equation ) zdu+ u = o .

(1.25.4)

T h e condition (1.25.5) is met if (1.25.2) is satisfied and f --t 0 as t + 00. Hence, (1.25.4) has solutions which are asymptotically of the form u1

=

[I

u2 = [l

+ o(l)] cos s, + sin s.

(1.25.6)

0(1)]

This means that (1.25.1) has solutions of the form f 1.25.7)

If, in addition, we suppose that u1

=

[I

u2 = [I

+ 0(1)3

s" f COS

[t

dt

< m, we can write

+ Jtfdt1:2],

+ o(1)l sin [I+ jtfdt1/21. 0

Exercise

1. Determine the asymptotic behavior of the solutions of U"

4-(1

+ l i t u )u = 0

for

a

> 0.

(1.25.8)

1.26. ASYMPTOTIC SERIES

35

1.26. Asymptotic Series

T h e techniques discussed in the foregoing pages enable us to find the principal terms in the asymptotic expressions for solutions of equations of the form 1 U" - ( 1 ta) u = 0. (1.26.1)

+

Suppose that we wish to obtain more refined estimates. I t is tempting to employ the following procedure. Set (1.26.2)

where the ai are constants, and substitute in (1.26.1). Equating coefficients of powers of l / t we obtain the recurrence relation n 3 1,

(1.26.3)

with a, = 1. Since

(

n2 a76 >I n ) a 7 6 - l = .(

-

1) % - I

>

(1.26.4)

we see that a, >, (n - I)! Consequently, the series in (1.26.2) diverges for all t. Nonetheless, it can be used to calculate numerical values of u(t) for large t! It is a particular example of an asymptotic series. These series of paramount importance were introduced into analysis in a systematic fashion by PoincarC and Stieltjes. Let us now give a formal definition. We say that a series S ( t ) = a,

a +2 + -a$ + ...

(1.26.5)

is an asymptotic series for a functionf(t) as t -+ co if the coefficients are obtained in the following fashion: a,

=

limf(t),

a,

=

l i m t ( f ( t )- a,),

an

=

l i m t n [ j ( t )- a, - ... - an-1

t-tm

t+m

t+m

31.

(1.26.6)

FIRST- A N D SECOND-ORDER DIFFERENTIAL EQUATIONS

36

We write (1.26.7)

When we use the notation (1.26.8)

we mean that e - t f ( t ) possesses the asymptotic series Cn=, a,tpn. T h e concept of asymptotic series is of particular significance in the theory of differential equations, where it is often not difficult to show the existence of generalized solutions in the form of asymptotic series. A fundamental problem is then that of establishing, under suitable hypotheses, that these correspond to actual solutions. m

- En=, m

1. Show that i f f

and thatf,

-

Clf

ant-*, g

+ czg -

+

-

[ao/+p (a,&

2. If a, # 0, show tl1atf-'

-

Exercises

+

-+

-Zr=,

m

(clan 11=0

b,t-",

then

+ czb,) t r n

+ a,b,)/t + -..I. [lla,, - a,/a,t

SL

m

+ ...I.

-

a,/t2 a,/t3 *.-, show that f dt, -a,/t - a,/2t2 *... Hence, show that i f f a, -1- a,/t ... and f ' possesses an asymptotic series, then f ' ---ul/t2....

3 . If f

+

4. Show by means of a specific example that f may possess an asymptotic series without f ' possessing an asymptotic series. Hint: Consider a case wheref 0 -1O / t ....

-

+

5. Let

for t

3 0. Show that

and obtain a linear differential equation satisfied by f ( t ) .

1.27. THE EQUATION

0‘

= P(u, t)/q(o, t )

37

6. Show that u” - (1 + I/t2)u = 0 possesses two solutions with the asymptotic series obtained formally by converting the differential equation into a suitable integral equation and proceeding as in the foregoing pages.

7. Obtain the detailed asymptotic behavior of the solutions of U“ - (1 + I / t ) u = 0 . For the foregoing and additional references, see R. Bellman, Stability Theory of DafJerential Equations, Dover Publications, New York, 1969. For asymptotic series associated with differential-difference equations, see the book by Bellman and Cooke previously referred to. For the delicate question of deciding how to use asymptotic series effectively, see

G. M. Roe, “An Accurate Method for Terminating Asymptotic Series,” J. SOC.Indust. Appl. M a t h . , Vol. 8, 1960, pp. 354-367.

1.27. The Equation u’

= p(u, t)/9(u, t)

Let us now show that we can analyze the nature of the solutions of

where p and q are polynomials in their arguments, using quite simple analytic and geometric ideas, provided that we agree to restrict our attention t o those solutions which are continuous for all large t. T h e methods are quite different from those we applied to study the Riccati equation. Our first result is that every such solution is ultimately monotonic. We must show that ti’ cannot vanish for a series of t-values of arbitrary magnitude, unless u = c, a constant. T h e proof is by contradiction. Suppose that u‘ vanishes at the sequence of points (tk>,where t,+ GO. Then the solution u and the curve z, = v ( t ) defined by the equation p(v, t ) = 0 intersect at these points. Since p is a polynomial in z, and t , the curve defined in this fashion possesses only a finite number of branches. Hence, the solution u must intersect one of these branches infinitely often. Without loss of generality, we may suppose that p and q possess no common factor and thus, that q possesses a constant sign in the immediate neighborhood of a solution of p = 0.

FIRST- AND SECOND-ORDER DIFFERENTIAL EQUATIONS

38

T h e branches of the form

p

=

0 which extend to infinity consist of curves of v

(a)

= c,

(1.27.2)

where q(t) is ultimately monotonic. Let us consider the second case first and show that the solution cannot intersect one of these branches infinitely often. For t sufficiently large, the points of intersection cannot be maxima or minima. Drawing a figure will convince the reader of this. Hence, the points of intersection of u and v are points of inflection. But again, a simple diagram will show that two points of intersection which are points of inflection must be separated by a point of intersection which is not an inflection point-a contradiction. We will provide an analytic proof below to supplement this intuitive geometric proof. 1,et us next consider the possibility of intersection of u with curves of the form v = c. These intersections must again be points of inflection of u and have one of the four forms shown in Fig. I . 1. We can eliminate (a) and (b) since u' changes sign in the neighborhood, but p(u, t)/q(u, t ) does not change sign if u c or u 3 c. If (c) and (d) occur, they can occur only a finite number of times since u can return to intersect v = c only by means of intersections with curves of the form v = v(t)or by means of intersections of type (a) and (b) with curves of the form = c1 .


c1 > ...,

a,

(1.28.3)

#0

At the intersection 1bltd'

+ ...,

do

> dl > ..., 6,

f 0

(1.28.4)

upon using the series in ( .28.3). Returning to (1.28.3), we have dv dt

- = aoc0tC"--l

+ alcltC1-l + -...

(1.28.5)

From this we see that for large t we have one of the following conditions: (1.28.6)

The persistent inequalities lead to a contradiction, as we see most easily from a figure. Hence at the points of intersection we must have duldt = dv/dt.If this holds for infinitely many t, we must have bo

= a,~,,

do

= C,

-

1,

6,

=

alcl

, dl = ~1

-

I ,...,

(1.28.7)

which means that dujdt = dvjdt. Hence, Y ( U , t) = 0 contains a solution of (1.27.1), which means that h(u, t) is constant for this solution. T h e case where dhjdt becomes infinite infinitely often is treated in exactly the same way. 1.29. Asymptotic Behavior of Solutions of u'

= p(u, t ) / q ( u , t)

We are now ready to demonstrate that any solution of (1.29.1)

40

FIRST- A N D SECOND-ORDER DIFFERENTIAL EQUATIONS

continuous for t 2 to , is ultimately monotonic, as are all of its derivatives, and satisfies one or the other of the relations (1.29.2)

where p is a polynomial in t, and c is an integer. Consider the expression q(u, t)u' - p(u, t ) which contains terms of the form altmu7' or bltnLunu'.From the foregoing discussion we know that the ratio of any two such terms is ultimately monotone and thus approaches a limit as t co. T h i s limit may be 0, fCQ, or a nonzero constant. Since the equation in (1.29.1) holds, there must be at least two terms whose ratio approaches a nonzero constant. If only one contains u', we obtain --f

UIUnpf

N

c1

.

(1.29.3)

If both or neither contain u', the result is u

N

(1.29.4)

c2t"/Q,

where p , q are rational. We will present the detailed analysis as a series of exercises. Exercises

1. Show that (1.29.3) leads to different results where n n n n

f-1, = -1 , =-I, f - I ,

m#+I, m f f l ,

m=+1, m=+1.

- + --

2. Show that Exercise ](a) leads to unt1/(n I ) and Exercise I(d) to un+'/(n 1 ) c1 log t .

+

3 . Show that Exercise l(b) leads to log u

which requires further consideration. Take 1 the equation

where the

cltl-m/(l

+ dl

+

cltl-m/(l - m ) d, , m > 0 and consider

p i and qi are polynomials in t . Show that

> 0,

m)

-

Y =s

c1 > 0. (Why is it sufficient to consider only the case I

4. Hence if c,

-

-

m

+ 1 if

> 0 ?)

1.29. ASYMPTOTIC BEHAVIOR OF SOLUTIONS

OF

U’

= P(u. t)/q(u, t)

41

for some a. Integrating, show that log Ic

=p(t)

+ CQ log t + O ( l / t ) .

