CONTENTS
Preface vii
Chapter 1. First- and Second-order Differential Equations
. 1. Introduction 1
.2. The First-order Linear Differential Equation 2
.3. Fundamental Inequality 3
.4. Second-order Linear Differential Equations 5
.5. Inhomogeneous Equation 7
.6. Lagrange Variation of Parameters 8
.7. Two-point Boundary Value Problem 10
.8. Connection with Calculus of Variations 11
.9. Green’s Functions 12
.10. Riccati Equation 14
.11. The Cauchy—Schwarz Inequality 16
. 12. Perturbation and Stability Theory 18
. 13. A Counter-example 20
.14. J°° |/(t)l dt < oo 21
.15. J°°|/’@l* < oo 22
. 16. Asymptotic Behavior 23
.17. The Equation u” — A /@)« = 0 24
.18. More Refined Asymptotic Behavior 26
.19. J°°/>A < oo 27
.20. The Second Solution 29
.21. The Liouville Transformation 30
.22. Elimination of Middle Term 31
.23. The WKB Approximation 33
.24. The One-dimensional Schrodinger Equation 33
.25. u” A f{t))u = 0; Asymptotic Behavior 33
.26. Asymptotic Series . 35
.27. The Equation «’ = p(u, t)jq(u, t) 37
.28. Monotonicity of Rational Functions of ? and t 38
.29. Asymptotic Behavior of Solutions of u’ = p(u, t)jq(u, i) 39
Misrellaneous Exercises 42
UiMioRraphy and Comments 51
Chapter 2. Matrix Theory
2.1. Introduction 54
1.2. I)ctcrminantal Solution 55
2.3. Elimination 58
2.4. Ill-conditioned Systems 59
2.5. The Importance of Notation 60
2.6. Vector Notation 60
2.7. Norm of a Vector 61
2.8. Vector Inner Product 61
2.9. Matrix Notation 63
2.10. Noncommutativity 64
2.11. The Adjoint, or Transpose, Matrix 65
2.12. The Inverse Matrix 65
2.13. Matrix Norm 67
2.14. Relative Invariants 68
2.15. Constrained Minimization 71
2.16. Symmetric Matrices 72
2.17. Quadratic Forms 74
2.18. Multiple Characteristic Roots 75
2.19. Maximization and Minimization of Quadratic Forms 76
2.20. Min-Max Characterization of the Xk 77
2.21. Positive Definite Matrices 79
2.22. Determinantal Criteria 81
2.23. Representation for A’1 82
2.24. Canonical Representation for Arbitrary A 82
2.25. Perturbation of Characteristic Frequencies 84
2.26. Separation and Reduction of Dimensionality 85
2.27. Ill-conditioned Matrices and Tychonov Regularization 86
2.28. Self-consistent Approach 88
2.29. Positive Matrices 88
2.30. Variational Characterization of ?(?) 89
2.31. Proof of Minimum Property 91
2.32. Equivalent Definition of ?(?) 92
Miscellaneous Exercises 94
Bibliography and Comments 101
Chapter 3. Matrices and Linear Differential Equations
3.1. Introduction 104
3.2. Vector-Matrix Calculus 104
3.3. Existence and Uniqueness of Solution 105
3.4. The Matrix Exponential 107
3.5. Commutators 108
3.6. Inhomogeneous Equation 110
3.7. The Euler Solution 111
3.8. Stability of Solution 113
3.9. Linear Differential Equation with Variable Coefficients 114
3.10. Linear Inhomogeneous Equation 116
3.11. Adjoint Equation Ii8
3.12. The Equation X’ = AX XB 118
3.13. Periodic Matrices: the Floquet Representation 120
3.14. Ciilculus of Variations 121
3.15. Two-point Houndury Condition 122
3.16. Green’s Functions 123
3.17. The Matrix Riccati Equation 123
3.18. Kronecker Products and Sums 124
3.19. AX XB = ? 125
3.20. Random Difference Systems 127
Miscellaneous Exercises 127
Bibliography and Comments 131
Chapter 4. Stability Theory and Related Questions
4.1. Introduction 134
4.2. Dini-Hukuhara Theorem—I 135
4.3. Dini-Hukuhara Theorem—II 138
4.4. Inverse Theorems of Perron 140
4.5. Existence and Uniqueness of Solution 140
4.6. Poincare-Lyapunov Stability Theory 142
4.7. Proof of Theorem 143
4.8. Asymptotic Behavior 146
4.9. The Function 4.10. More Refined Asymptotic Behavior 149
4.11. Analysis of Method of Successive Approximations 150
4.12. Fixed-point Methods 152
4.13. Time-dependent Equations over Finite Intervals 152
4.14. Alternative Norm 155
4.15. Perturbation Techniques 156
4.16. Second Method of Lyapunov 157
4.17. Solution of Linear Systems 157
4.18. Origins of Two-point Boundary Value Problems 158
4.19. Stability Theorem for Two-point Boundary Value Problem 159
4.20. Asymptotic Behavior 160
4.21. Numerical Aspects of Linear Two-point Boundary Value Problems 161
4.22. Difference Methods 163
4.23. Difference Equations 165
4.24. Proof of Stability 165
4.25. Analysis of Stability Proof 166