5 . Show that c1 < 0 may be treated by replacing original equation.

zi

by l/u in the

6 . Consider Exercise I(c). There are two terms, atbucu’and dtb--luc+l, of equal order. Show that there is no other term of equal order.

7. Let

7 be any third term. Th en the quotient (atbuCu’ - dtb-1uc+1)/7 tends to a limit as t + CO. There are now two possibilities: (a) There is a third term whose order is equal to that of the difference between the two principal terms. (b) There is no such third term.

8. Consider the first possibility. T h e n either

or

Show that both of these lead to the stated result in (1.29.2).

9. Complete the analysis for part (b) of Exercise 7. More detailed

results can be obtained both for the solutions of polynomial equations of the form p ( t , u , u’)= 0 and for equations of the form u” = p(u, t)/q(u,t ) . See

G. H. Hardy, “Some Results Concerning the Behavior at Infinity

of a Real and Continuous Solution of an Algebraic Differential Equation of the First Order,” Proc. London Math. SOC.,Vol. 10, 1912, pp. 4 5 1 4 6 8 . R. H. Fowler, “Some Results on the Form Near Infinity of Real Continuous Solutions of a Certain Type of Second Order Differential Equation,” Proc. London Math. SOC.,Vol. 13, 1914, pp. 341-371. R. Bellman, Stability Theory of Da8erential Equations, Dover Publications, New York, 1969.

10. Use the foregoing techniques to analyze the behavior of the solutions of 71’ + 7 1 ~- a2(t)= 0 under various assumptions concerning a(t).

FIRST- A N D SECOND-ORDER DIFFERENTIAL EQUATIONS

42

Miscellaneous Exercises

1. Let p , ( t ) and q,,,(t) be polynomials of degrees n and m respectively whose coefficients are determined by the condition that

+

where f ( t ) = f,,+ f i t .--.These polynomials determine a Pad6 table and f ( t ) q,,,/p, is called a Padk approximation. Are p , and q,,, uniquely determined by this condition ?

2. Determine Pad6 approximants for p , ( t ) et

el.

Hint: Start with

+ qm(t) = tm+n+l + ...,

and differentiate to obtain recurrence relations.

+

+

+

+

3. Determine Pad6 approximants for (1

a,t a2t2)1/2.Hint: Uniformize by means of trignometric functions and count zeros and poles.

4. Determine Pad6 approximants for (1

+

a,t a,t2 u $ ~ ) ‘ / Hint: ~. Uniformize by means of elliptic functions. For the foregoing and additional references, see

R. Bellman and E. G. Straus, “Algebraic Functions, Continued Fractions and the Pad6 Table,” Proc. Nut. Acad. Sci., Vol. 35, 1949, pp. 472476.

+

+

5. Consider the differential equation U” au’ bu = 0, a, b > 0. Show that constants a, , a2 can be found such that u2 aim' U ~ U ’ ~ is positive for all nontrivial values of u and u’and, in addition, such

+

+

that

where a 3 , a4 > 0.

u’+ 0 as t -+co and, indeed, that some positive constants, a 5 , a 6 , without calculating the explicit solutions. This is a particular application of the powerful “second method” of Lyapunov we shall discuss in Volume I1 in connection with the topic of differential inequalities.

6. Hence, show that u2

+ ur2< a5ecaG1 for

u,

MISCELLANEOUS EXERCISES

7. Consider the equation u”

+

au’ and suppose that the roots of r2 parts. From the relations

s

m

+ + bu) dt = 0,

~ ( u ” au

0

43

+ bu 0, u(0) = c1 , u’(0) = c, + ar + b = 0 have negative real :

s

m

u’(u”

0

+ au’ + bu) dt = 0,

Sr

plus integration by parts, obtain expressions for ST u2 dt, uI2 dt as quadratic forms in c, ,c2 without using the explicit form of u.

sr

u2 dt as a quadratic form in c, and c2 , derive a necessary and sufficient condition for the roots of r2 ar b = 0 to have negative real parts in terms of a and b.

8. From the expression for

+ +

9. Similarly, derive necessary and sufficient conditions that the roots of r3

+ a1r2+ a,r + aR= 0 have negative real parts. See

A. Hurwitz, “Uber die Bedingungen unter welchen eine Gleichung

nur Wurzeln mit negativen reellen Teilen Besitzt,” Math. Ann., Vol. 46, 1895 (Werke, Vol. 2), pp. 533-545. H. Cremer and F. H. Effertz, “Uber die Algebraische Kriterien fur die Stabilitat von Regelungssystemen,” Math. Ann., Vol. 137, 1959, pp. 328-350.

10. Show that if a(t) 3 0 then all solutions of remain bounded as t --t co. Hint: d(u’2

+ u2)/dt = 2u‘u“ + 2uu’

=

2u’(--a(t)

u”

14‘ -

+ a(t)u’ + u = 0

u)

+ 2uu‘.

11. By consideration of the equation u“ + (2 + d)u‘ + u = 0, show that the condition that a(t) 3 a, > 0 is not sufficient to ensure that all solutions approach zero as t -+ a.

12. By means of a change of variable, show that f ( t ) - 0

as t + co cannot be a sufficient condition to ensure that all solutions of f ( t ) u ” u‘ u = 0 approach a solution of u’ u = 0 as t -+ co.

+ +

+

13. If a ( t ) > 0, show that no solution of u” - a(t)u = 0 can have more than one zero. Hint: Without loss of generality, let u ( t ) be negative between the two points t, and t, where u(tl) = u(tJ = 0. Let t, be

a point where u ( t )assumes a relative minimum value for t, < t < t , . Show that this leads to a contradiction upon using the fact that u“ - a(t)u = 0.

+ +

+

(1 E)U’ u = 0, u(0) = 1, u’(0) = 0, where E is a positive constant. Write the solution as u(t, E ) to indicate

14. Consider the equation EU’‘

FIRST- A N D S E C O N D - O R D E R DIFFERENTIAL EQUATIONS

44

thc dependencc on c. Does lime+ou(t, c) exist? Does it satisfy the differential equation u‘ u = 0 ? What is the initial condition ?

+

15. Consider the Green’s function k ( t , t l ) associated with u” =f, zi(0) -- ZL( T ) 0. Show that T


0, andf,(t),fi(t) are monotone increasing in

[a,

01, we have

MISCELLANEOUS EXERCISES

45

T h e upper sign holds if fi and f2 are both increasing or both decreasing. T h e lower sign applies if one function is decreasing and the other increasing (Cebycev). For some applications, see

J. A. Shohat and A. V. Bushkovitch, “On some Applications of the Tchebycheff Inequality for Definite Integrals,” J. Muth. Phys., Vol. 21, 1942, pp. 211-217.

+

21. T h e equation u’ - u2 = -2/t2 O ( l / t ) as t $ 0 implies that either u = l/t 0(1) as t $ 0 or u = -2/t O(1) as t $ 0 (Korevaar).

+

22. T h e equation

u‘ - u2 =

+

O(l/t) as t $ 0 implies that either as t $ 0.

u = O(1og l / t ) as t $ 0 or u = - 1 j t

2. Koshiba and S. Uchiyama, “On the Existence of Prime Numbers in Arithmetic Progression,” Proc. Japan Acud., VoI. 42, 1966, pp. 696-701.

23. Let {xk>, {fJ, and {zk},k = 0, 1,..., m be real valued sequences and let { z k )be nonnegative. If for k = 0, 1 ,..., m, X k 0. Using the fundamental )) x

-

lemma in Sec. 1.3, we have

y /) ,< eetllRII.

(3.3.9)

Since this holds for any E > 0, we must have 1) x - y 1) that x = y , the desired uniqueness.

=

0. T h is ' means

3.4. The Matrix Exponential

T h e formal analogy of the series in (3.3.6) to what is obtained in the scalar case, where the solution is eatc, prompts us to introduce the matrix exponential defined by the series (3.4.1)

T h e estimate used in (3.3.4) shows that the matrix series converges uniformly in any finite t-interval. Hence, we see that e A t satisfies the matrix equation -dX _

dt

-

AX,

(3.4.2)

X(0) = I.

T h e uniqueness of solution of (3.4.2) follows from the corresponding result for the vector equation. T h e functional equation eA(t+s)

~

(3.4.3)

eAteAs

can be demonstrated as in the scalar case by series expansion in powers of the exponent and appropriate grouping of terms using the absolute convergence of the series involved. From (3.4.3) we derive the basic result I

Hence,

eAt

= e A ( t - t ) = eAte-At.

is never singular and its reciprocal is

(3.4.4) erAt.

MATRICES A N D LINEAR DIFFERENTIAL EQUATIONS

108

Exercises

1. I t is instructive to regard (3.4.3) as an expression of the law of causality. Derive it from the uniqueness of solution of the equation dX

-~ AX,

X ( 0 ) = eAs.

-

dt

2. Show that we can establish uniqueness for (3.4.2) by viewing it as N vector equations of dimension N or as one vector equation of dimension N 2 .

3 . Using the functional equation of (3.4.3), establish the functional equations for cos t and sin t. 4. Show ~ ( 1 " )

that

-1

we

alu(N-l)

4-

can

establish

+ a,zi

7

existence

and

0, ~ ( ~ ' = ( 0ci ) ,i

by viewing it as equivalent to the system 21' =

u1 , ul'

= u2

,..., u;YPl =

u(0) = co ,

5. Show that

eAf

= lininim ( 1

-aluN--l

Ul'(0)

--

Ant.

uniqueness

0, 1 ,..., N

a2uN--2...

= c1 ,...

~

for 1,

-

aNu,

.

+ At/@)".

6. Consider the differcnce equation xntl X,

-

=

=

Ax,, x,,

= c.

Show that

+

7. Consider the difference equation y ( t A ) = Ay(t),y ( 0 ) = c, where t 0, 4 , 2 4 , .... Examine the convergence of y ( N A ) , where NA = t , as 4 0. ~

---f

3.5. Commutators

T h e relation in (3.4.3) is an analogue of what holds in the scalar case. On the other hand, eAleBl # e ( A + B ) t (3.5.1) for general A and B. If equality holds in (3.5.1) for -co we have, upon expanding,

< t < 00,

3.5. COMMUTATORS

109

Examining the coefficient of t2 on both sides, we have

B2 A2 -+-+AB= 2 2

A2

+ AB + B A 2

+B2

(3.5.3)

or AB

=

(3.5.4)

BA.

Thus, equality can hold only if A and B commute. If A and B do commute, we can group terms as in the scalar case to show that equality holds in (3.5.1). Since eAteBtis nonsingular, we know that it can be written as an exponential (see end of Sec. 3.7). The task of determining the “logarithm” of eAteBtis an interesting and important one. We leave it to the reader to show formally that eAteBt =

exp[(A

+ B)t -1(AB

-

+ ....I

(3.5.5)

BA)t2/2

T h e expression [ A , B ] = A B - B A is called the commutator of A and B and is a basic matrix function of A and B . T h e series obtained in the exponent is called the Baker-Campbell-Hausdorff series and plays a vital role in many investigations. See the exercises after Sec. 3.12 for some results in this area. Exercises

1. Let A and B be matrices which do not necessarily commute. Then &A+B)

=

lim

(eAt/neBt/n)n.

n-tm

See

H. F. Trotter, “On the Product of Semigroups of Operators,” Proc. Amer. Math. SOC.,Vol. 10, 1959, pp. 545-551. This has interesting applications to the study of the Schrodinger equation with Feynman integrals. See W. G. Faris, “The Trotter Product Formula for Perturbations of Semibounded Operators,” Bull. Amer. Math. SOC.,Vol. 73, 1967, pp. 211-215.

+

+

2. Consider the equation X = C € ( A X X B ) , where m parameter. Write X = C + enyn(A,B), where yo(A, B )

=

C,

yn(A, B) = Ayn-i(A, B )

+ ~ n - i ( AB)B, ,

E

is a scalar n

2 1.

110

MATRICES A N D LINEAR DIFFERENTIAL EQUATIONS

Show inductively that B ) = AnC

where

+ (7)A"-lCB + );(

An-2CB2

+ ... + CB",

(2) are the binomial coefficients.

3. Introduce a position operator P with the property that when it operates on a monomial consisting of powers A and B in any order with C somewhere, it shifts all powers of A in front of C and all powers of B after C. Thus, P(AaiBB1 . .. AakBBkBBkCAak+iBflk+i .. . Aa~Bfl,)= AzUiCBz@i.

Further, define P to be additive, P(mi(A, B )

+ m2(4 B ) )

=

P(m,(A, B ) )

+ P(m2(A9 B)),

where m1 and m, are monomials of the foregoing type. Show that %(A, B ) = P ( ( A qnC).

+

+ B)]-l C . E + , 0 for u, w 3 0. Show formally that limz+m y(u, x) = cp(u). 13. Prove that if (1) has a unique positive solution y(u), then (2) possesses a unique solution which converges monotonically to ~ ( u as ) x + co. 14. For what matrices A and B do we have eAeB= eA+B,in addition to those for which A and B commute ? For the foregoing and additional references, See

R. F. Rinehart, “The Equation eXeY = fl+y in Quaternions,” Rend. Circ. M a t . Polermo. Ser. 11-Tomo VII, 1959, pp. 1-3. Bibliography and Comments 03.1. For the more extensive coverage of the material in this chapter, see Chapters 10 through 13 of

R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1960. (Second Edition, in preparation.)

For a survey of numerical techniques, see

G . E. Forsythe, “Today’s Computational Methods of Linear Algebra,” SIAM Review, Vol. 9, 1967, pp. 489-515.

93.3. See B. W. Helton, “Integral Equations and Product Integrals,” Pac. J. Math., Vol. 16, 1966, pp. 277-322.

93.5. See W. Magnus, “Algebraic Aspects of the Theory of Systems of Linear Differential Equations,” Comm. Pure Appl. Math., Vol. 7, 1954. H. Wilcox, “Exponential Operators and Parameter Differentiation in Quantum Physics,” J. Math. Phys., V O ~8,. 1967, pp. 962-982.

132

MATRICES A N D LINEAR DIFFERENTIAL EQUATIONS

Kao-Tsai Clien, “Integration of Paths, Geometric Invariants, and a Generalized BakerIIausdorff I:ormula,” Ann. Math., Vol. 65, 1957, pp. 163-178. Scc also

W. 1,. Miranker and B. iVeiss, “The Feynman Operator Calculus,” S I A M Review, L7oI.8, 1966, pp. 224-232.

43.8. For ;I detailed discussion of what can be done using Kronecker sums and Lyapunov matrices, see A. T. Fuller, “Conditions for a Matrix to IIave Only Characteristic Roots with Negative I, 0 we have

where b, is a constant. We can do this inductively using the recurrence relation of (4.7.2). T h e result is valid for x1 = y = eAtc, and the induction can be established using the arguments employed in Sec. 4.7. Alternatively, we can start from

(4.8.5)

for some scalar b, independent of c. Hence (4.8.6)

Applying the fundamental inequality, we obtain


, 0, plus that of (4.8.4), allows u s to conclude that the integral e+lSg(x) ds converges. f

Jr


follows a familiar route. Hence, we have established the following theorem: Theorem.

Let the equation (4.13.1 5)

4.14. ALTERNATIVE N O R M

155

possess a solution over [0, TI. Let g(x) be analytic in x (guaranteeing uniqueness), and let 11 h l l M , as defined above, be suficiently small. Then the equation

2 = g ( y ) 3- h(t), dt

(4.13.16)

y(0) = c,

possesses a unique solution over [0, TI and

The uniqueness of the solution of (4.13.16) is a consequence of the postulated analyticity of g( y). We can reduce the requirement of analyticity, but this is of little interest since this condition is usually fulfilled in important applications.

Exercises

1. Suppose that x' = g(x), x(0) = c, possesses a unique solution over [0, TI, where g ( x ) satisfies a Lipschitz condition. Then y' = g( y ) + h( y ) , y(0) = c , possesses a unique solution provided that h( y ) satisfies a Lipschitz condition in the neighborhood of y

)I h(x)/l < E where c is sufficiently small.

=

x, and

2. Extend the result of preceding section to cover the case where dx/dt = g(x, t ) , dy/dt = g ( y , t ) h(t). 3. Consider the general case

+

under suitable assumptions. 4. Consider the case where x(0)

=

c, y ( 0 ) = d, (1 c - d (1

< E.

4.14. Alternative N o r m

In some cases, we want to use a different norm. For example, we may want to use the norm (4.14.1)

STABILITY THEORY AND RELATED QUESTIONS

156

T h e same type of argument as before establishes the existence of y and the inequality (4.14.2) II x - Y ll < 6, II h I12 or (4.14.3) I/ x Y 112 6 4 I/ h 112

e

-

7

under the assumption that (1 h /I2 is sufficiently small. We shall employ these results in connection with both differential approximation in Chapter 6 and the Bubnov-Galerkin method in Chapter 5. 4.15. Perturbation Techniques

It is also possible to study the equation

+ g(x),

dx --A x dt

x(0) = c,

(4.15.1)

by means of perturbation techniques. We can often profitably regard the coefficients in the expansion of g(x) as parameters. l’hus, for example, given the scalar equation u

we can write u

=

uo

+ €U2,

+ EU1 +

u(0) = c,

E2U2

+ -.*,

(4.15.2)

(4.15.3)

and thereby obtain an infinite sequence of differential equations

(4.15.4)

These equations can be solved recurrently since the nth equation is linear in a,. I t is easy to establish the convergence of the series in (4.15.3) for E I small. I n general, we can assert: ~

Theorem.

Consider the equation dx dt

-

-=

Ax

+,.(g

E),

x(0)

= c.

(4.15.5)

4.17. SOLUTION OF LINEAR SYSTEMS

157

Ifg(x, E ) is analytic in x and E f o r 11 x (1 and 1 E I small, and the previous conditions concerning A and g are satis$ed, then x is analytic in E f o r 1 E I small and each of the functions rpk(c) introduced above is analytic in E , f o r t 3 0. Since we shall not use the result, we will not present the details of the proof. Exercise

1. Establish the convergence of the series in (4.15.3) for 1

E

1

< 1.

4.16. Second Method of Lyapunov

I n the previous pages we presented one powerful approach for the discussion of the stability of the solutions of an important class of differential equations. Another method of great importance is the “second method of Lyapunov.” This method can be applied to many situations where the nonlinear term is of unconventional structure and also quite easily to more complex functional equations. On the other hand, it furnishes much less information concerning the nature of the solution. Since the technique is closely allied with the theory of differential inequalities, we will defer any discussion until the appropriate chapter in Volume 11. 4.17. Solution of Linear Systems

T h e foregoing discussion of the nonlinear equation dx dt

-

= A X +g(x),

~ ( 0= ) c

(4.17.1)

implicitly assumes that there are various attractive and simple aspects to the linear equation dY = A y ,

dt

y(0) = c.