4.26. The General Concept of Stability 168
4.27. Irregular Stability Problems 168
4.2H. The Emden—Fowler-Fermi—Thomas Equation 170
Miscellaneous Exercises 171
HihlioKraphy and Comments 182
Chapter 5. The Bubnov-Galerkin Method
VI. Introduction 187
V2. Kxample of the Bubnov-Galerkin Method 188
V3. Validity of Method 189
V4. Discussion 190
V5. The General Approach 190
V6. Two Nonlineur Differential Equations 192
5.7. The Nonlinear Spring 193
5.8. Alternate Average 196
5.9. Straightforward Perturbation 196
5.10. A “Tucking-in” Technique 198
5.11. The Van der Pol Equation 198
5.12. Two-point Boundary Value Problems 200
5.13. The Linear Equation L(u) = g 200
5.14. Method of Moments 202
5.15. Nonlinear Case 202
5.16. Newton-Raphson Method 204
5.17. Multidimensional Newton-Raphson 207
5.18. Choice of Initial Approximation 208
5.19. Nonlinear Extrapolation and Acceleration of Convergence 210
5.20. Alternatives to Newton-Raphson 211
5.21. Lagrange Expansion 212
5.22. Method of Moments Applied to Partial Differential Equations 214
Miscellaneous Exercises 215
Bibliography and Comments 222
Chapter 6. Differential Approximation
6.1. Introduction 225
6.2. Differential Approximation 225
6.3. Linear Differential Operators 226
6.4. Computational Aspects—I 226
6.5. Computational Aspects—II 227
6.6. Degree of Approximation 228
6.7. Orthogonal Polynomials 229
6.8. Improving the Approximation 231
6.9. Extension of Classical Approximation Theory 231
6.10. Riccati Approximation 232
6.11. Transcendentally-transcendent Functions 233
6.12. Application to Renewal Equation 233
6.13. An Example 236
6.14. Differential-Difference Equations 238
6.15. An Example 239
6.16. Functional-Differential Equations 240
6.17. Reduction of Storage in Successive Approximations 242
6.18. Approximation by Exponentials 242
6.19. Mean-square Approximation 242
6.20. Validity of the Method 243
6.21. A Bootstrap Method 244
6.22. The Nonlinear Spring 244
6.23. The Van der Pol Equation 246
6.24. Self-consistent Techniques 248
6.25. The Riccati Equation 248
6.26. Higher-order Approximation 250
6.27. Mean-square Approximation—Periodic Solutions 251
Miscellaneous Exercises 253
Bibliography «nd Comment» 255
Chapter 7. The Rayleigh-Ritz Method
7.1. Introduction 259
7.2. The Euler Equation 259
7.3. The Euler Equation and the Variational Problem 260
7.4. Quadratic Functionals: Scalar Case 261
7.5. Positive Definiteness for Small T 263
7.6. Discussion 264
7.7. The Rayleigh-Ritz Method 265
7.8. Validity of the Method 265
7.9. Monotone Behavior and Convergence 267
7.10. Estimation of | ? – v in Terms of J(v) — J(u) 268
7.11. Convergence of Coefficients 269
7.12. Alternate Estimate 270
7.13. Successive Approximations 271
7.14. Determination of the Cofficients 272
7.15. Multidimensional Case 273
7.16. Reduction of Dimension 274
7.17. Minimization of Inequalities 275
7.18. Extension to Quadratic Functionals 277
7.19. Linear Integral Equations 279
7.20. Nonlinear Euler Equation 280
7.21. Existence and Uniqueness 281
7.22. Minimizing Property 282
7.23. Convexity and Uniqueness 282
7.24. Implied Boundedness 283
7.25. Lack of Existence of Minimum 284
7.26. Functional Analysis 284
7.27. The Euler Equation and Haar’s Device 286
7.28. Discussion 287
7.29. Successive Approximations 288
7.30. Lagrange Multiplier 288
7.31. A Formal Solution Is a Valid Solution 289
7.32. Raising the Price Diminishes the Demand 289
7.33. The Courant Parameter 290
7.34. Control Theory 291
Miscellaneous Exercises 291
hihliography and Comments 301
Chapter 8. Sturm-Liouville Theory
8.1. Equations Involving Parameters 304
8.2. Stationary Values 305
8.3. Characteristic Values and Functions 306
8.4. Properties of Characteristic Values and Functions 307
H.5. Generalized Fourier Expansion 312
8.6. Discussion 313
8.7. Rigorous Formulation of Variational Problem 314
H.8. Kayleigh-Ritz Method 315
?.?. Intermediate Problem of Weinstein 316
8.10. Transplantation 316
8.11. Positive Definiteness of Quadratic Functionals 317
8.12. Finite Difference Approximations 318
8.13. Monotonicity 319
8.14. Positive Kernels 320
Miscellaneous Exercises 322
Bibliography and Comment 329
Author Index 331
Subject Index 337