(4.17.2)

If the dimension of y is small, we can reasonably think of using the explicit analytic form of y to calculate values of y(t).If the dimension is large, it may be far more convenient to calculate these values using some other approach. If a method based upon difference equations is employed,

STABILITY THEORY AND RELATED QUESTIONS

158

considerable difficulty can be encountered if the characteristic roots of A vary greatly in magnitude. This corresponds to the fact that (4.17.2) describes a physical process in which different subprocesses are going on with vastly disparate time scales. T h e problem encountered here is very close to the problem of ill-conditioning discussed in Chapter 2. We would like to emphasize the fact that there is nothing routine about the solution of high-dimensional linear systems of differential equations. Novel types of stability questions arise, particularly in connection with two-point boundary value problems.

4.18. Origins of Two-point Boundary Value Problems

We now wish to examine some properties of nonlinear differential equations subject to two-point boundary conditions. Equations of this nature arise in several important ways. I n the first place, in the process of minimizing a functional of the form =

subject to x(0)

= c,

X” -

Bx

1‘

[(x’, 3‘)

+ (x,Bx) + h(x)] dt,

(4.18.1)

we obtain the Euler equation -

g(x)

= 0,

x(0)

= c,

x’(T) = 0,

(4.1 8.2)

where g is obtained in a simple fashion from h. We will examine the rigorous details in the chapter on the calculus of variations (Chapter 7). T w o questions are now of interest. If a unique solution to (4.18.2) holds when g(x) = 0, the linear case, does a unique solution exist if 11 g(x)l( is small ? Secondly, if (1 g(x)ll = o(l1 x 11) as 11 x 11 -+ 0, and 11 c 11 is small, what is the asymptotic behavior of min, J ( x ) as T 4 00 ? These are important questions in connection with the theory of control processes. Questions of similar nature arise in the study of transport processes in plane-parallel slabs of finite thickness T . If we discretize both the angle and energy of the particle, we obtain a finitedimensional set of vector equations

4.19. STABILITY THEOREM FOR TWO-POINT BOUNDARY VALUE PROBLEM

159

where, as indicated in Fig. 4.1, x may be regarded as the right-hand flux and y the left-hand flux. The determination of limiting behavior as T -+ co is of interest in connection with the properties of very thick slabs and of planetary atmospheres which can often considered semi-infinite.

Figure 4.1

4.19. Stability Theorem for Two-point Boundary Value Problem

T o illustrate the kind of result that can be obtained, consider the scalar equation U" - u = g(u), ~ ( 0= ) C, u'(T) = 0. (4.19.1)

As before, we proceed to convert this into an integral equation (4.19.2)

where uo =

ccosh(t - T ) cosh T

(4.19.3)

and the Green's function is given by K(t, t, , T ) = - -

Observe that as T -+

cosh(T - t) sinh t, cosh T cosh(T - t,) sinh t , cash T 9

OC)

K(t, t, , t )

O 0, is w


1 . Is this condition necessary? 30. Show that u(t) = g ( t ) +

s:

e-alt-lll u(t dt, can be converted into a linear second-order differential equation subject to a two-point boundary condition, while the truncated equation, the equation for v, is an initial value problem.

31. Consider the equation

a > 1, with h not equal to a characteristic value, and the truncated equation v(t) = g(t>

If g ( t ) -+ 0 as t

+

+A S

t

0

e-alt-tll

u(t1) dt,

*

co, does I u(t) - v(t)l -+ 0 as t

-+

co ?

32. Let (un(t)} be a sequence of continuous functions defined over closed finite interval [a, b] which converges monotonically to

a a

continuous function u(t). Show that the sequence necessarily converges uniformly. (Dini.)

33. Let {un} be a sequence of values determined by the relation u, = N airin. Given the values {u,}, how does one determine N ? Hint: ~~satisfiesalinear differenceequationu,,,+b,u,+,-,+ bNun=0. See

+

MISCELLANEOUS EXERCISES

179

R. Bellman, “On the Separation of Exponentials,” Boll. U.M.I., (3), Vol. 15, 1960, pp. 38-39.

The problem is a particular case of a more general question of determining N and the parameters when it is known that u, = N L1 aig(ri , n). See

P. Medgyessy, Decomposition of Superpositions of Distribution Functions, Akademiai Kiado, Budapest, 1961. 34. Discuss the numerical stability of the procedure. 35. Let x ( t ) be a nonconstant periodic solution with periodp of x’ = F(x). If 11 F(x,) - F(x2)11 LJjx1 - x2 11 for all x1 , x2 , then p 2 24L.


1. 38. What is the asymptotic behavior of solutions as h 3 hp(t),p ( t ) > 0. Consider first the case p = 1 . See

oc)

if r2(t) =

J. Canosa and J. Cole, “Asymptotic Behavior of Certain Nonlinear Boundary-value Problems,” J. Math. Phys., Vol. 8, 1968, pp. 1915-1921.

+

39. Consider the Riccati equation u‘ = -u u2, u(0) = c . Show that it is equivalent to the infinite-dimensional linear system uh,, = -(n 1) u,+~ (n 1) u , + ~ ,U,+~(O) = cn+l, n = 0, 1,... . Hint: u, = un. This type of linearization was first used by Carleman.

+

+ +

40. Show similarly that any nonlinear differential equation of the form u‘ = g(u), u(0) = c, where g(u) is a power series in u, may be trans-

formed into a linear system of differential equations of infinite dimension.

41. Show that in place of the sequence {u”} we can use the sequence (P,(u)} of Legendre polynomials for the same purpose. Are there any advantages to using the sequence of Legendre polynomials as opposed to the sequence of powers ?

180

STABILITY THEORY AND RELATED QUESTIONS

42. Consider the truncated system Urt’ = - 7 W n

+ nu,+,

V,(o)

?Z =

== Cn,

1, 2,...,N - 1,

V N ( 0 ) = CN,

--Nu,,

u”

,

where 1 c I < 1. Write v n = vkN,”’. Show that limN+may’ = %l43. Consider the vector equation x’ = A x g(x), x(0) = c, where the components of g ( x ) are power series in the components of x, lacking constant and linear terms. Obtain the analogue of the Carleman linearization of Exercise 39.

+

44. Examine the validity of the truncation scheme analogous to that of Exercise 43 under the assumption that A is a stability matrix and that 11 c 11 1. For some results concerning truncation, see


(t - to>,

+

5.16. NEWTON-RAPHSON METHOD

205

Figure 5.1

as indicated. Let t,, the next approximation to r, be determined by the intersection of the tangent with the t-axis. Thus, tl

=

to -f(to)lf’(to).

(5.16.2)

Continuing in this way, we obtain the sequence (t,} generated by the recurrence relation tn+l = t n

-f(tn)if’(tn).

(5.16.3)

I t is clear from the figure that if u(t) is convex as indicated in the neighborhood of the zero, then t , converges to r, and indeed monotonically. But even more is true. Without loss of generality, take r = 0 which means that u(t) has the expansion

+ a,t2 + ..-, u’(t) = a, + 2a,t + .-., u ( t ) = alt

in the neighborhood of t tnfl = t,

-

=

[a$,

(5.16.4)

0. Then, from (5.16.3),

+ aztn2+ ...] / [ a l+ 2a2t, + ...I (5.16.5)

This is quadratic convergence. Computationally, it means that the number of significant figures essentially doubles on each iteration when we get sufficiently close to the root. Analytically, this type of approximation often provides a crucial additional degree of freedom in circumventing some troublesome difficulty. A rigorous proof of the convergence of t, can readily be obtained under various assumptions. If U ( t ) is not convex, the choice of the initial approximation is of the utmost significance.

206

THE BUBNOV-GALERKIN METHOD

Exercises

1. Consider the problem of determining a square root using the ancient 1 u,), ug = y, where 0 < x ,y . Show algorithm untl = F(x2/u, that u, converges to x as n + 00 for any initial choice of y and that the convergence is quadratic.

+

+ v,)

2. Set v, = u, - x. Show that v,+~= vn2/2(x v7L3 0 for n 3 I , and that u,+~ < v,.

and thus that


1. I n place of this approach, we can set y =

M

(7.16.2)

ckwk

k=l

as a trial function for the minimization of

and minimize over the ck subject to the initial condition ~ ( 0 = ) c. This N , this leads to a system of M linear algebraic equations. I< leads to a considerable simplification. Secondly, we can take as a trial function


, 1.

(7.21.3)

Under the foregoing assumptions {urn}converges uniformly to a solution of (7.20.1). Uniqueness is a different matter. As we have noted in Chapter 4, there is a unique solution in the class of functions satisfying a constraint such as I u 1 k, I c1 I. I n general, we cannot expect unrestricted uniqueness of the solution of the Euler equation. Consider, for exampIe, the problem of determining a geodesic on a torus as in Fig. 7.1. There is a minimum distance between P and Q


0

(7.22.2)

for all u, and a relative minimum within an appropriate class of functions v to a class of otherwise. Thus, for example, if we restrict u and u k, , it is clear that (7.22.2) will hold for functions such that 1 g”(6)l T < 1. An important consequence of the foregoing argument is that (7.22.2) holds trivially if g”(6) > 0, which is to say, provided that g(u) is convex in u.


- co for all u, and thus

I(.)

(7.26.2)

>, R, > --co

for all admissible u. T h u s lim J(u) = R, -

> --co

(7.26.3)

exists. Let {urn)be a sequence of admissible functions with the property that lim j(un) = R,. (7.26.4) n-m

We now show that actually {urn'} converges strongly in LZ(0, T ) . We have

Hence

+ 1: ( un

z

u r n ) dt

-

( un

urn)

.

(7.26.6)

by virtue of convexity, and (7.26.8)

we have

(7.26.9)

286

THE RAYLEIGH-RITZ METHOD

Hence, as m and n +

00, we

have

)

J: (

(7.26.10)

dt -+ 0,

the desired strong convergence. It follows that u,' converges strongly in L"0, T ) to a function zj €L2(0,T ) , and thus Jiu,' dt converges 1 strongly to a function TI dt, = u, for which the !&I is attained. Since g(u) is convex, J ( u ) is also convex, which means that u is the unique admissible minimizing function.

so

7.27. The Euler Equation and Haar's Device

I t is interesting to show that the function u singled out in this fashion satisfies the Euler equation. Let v be a function such that v' eL2(0,T ) with v(0) = 0. Then u 4-ZI is an admissible function and

(Replace g(u) by 2g(u) to simplify the arithmetic.) Since J(u) is the absolute minimum, we must have the variational equation

j n [2u'v' + 2g'(u) v7 dt = 0 T

(7.27.2)

for all functions of the foregoing type. Integrating by parts, we have ~u'v]:

+2

JT

n

+ g'(u)] dt = 0,

v[-u"

(7.27.3)

whence we would like to conclude that -u"

+ g'(.)

0,

u'(T)

0.

=

(7.27.4)

T h e difficulty with this direct approach resides in the fact that we do not know a s yet that u" exists. We know only that u' E L2(0,T ) . T o circumvent this difficulty, following Haar, let us integrate by parts in the reverse fashion. Namely, T

,$(Id)

v dt

I'

T

=

=

r:

---v

T

g'(u)]

0

[d j : g ' ( u )

+j T

0

[TI'

ds] dt,

I

T

t

g'(u) ds] dt

(7.27.5)

7.28. DISCUSSION

since v(0)= 0. Hence

1%

o' [u'

+ Sf g'(u) ds] dt = 0

for all v' eL2(0,T ) . But 2)'

287

=

u'

+J

T t

(7.27.6)

(7.27.7)

g'(u) ds

is itself a function of this class if u' €L2(0,T ) . Hence (7.27.6) yields (7.27.8) whence u'

+ ITg'(.) ds = 0

(7.27.9)

t

almost everywhere. Without loss of generality, redefine u' so that the relation holds everywhere. From (7.27.9) it follows that u' is the integral of a continuous function. Hence, we may differentiate, obtaining the desired Euler equation. Since every solution of the equation u"

-

g ' ( . )

=

0,

u(0) = c,

u'(T) = 0,

(7.27.10)

yields an absolute minimum of J(u), (7.27.10) has only one solution, the strong limit obtained above. This emphasis upon u' rather than u as the basic function appears rather strange from the standpoint of the calculus of variations but is completely understandable from the standpoint of dynamic programming, as we shall discuss in Volume 11. 7.28. Discussion

We have thus illustrated one of the fundamental services of the calculus of variations, that of providing a firm basis for the existence and uniqueness of solutions of functional equations. Often complex equations cannot be approached in any other fashion. Once existence and uniqueness have been established, the RayleighRitz method can be applied with confidence. Using an appropriate trial N function, u = C k Z l a k w k , we have (7.28.1)

288

THE RAYLEIGH-RITZ METHOD

a function of a finite number of variables. T h e various methods discussed in Chapter 5 , as well as a number of methods we have not mentioned, can be applied to the task of obtaining numerical values. There is no difficulty in applying the same ideas to the multidimensional versions of the variational problem. Naturally, the computational complexity increases as the dimension of the system increases. Furthermore, the relation is nonlinear. For example, the time required to solve a system of linear algebraic equations is roughly proportional to the cube of the dimension; hence, our constant reference to the size of N . 7.29. Successive Approximations

T h e rclativc facility with which we can handle quadratic variational problems suggests that there may be considerable advantage to treating more general questions by means of a sequence of carefully chosen approximations in terms of quadratic functionals. We shall pursue this in Volume I1 in connection with the theory of quasilinearization and duality. 7.30. Lagrange Multiplier

I n a number of important processes we meet the problem of minimizing a functional (7.30.1)

subject to an auxiliary condition such as H(u) =

J

T

h(u, u ' ) dt

=

k,

(7.30.2)

0

and an initial condition u(0) = cl. These problems are considerably more difficult than those so far considered, both analytically and computationally. There is, however, one general method which can be successfully employed in a number of cases, the method of the Lagrange multiplier. I n placc of the original problem, consider the problem of minimizing thc modified functional J(u, A)

=

ST 0

g(u, u') dt

+ X 1' h(u, u') d t , 0

(7.30.3)

7.32. RAISING THE PRICE DIMINISHES THE DEMAND

289

where A is the Lagrange multiplier. T h e idea of the method is to minimize J(u, A) subject only to the restriction u(0) = c, obtaining a minimizing function u(t, A) dependent upon the parameter A. A suitable value of A is then determined by use of the original integral constraint, leading to an equation

1: h(u(t,

A), ~ ' ( tA)), dt

=

k, .

(7.30.4)

T h e method has a simple geometric interpretation, as can be seen by considering the values of J and H as Cartesian coordinates as u assumes all admissible functional values. It is, however, perhaps most intuitive from the economic or engineering point of view. T h e constraint in (7.30.2) often arises as a restriction on the quantity of resources that can be employed to attain a certain objective. Alternatively, we may consider that we have an unlimited supply of these resources that can be used at a certain unit price. T h e parameter A then represents this price. It is reasonable to suspect that this price can be adjusted so that the total consumption of resources is equal to the initial quantity available. I n many important cases, this is true. 7.31. A Formal Solution I s a Valid Solution

As mentioned above, a detailed discussion requires a considerable amount of effort. Let us content ourselves then with the proofs of two simple, but highly important, results. T o begin with, let us suppose that u = u(t, A) furnishes the absolute minimum of J(u, A) and that (7.30.4) is satisfied. T h e n we maintain that u minimizes J(u) in (7.30.1) subject to (7.30.2). We proceed by contradiction. Suppose that

(7.3 I .l) Then a contradiction to the assumption that u provided the absolute minimum. 7.32. Raising the Price Diminishes the Demand

I n certain cases we can obtain u(t, A ) explicitly. I n general, however, it is necessary to solve the problem computationally for a set of values

290

THE RAYLEIGH-RITZ METHOD

of X and then to use some type of search, extrapolation, or interpolation procedure to obtain more accurate results. Intuitively, we would expect that as A, the “price,” increases, the functional H(u(t, A)) of (7.30.2), the quantity of resources used, will decrease. Let us demonstrate this. Let u = u(t, a ) correspond to X = a and 21 = u(t, b ) correspond to X = 0 . Then, by virtue of the minimizing property,

H(.)



r)l*

Show that Bl > B2 > ... > B , > A-l, and consider the possibility of the use of extrapolation techniques to obtain appropriate elements in A-l.

+

+

+

6. Let J(xI Y ) (x, B x ) 2(x, A y ) ( y , BY) - 2(a, x) 2(b, y ) , where B > 0 and A is real, not necessarily symmetric. Show that the variation equations are Bx Ay = a, A‘x By = b, when we minimize over both x and y . Under what conditions does an absolute

+-

minimum exist ?

+

+

7. Let J ( x , Y ) (x,C X ) 2 ( X , BY) - ( y , CY) - 2(x, 6 ) - 2(a, y ) , where B and C are real and symmetric, with C > 0. Show that min, max, J(x, y ) = maxu min, J(x, y ) . ~

8. Consider the problem of minimizing J&)

= J T (u’2 0

+ 9 )dt + h (ITu dt 0

-

.IZ

over all admissible ~ ( tsubject ) to u(0) = c, where X 3 0. What is the limiting behavior of the minimum value of JA(u)and the minimizing function as A + 00 ?

+

9. Consider the functional Jn(u) JOT (Xuf2 u2)dt, where A 3 0 and ZL is subject to u(0) c. Write g(X) = min, JA(u).Is g(h) continuous for X > 0 ? Is the minimizing function continuous for X 3 0 ? 7

+

-

-

10. Consider I( =. J) ~ ( U ’ ~ u2)dt, where u is subject to u(0) = c l , u( T ) cg . Show that T 4 0, u (cl + t(c,--c,)/T), u’ (cZ-cl)/T. :

+

11. Consider the minimization of J(u) subject to u(0) = c, u ( T ) bu’( T ) = a. Obtain the analytic representation of min, J ( u ) and the minimizing function, and study the behavior as a + co and b + CO. Could one predict this behavior without use of the explicit analytic representation ?

MISCELLANEOUS EXERCISES

293

12. Consider Jh(u)=

J

T 0

+

h

(M2 u2) dt,

> 0,

u(0) = c.

What is the behavior of min, fh(u)as T + co ? Does lim lim JA(u) = lim lim JA(u)? A-0

T+m 1-0

T+m

13. Determine the function which minimizes

where u(0) = u( T ) = 0.

14. Show that the minimum value of K(u>=

JT

+ u2 - 2h(t)u ) d t ,

(d2

0

subject to u(0) = u( T ) = 0, is a quadratic functional ah)=

-

ST 0

JT 0

k ( t , t l ) h(t) h(t,) dt d2,

Hence, show that K(t, t l ) is a positive definite kernel, that is,

for all nontrivial h(t). Hint: u

=0

is an admissible function.

15. Consider the problem of minimizing J(u) = ~ ~ udt,1 subject 2 to J t u 2dt = K , K > 0, with u(0) = c. Consider the associated functional

where u(0) is again equal to c, A 3 0. Show explicitly that J i u 2 dt decreases as h increases.

16. Consider the problem of minimizing J(u, o) =

I

T

0

+ v2)dt

(uz

294

THE RAYLEIGH-RITZ METHOD

+

over v, where u' = au v, u(0) = c. Show that v satisfies the equation v' + av = u, v ( T ) = 0, and that u and ZI are uniquely determined in this fashion.

17. Consider the minimization of

IT

+

( u ' ~ u2)d t ,

0

subject to u(0) = c1 , u( T ) = cz . Is it true that min J(u)

=

u(0)=c1

min [ min J(u)] ? c2

u(0)=c1 u(T)=c,

18. Show that the solution to the problem of minimizing

j T(u2 + v2)d t 0

over all v, where u' u - c [

v

c

=

au

+

+ v, u(0) = c, is given by

a sinh[dl u2(T t ) ] a sinh[dlfT]

[

~

~

~

dl d1

+ a2 c o s h [ d r + a2 cosh[dl + a2T] ~

~

s i n h [ w ( T - t)] u s i n h [ d c t 22'1 - d l - 4 - u 2 cosh[dl

+ a2T]

Determine the limiting behavior as T + 00 and as a + 00.

19. Consider the problem of minimizing J(u) =

lT

(u2 -{- u f 2 )d t ,

0

u(T)

u(0) = c,

Show that the minimizing function is given by u

--

c1 sinh t

and min J 11

=

(c2

+ c sinh(T sinh T

+ cI2)coth T

-

~

t)

2cc, cosh T

=

cI .

295

MISCELLANEOUS EXERCISES

20. Let R(c, T , A)

=

min u

[IT o

(uI2

+ u2)dt + h(u(T) - c ) ~ ],

u(0)

=

c.

Show that the minimizing function is given by u=

sinh T

X(c2

+ Xc sinh(T t ) + c cosh(T X sinh T + cosh T -

-

t) f

+ c12)cosh T + c2 sinh T - ~ X C C ,

+ cosh T Consider the problem of maximizing JOT u2d t , subject to the condition R(c, T , A)

21.

Xc,

=

1:

X sinh T

u'~ dt = A,,

u(0) = c.

Letf(c, T ) denote this maximum value. Is there a choice of c which maximizes, or which minimizes ?

22. Consider the problem of minimizing the function

over all u and v, with u(0) = c. What is the limiting behavior of the minimizing functions as h + ? For the preceding Exercises (8-22), see

R. Bellman, Introduction to the Mathematical Theory of Control Pyocesses, Volume I, Academic Press, New York, 1967.

23. Show that the condition that u € L 2 ( 0 ,a),where h eL2(0,a),implies that

for some constant c2 . Hence, show that if u , ELZ(0, m).

U" -

U"

u = h, with

€ L 2 ( 0 ,m), then

u'

24. Establish corresponding results for the case where I u 1, 1 U" I are uniformly bounded for t

3 0.

25. Using the identity d / d t [e&(u' u' = -u

- et

+ u)] = ect(u"

1

m

t

e-s [u"

-

-

u] ds.

u),show that

T H E RAYLEIGH-RITZ M E T H O D

296

Hence, show that max I u’ 1 >o

I

< 2 max I u 1 + max I U” I . 20 f>O

26. Replace u ( t ) by u(rt), where r

I

r max I u t >o for any r

t

> 0, and deduce that

< 2 max 1 u I + r2 max 1 u” 1 20 t>O

t

> 0. Hence, conclude that (max j 120

< 8(max I u l)(max I u” I). >o

u

t>O

t

sr

sr

27. Obtain corresponding inequalities connecting u2 dt, uf2dt, and un2dt. T h e constant 8 is not the best possible constant. For further results along these lines, and for additional references, see

Sr

G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, New York, 1934. E. F. Beckenbach and R. Beilman, Inequalities, Springer-Verlag, Berlin, 1961.

+

+

28. Show that if (u” a,(t)u’ a2(t)u)eL2(0,a),u €L2(0,a),and ~ u l ~ ~< ca l 2< \ m f o r t & 0 , t h e n u ‘ E L 2 ( 0 , a ) . 29. Assume that all of the following integrals exist. Then

I

co

(u“

0

4-u ) dt~

=

Srn un2d t + I‘“ dt u2

0

0

-

2

Srn

~ ‘ d2t ,

0

provided that either u(0) or u’(0) is zero. I n this case, for any positive r , r4

jm un2d t + 0

and thus

(s‘“ 0

Jco u2 0

dt - 2r2

unZdt)(Jm u2 d t ) 0

1

co

uI2 dt

2 0,

0

3

(1; d 2dt)’,

if either u(0) = 0 or u’(0) = 0.

30. Determine the minimum of 1 ; (u2 + blv2) dt over v, where au” 6u’ 4-cu = ZI, u(0) = c l , u’(0) = c 2 . See

+

0. I,. R. Jacobs, “The Damping Ratio of an Optimal Control System,” I E E E Trans. Automatic Control, Vol. AC-10, 1965, pp. 473476.

297

MISCELLANEOUS EXERCISES

31. Determine the minimum over v of J:

(uz - b2(u”)2) dt.

See

G. C. Newton, L. A. Could, and J. F. Kaiser, Analytic Design of Linear Feedback Controls, Wiley & Sons, New York, 1957, Chapter 2.

32. Determine the minimum over

u of

Jr + d 2 )dt. See (uz

R. E. Kalman and J. E. Bertram, “Control System Analysis and Design via the Second Method of Lyapunov,” Trans. A S M E , Ser. D., Vol. 82, 1960, pp. 371-373. C. W. Merriam 111, Optimization Theory and the Design of Feedback Control Systems, McGraw-Hill, New York, 1964. 33. If one assumes in all these cases that the optimal control (u is the control variable) is linear, z, = b,u + b&, where 6 , and 6, are constants, how does one determine these constants ?

- u - 2u3 = 0, u(0) = 0, u’(0) = c1 > 0. Show that u is nonzero for t > 0 in two ways: (a) Assume that u ( T ) = 0 and consider Jlu(urr- u - 2u3) dt. (b) Assume that T is the first value for which u ( T ) = 0, and examine the consequence of the fact that u has at least one relative minimum in [0, TI.

34. Consider the equation U”

35. Suppose that we employ a perturbation procedure to minimize J(u, E ) =

s:

+ + cu4)dt,

( u ’ ~ u2

where E > 0, and u(0) = c. T h e Euler equation is u”- u - 2 ~ u 3= 0, u(0) = c, u’(T)= 0. Set u = uo cul .-*, where u o , u1 ,..., are independent of T . Then

+

+

u;; - uo = 0, 241

- u1 - 224: = 0,

u0(O) = c,

u,’(T) = 0,

0,

ul‘(T)= 0,

u,(O)

=

and so on. Show that u o , u1 ,..., are uniquely determined, and exhibit uo , u1 explicitly. Does the perturbation series converge for any negative value of E ? Does the minimization problem have a meaning for any negative value of E ?

36. Consider the problem of minimizing

THE RAYLEIGH-RITZ M E T H O D

298

where E is a small parameter and x(0) = c. Show that the associated Euler equation x” - x - cg(x) = 0, x(0) = c, x‘( T ) = 0, has a solution by considering the Jacobian of ~ ’ ( tat ) t = T for E = 0. 37. What is the behavior of

as

E + 0,

where x(0)

=c

and A is positive definite ?

38. Consider the problem of minimizing

+

with respect t o y , where x’ = A x y and x(0) = c1 , x( T ) = c2 . To avoid the problem of determining when y can be chosen so as to meet the second condition, x ( T ) = cp , consider the problem of minimizing

0, where the only constraint is now x(0) = c1 . Study the for X asymptotic behavior of min J(x, y , A) as h -+ co, and thus obtain a sufficient condition that a “control” y exists such that x( T ) = c2 . For a discussion of the significance of questions of this nature, see the book on control theory by R. Bellman, cited in Exercise 22. 39. Consider the following generalization of the Riesz-Fischer theorem. 1 Let (cp,] be an almost-orthogonal sequence, that is, So vmyndt = amn , m # n, where

If

ni,n=O

I< c m

amnUmUn

c1

Un2-

n=O

Show that, given any sequence (bk} with function f such that

xr=ob,

< 00,

there is a

See

R. Bellman, “Almost Orthogonal Series,” Bull. Amer. Math. Soc., Vol. 50, 1944, pp. 517-519.

299

MISCELLANEOUS EXERCISES

There are very interesting applications of the concept of almost orthogonality in probability theory and analytic number theory.

40. Let L(u) be a linear functional on the space of functions u €L2(0,1). By considering the minimization of the quadratic functional J(u) =

s’

u2 dx

0

+ L(u)

over u E L2(0,I), establish the Riesz representation theorem, namely, L(u) =

uz, dx, 0

for some See

ZI

eL2(0,1)

R. Bellman, “On the Riesz Representation Theorem,” Boll. U.M.I., Vol. 20, 1965, p. 122. 41. Show that the minimum of J(u) =

SYn

+ zi2) + 2(u2 + zi2)li2]dt,

[u6/(u2

0

Jr

subject to the conditions that the integral exists and u2 dt = k, is furnished by the solutions of (uz zi2)1i3 = u4(u2 zi2)-2/3, and the minimum value is 3k, . Hint: Apply Young’s inequality,

+

s‘“f 0

+

+ 2 r n g 3 i 2dt > 3 1 fg dt, 2n

dt

0

0

for suitable f and g. See R. Bellman, “On a Variational Problem of Miele,” Astronautica Acta, Vol. 9, 1963, No. 3. A. Miele (editor), Theory of Optimum Aerodynamic Shapes, Academic Press, New York, 1965.

For other applications of the theory of inequalities to the solution of variational problems, see D. C . Benson, “Solution of Classical Variational Problems Using an Elementary Inequality,” Bull. Amer. Math. SOC.,Vol. 67, 1967, p. 418.

42. If u(t) is continuous in [a, 61, show that

THE RAYLEIGH-RITZ M E T H O D

300

43. Using the fact that max

a 0. Show that a search for particular solutions of the equation of the form u(x,t ) = e-+w(x) leads to the hk(x)w = 0, w(0) = w(1) = 0. Sturm-Liouville equation W" 1

+

304

8.2. STATIONARY VALUES

305

8.2. Stationary Values

Let us now indicate a connection between the Sturm-Liouville equation of (8.1.1) and the calculus of variations. Consider the homogeneous functional K(u) =

SoT

uf2dt Jf ~ ( tu2) dt

(8.2.1)

Let us determine necessary conditions satisfied by functions satisfying the conditions u(0) = u ( T ) = 0, u' eL2(0,T ) , which render K(u) stationary. As before, we suppose that I y 1 EL(O,T ) . We proceed at first in a formal fashion. Set u = ii EV, where u is a function in this class, E is a scalar, and ZI is an arbitrary admissible function, that is, v' cL2(0,T ) , v(0) = v ( T ) = 0. Starting with

+

(8.2.2)

take the derivative with respect to E and set it equal to zero. We obtain in this fashion the variational equation (8.2.3)

for all admissible v. Introduce the parameter (8.2.4)

Then (8.2.3) reads

1'

U'V' dt

0

-

h

j"

T

F(t) 0

uv

dt

=

0,

(8.2.5)

for all admissible v. Integration by parts yields the Euler equation

u"

+ A&)

u=0

(8.2.6)

and provides an interpretation for the constant A. Observe that the introduction of this parameter enables us to replace a nonlinear equation by a linear equation.

STURM-LIOUVI LLE THEORY

306

Exercise

1. What is a corresponding variational problem yielding U"

+ (A + p)(t)) u = 0,

u(0) = u ( T ) = O ?

8.3. Characteristic Values and Functions

I t remains to establish, as before, precise connections between the differential equation and the original variational problem. If we can, we are in a position once again to apply the Rayleigh-Ritz method. T o this end, consider the initial value problem

+ hp(t) u = 0,

u"

(8.3.1)

u'(0) = 1.

u(0) = 0,

Since we want u to be nontrivial and since the equation is homogeneous, there is no loss of generality in using the initial condition u'(0) = 1. This is a normalization. Write

c hnu,(t). m

u=

(8.3.2)

n=O

We wish to show that u is an entire function of h for each t in the interval [0, TI. Substituting in (8.3.1), we have a sequence of equations determining the coefficients

u;

u; =

0,

+ p(t) un-l

=

U,(O) =

0,

u,'(O) = 1,

0,

U,(O) =

u,'(O)

=

0,

n

2 1.

(8.3.3)

We see that (8.3.4)


0, then:

The values of h are real and positive. Hence, we can write A, < A, < .... (b) The associated solutions ul(t), ug(t),..., may be chosen (8.4.1) real and they are orthogonal with the weight function ~ ( t )that , is, q ( t ) u p j dt = 0, i # j .

(a)

JOT

Let us start with the orthogonality from which the other properties follow. From u; A,&) 24, = 0, (8.4.2) u; + Ajfp(t) uj = 0,

+

we have uiu;- uju;

+ (Aj

- hi) v(t)uiu3

=

0.

(8.4.3)

308

STURM-LIOUVILLE THEORY

Integrating between 0 and T , we obtain (8.4.4)

T h e boundary conditions satisfied by ui and uj reduce this to (Aj

-

Xi)

J Ty ( t ) U*Uj dt =

0.

0

(8.4.5)

If Xi # X i , we have

J:

p)(t) UiUj dt =

0,

(8.4.6)

the desired orthogonality. This holds with no restriction on p)(t). Henceforth, let us assume that cp(t) > 0. This assumption will enable us to use the orthogonality to good effect. Before proceeding further, let us show that u( T , A) = 0 cannot have a double root. If u(T, A) = 0, u,(T, A) = 0, then v(t) = u(t, A) and w ( t ) = u,(t, A) are simultaneously solutions of V''

w"

++(t) v + Xy(t) w

=

0,

v(0) v,

= -p)(t)

=

v ( T ) = 0,

w(0)

=

w(T)= 0.

(8.4.7)

€Ience wv" - vw" = p)(t) 02.

Integrating, 0 = [wv'- .w'],'=

s:

y ( t )212 dt,

(8.4.8)

(8.4.9)

a contradiction if ~ ( t>) 0, v # 0. Hence, hi # hi for i # j , and (8.4.6) holds for i # j . From this we can easily show that the A, are real and thus that the ui can be chosen real. For if A, , u, , ui are respectively characteristic values and functions with Xi # Xi (here, iii denote the complex conjugates of X i and ui respectively), we have, from (8.4.6),

xi

~

xi,

(8.4.10)

a contradiction if ~ ( t>) 0, as we have assumed,

8.4. PROPERTIES OF CHARACTERISTIC VALUES AND FUNCTIONS

309

Let us note for further reference that (8.4.6) plus integration by parts yields the additional orthogonal property

sr

(8.4.1 1)

i # j.

uiujl dt = 0,

So far, we have used the normalization ~ ~ ’ (= 0 1) , i = 1, 2, .... Let us replace this by the more convenient S:&)uiZdt

=

i = 1 , 2)....

1,

(8.4.12)

Then (8.4.13)

T h e second result follows by means of integration by parts. Exercises

1 . Assume that the function y is positive for 0

<
h > b i l l k , k = 1, 2, ..., and that {bk/bk+l} is monotone decreasing while {bkllk> is monotone increasing. Show that

b

- lim k = lim b-"k - k-.m bk+l k-.m

'

'

8. For large k which estimate furnishes a better approximation to h, , bk/bk+l or bk1lk ?

9. Applying these results to the equation U" + A( 1 + t)u = 0, u(0) = U( 1) = 0 (which can be treated explicitly by means of Bessel functions), show that we obtain the values in Tables I and 11. TABLE I .v"(l)

n

0

1 2 3 4 5

6 7 8 9 10

1. 1. 2. 3. 4. 7. 10. 16. 24. 36. 53.

=

(-1)"(2n

+ l)!ZP(l)

000 500 238 333 960 378 97 1 31 1 248 043 572

0000 0000 0952 3333 730 2054 506 836 509 599 540

TABLE I1

k 1 2 3 4 5

6 7 8 9 10

bk/bk+l

.25000000 .025198413 3.6210317 5.4595362 8.3125740 1.2685019 1.9367810 2.9575153 lo-' 4.5163492 lo-* 6.8968612 lo-a

9.921260 6.958904 6.632490 6.567805 6.553064 6.549537 6.548676 6.548465 6.548412

b;'lk

4.000000 6.299606 6.512122 6.542008 6.547159 6.548143 6.548342 6.548384 6.548393 6.548395

312

STURM-LIOUVILLE THEORY

10. Obtain an improved estimate for A, by use of the Newton-Raphson technique.

11. Obtain an estimate for X,h, using the determinant

. Is the sequence {(bi2))lii}monotone ? Show that 6 ~ 2 ) / b> ~ ~b::),/6:2:, , See for the foregoing R. Bellman, “On the Determination of Characteristic Values for a Class of Sturm-Liouville Problems,” Illinois J. Math., Vol. 2, 1958, pp. 577-585. P. F. Filchakov, “An Effective Method for Determining the Eigenvalues for Ordinary Differential Equations,” A N UkrRSR, Scriya A, Fizyko-teknichni te matematychni nauky, no. 10, 1967, pp. 883-890. C. Shoemaker, Computation of Characteristic Values of SturmLiouville Problems with a Digital Computer, University of Southern California, USCEE-267, 1968. T h e point of the foregoing exercises is that the higher characteristic values can usually be determined quite accurately from their asymptotic expansions and that it is the first few characteristic values which are difficult to determine accurately. Consequently, it is worth developing special techniques specifically for the determination of the small characteristic values. We shall subsequently discuss methods for determining the first N characteristic values at one time. 8.5. Generalized Fourier Expansion

T h e orthogonality relation of Sec. 8.4 (8.4.13) makes it tempting to expand a function u(t) in an orthogonal series using the uk(t). If we set (8.5.1)

the coefficients will be determined by means of the relations (8.5.2)

8.6. DISCUSSION

313

Formally, we have the Parseval relation

J

T 0

C m

p2dt

=

ak2

(8.5.3)

k=l

and, by virtue of the second relation in (8.4.13), the additional result

j

T

C X,ak2. m

d2dt=

k=l

0

(8.5.4)

Indeed, the principal purpose of the foregoing sections has been to obtain these relations, (8.5.3) and (8.5.4), which enable us to discuss the minimization of K(u) =

j r ur2d t / j

T

yu2 dt

0

(8.5.5)

in a simple, rigorous fashion.

8.6. Discussion

For the case of second-order linear differential equations, of the type discussed above, the behavior of the characteristic functions and characteristic values can be analyzed in detail, as indicated in the exercises at the end of Sec. 8.4. This enables us to use equiconvergence theorems of the type due to Haar to conclude that essentially what is valid for ordinary Fourier series is valid for Sturm-Liouville series. This approach can be pursued for higher-dimensional ordinary differential equations. For partial differential equations, the situation is quite different. I t is necessary to use the Fredholm theory of integral equations to develop a theory of characteristic values and functions and, indeed, the Fredholm theory was created specifically for this purpose. T h e basic idea is to convert an equation such as u”

+ Xcp(t) u = 0,

u(0) = u(T) = 0,

(8.6.1)

into the homogeneous integral equation

j K(t, T

=

0

s) g)(s) u(s) ds,

by means of the appropriate Green’s function.

(8.6.2)

STURM-LIOUVILLE THEORY

314

T h e theory of integral equations, a la F. Riesz, is part of the general theory of linear operators in Hilbert space. We have not mentioned the theory, nor used the notation, since we have no particular need of it in this or the second volume. Rather than provide any ad hoc proof of the Parseval relations of (8.5.3) and (8.5.4), we shall assume their validity under the condition that .p(t) is a positive continuous function and proceed from there. 8.7. Rigorous Formulation of Variational Problem

Using the results of (8.5.3) and (8.5.4), we can write

-x.,"=,

for u €L2(0,T ) , u(0) = u( T ) = 0, u sentation of K ( u ) , it is immediate that

(8.7.1)

a,u,(t). From this repre-

rnin K(u) = A,, U

(8.7.2)

where u runs over the admissible functions, and further that the minimum is attained uniquely for u proportional to u1 . Furthermore, this representation in (8.7.1) clearly shows the significance of the higher characteristic values as relative minima. Thus, A,

=

m:, l(4,

(8.7.3)

where R, is the region of function space determined by u(0)

=

u ( T ) = 0,

(8.7.4)

u' E L2(0,T ) ;similarly,

A,

=

min J(u), RZ

over the region R, determined by u'

(8.7.5)

E L2(0,T ) , and

u(0) = u(T) = 0,

1;

p(t) uu, dt = 0,

(8.7.6)

8.8. RAYLEIGH-RITZ METHOD

315

Once (8.7.1) has been established, we can invoke various min-max representations of the higher characteristic values which are far more convenient for many analytic purposes. 8.8. Rayleigh-Ritz

Method

As noted above, in many important scientific investigations only the characteristic values are of interest. They correspond to natural frequencies, energy levels, and so forth. Hence, it is advantageous to process a variational representation which provides a quick way of obtaining approximate values in terms of relatively simple algebraic operations. Given the expression X

min

- u(O)=u(T)=O

S,'

ut2dt J,' ~ ( tu2) dt '

(8.8.1)

we can use the Rayleigh-Ritz method in the expected fashion to obtain upper bounds for A, and, with enough effort, accurate estimates. Set N

==

(8.8.2)

>

k=l

where vk(0)= vk( T ) = 0, with the v k conveniently chosen functions and the ak free scalar parameters. We can obtain an upper bound for A, by minimizing the expression (8.8.3)

with respect to the ak , the components of the vector a. Here A and B are symmetric matrices, A

=

(l'

z~;w~' d t )

0

, (8.8.4)

This is an algebraic problem which, as we know, leads to the problem of determining the characteristic roots of a determinantal equation of order N , namely I A - AB I = 0. Let the characteristic roots obtained -.- A',". in this fashion be hiN) A$'" Not only does X i N ) yield information concerning A,, but the other characteristic roots, A i N ) ,..., A("), yield information concerning A, , A, ,....


.

(8.10.2)

Frequently, we want estimates of the variation of A, as E changes, or perhaps as T , the interval length, varies. Consider the first question. Set K(u, €)

=

J;

u'2

d t / J T g)(t, €) 242 dt. 0

(8.10.3)

I n order to obtain estimates, corresponding say to E = 0, E = e l , we use the solution for E = 0 as an estimate in K(u, E , ) and the solution for E = el as an estimate in K(u, 0). This is called transplantation. It often yields estimates precise enough to use for many purposes. Thus, (8.10.4)

8.11. POSITIVE DEFINITENESS OF QUADRATIC FUNCTIONALS

317

Exercises

1. Consider the case where ~ ( tc), = p)(t) + ~ + ( t and ) , obtain an estimate for I A,(€) - Al(0)l. 2. Show that A,( T ) is monotone decreasing as T increases. 3. Obtain in this fashion some estimates of the variation of A,( T ) as T , the interval length, changes. 4. Obtain an estimate for min U

where

E

>, 0 and

'.T

J

+ + 4 dt,

( u ' ~ u2

u is subject to the conditions u(0) = u( T ) = 0.

8.11. Positive Definiteness of Quadratic Functionals

With the aid of Sturm-Liouville theory we are in a position to determine the precise interval [O, TI over which the absolute minimum of the quadratic functional J(u) =

I

T

( u ' ~- ~ ( tuz) ) dt,

0

(8.11.1)

subject to u(0) = c, u( T ) = d, exists and is determined by the solution of u"

+ T ( t ) u = 0,

u ( T ) = d.

u(O) = C,

(8.1 1.2)

We suppose that g)(t) > 0. Let {An}, {un) be respectively the characteristic values and functions associated with the Sturm-Liouville equation u(0) = u ( T ) = 0. (8.11.3) u" + + ( t ) u = 0, Returning to (8.11.1), let us write u = g(t) + v, where g ( t ) is the linear function of t satisfying g(0) = c, g( T ) = d. T h e n

J@) = J(g

+4

=

J(g)

+2

T

(g'v' - v(t)gv)dt

+ .I(.(8.11.4) ).

T h e positive definite nature of J(u) depends then upon the positive definite nature of J(v). Writing

-

>

k=l

(8.1 1.5)

318

STURM-LIOUVI LLE THEORY

we have (8.11.6)

Hence if A1 > I , the functional J(u) is positive definite. Since Al(T) decreases monotonically as T increases, it is possible that there is a critical value of T for which A,( T ) = 1 . Call this value T, . For T > T, , J(u) does not possess an absolute minimum. Exercises

1. If J ( u ) = S value ?

2. If J(u) = value ?

~ ( U -’ ~u z ) dt,

J ~ ( U ’~

u(0) = c, u ( T ) = 0, what is the critical

u2)dt, u(0) = c, u’(T) = 0, what is the critical

3. What happens if q ( t )changes sign over [0, T ] ? 8.12. Finite Difference Approximations

We can reduce the problem of obtaining estimates for the Ai to an algebraic problem in another fashion. Use the simple quadrature approximations

j: ~ ( tu2) dt g A 1 &l)u2(kA), N-1

,:

k=O

dt g d

(8.12.1)

N

1 [u(kA)

~

u((k

1) A)I2.

~

k=l

Let ufkd) = Uk ,

&?A)

=

cpk

.

(8.12.2)

T h e n the new problem is an algebraic one of minimizing the quotient (8.12.3)

Here uo = a,,, = 0. T h i s again leads to the calculation of the characteristic values of a matrix.

8.13. MONOTONICITY

319

8.13. Monotonicity

As d -+ 0, we can expect that the characteristic values and characteristic functions obtained in this fashion will approach those associated with the differential equation. This may again be regarded as a stability problem, which we will not pursue here. Let us show, however, how we can obtain monotonicity by a slight change in the foregoing formulation, Instead of using the approximations of Sec. 8.12, write U‘(t)=Uk,

k = o , 1 , ..., N - 1 .

kLl < t < ( k + l ) d ,

(8.13.1)

Th i s is an approximation in policy space, to use the terminology of dynamic programming. This will be discussed in detail in Volume 11. It reflects the idea previously stated that u’, rather than u, is the basic function in variational problems. Then

(8.13.2)

. (8.13.1) yields Write v k = ~ ( k d )Then (8.13.3)

whence = vk

vk+l

and p)(t) u2

at

=

I

+

(8.13.4)

ukA

(k+l)A RA

p)(t)[vk

$-

u k ( t - kLl)]2

at*

(8.13.5)

I n this way we obtain another algebraic problem. If we let h,(d) denote the smallest characteristic root obtained in this way, clearly

4 <W ) *

(8.13.6)

Once again we have monotone convergence of the approximating values as d +O. Extrapolation procedures, such as deferred passage to the limit, to estimate A1 can be used once a sequence of values of the form {Ai(d/2k)}, K = 1, 2, ..., R, has been obtained. Furthermore, it is easy t o see that we have convergence since we can approximate to the function u’(t) arbitrarily closely by a step function of the type described in (8.13.1).

320

STURM-LIOUVILLE THEORY

8.14. Positive Kernels

T h e representation

makes it easy to obtain upper bounds for A,. How do we obtain lower bounds? One approach already mentioned is through the theory of “intermediate problems” of Weinstein. Another approach is based on the theory of positive operators. I n Chapter 2 we indicated corresponding results for positive matrices. Let us here describe an application to integral equations. We begin with the conversion of the equation U”

+ XVU = 0,

(8.14.2)

u(0) = U(1) = 0

into the integral equation

where k, is the appropriate Green’s function as discussed in Sec. 8.6. Let us replace A by its reciprocal and consider the integral equation Xu(t) =

JU

(8.14.4)

R(t, s) U ( S ) ds,

0