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Differential-Difference Equations
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Differential-Difference Equations
Autorzy: Richard Bellman, Kenneth L. Cooke Wydawnictwo: Elsevier Science Data wydania: 1963 Język publikacji: Angielski Liczba stron: 483 Formaty publikacji: EAN: 9780080955148 ISBN: 9780080955148 Kategoria: Mathematical foundations Kolekcja specjalna: Rok matematyki 2019 Indeks wydawcy: S0076-5392(09)X6010-8 Nota bibliograficzna: -
Spis treści
- Front Cover 2
- Differential-Difference Equations 5
- Copyright Page 6
- Dedication 7
- Introduction 9
- Contents 13
-
Chapter 1. The Laplace Transform
19
- 1.1. Introduction 19
- 1.2. Existence and Convergence Exercises 19
- 1.3. The Inversion Problem 21
- 1.4. Behavior of the Dirichlet Kernel 22
- 1.5. Analytic Details Exercises 22
- 1.6. Statement of Result 25
- 1.7. Jump Discontinuity 26
- 1.8. Functions of Bounded Variation 26
- Front Cover 2
- Differential-Difference Equations 5
- Copyright Page 6
- Dedication 7
- Introduction 9
- Contents 13
-
Chapter 1. The Laplace Transform
19
- 1.1. Introduction 19
- 1.2. Existence and Convergence Exercises 19
- 1.3. The Inversion Problem 21
- 1.4. Behavior of the Dirichlet Kernel 22
- 1.5. Analytic Details Exercises 22
- 1.6. Statement of Result 25
- 1.7. Jump Discontinuity 26
- 1.8. Functions of Bounded Variation 26
- 1.9. Contour Integration. 27
- 1.10. Examples 28
- 1.11. The Fejér Transform 29
- 1.12. The Inverse Inversion Problem 30
- 1.13. The Convolution Theorem Exercises 31
- 1.14. The Fourier Transform 35
- 1.15. Plancherel-Parseval Theorem 36
- 1.17. The Post-Widder Formula 37
- 1.16. Application to Laplace Transform 37
- 1.18. Real Inversion Formulas Exercise 38
- Miscellaneous Exercises and Research Problems 39
- Bibliography and Comments 44
-
Chapter 2. Linear Differential Equations
45
- 2.1. Introduction 45
- 2.2. Linear Differential Equations 45
- 2.4. Successive Approximations 47
- 2.3. Fundamental Existence and Uniqueness Theorem 47
- 2.6. Uniqueness Theorem 49
- 2.5. A Fundamental Lemma 49
- 2.7. Fixed-point Techniques 50
- 2.8. Difference Schemes 50
- 2.9. The Matrix Equation 51
- 2.10. Alternative Derivation 51
- 2.12. The Adjoint Equation 52
- 2.11. The Inhomogeneous Equation 52
- 2.13. Constant Coefficients—I 53
- 2.14. Constant Coefficients—II 54
- 2.15. Laplace Transform Solution 54
- 2.16. Characteristic Values and Characteristic Funotions 55
- Miscellaneous Exercises and Research Problems 58
- Bibliography and Comments 59
-
Chapter 3. First-order Linear Differential-Difference Equations of Retarded Type with Constant Coefficients
60
- 3.1. Introduction 60
- 3.2. Examples Exercises 63
- 3.3. Equations of Retarded, Neutral, and Advanced Type 66
- 3.4. The Existence-Uniqueness Theorem Exercises 67
- 3.5. Exponential Solutions Exercises 71
- 3.6. Order of Growth of Solutions Exercises 76
- 3.7. Laplace Transform Solution Exercises 81
- 3.8. Solution of a Differential Equation in the Form of a Definite Integral 89
- 3.9. Solution of a Differential-Difference Equation in the Form of a Definite Integral Exercises 91
- Miscellaneous Exercises and Research Problems 98
- Bibliography and Comments 115
-
Chapter 4. Series Expansions of Solutions of First-order Equations of Retarded Type
116
- 4.1. The Characteristic Roots 116
- 4.2. Series Expansions 120
- 4.3. Other Forms of the Expansion Theorem Exercises 126
- 4.4. Asymptotic Behavior of the Solution Exercises 131
- 4.5. Stability of Equilibrium Exercises 135
- 4.6. Fourier-type Expansions Exercises 139
- 4.7. The Shift Theorem Exercises 144
- Miscellaneous Exercises and Research Problems 149
- Bibliography and Comments 155
-
Chapter 5. First-order Linear Equations of Neutral and Advanced Type with Constant Coefficients
157
- 5.1. Existence-Uniqueness Theorems Exercises 157
- 5.2. Solution by Exponentials and by Definite Integrals: Equations of Neutral Type Exercises 161
- 5.3. Series Expansions: Equations of Neutral Type Exercise 171
- 5.4. Asymptotic Behavior and Stability: Equations of Neutral Type 176
- 5.5. Other Expansions for Solutions of Equations of Neutral Type Exercises 177
- 5.6. Equations of Advanced Type 178
- Miscellaneous Exercises and Research Problems 179
-
Chapter 6. Linear Systems of Differential-Difference Equations with Constant Coefficients
182
- 6.1. Introduction 182
- 6.3. Classification of Systems 183
- 6.2. Vector-matrix Notation 183
- 6.4. Existence-Uniqueness Theorems for Systems Exercises 185
- 6.5. Transform Solutions: Retarded-Neutral Systems Exercises 191
- 6.6. Solution of Neutral and Retarded Systems by Definite Integrals Exercises 197
- 6.7. Series Expansions for Neutral and Retarded Systems Exercises 201
- 6.8. Asymptotic Behavior of Solutions of Neutral and Retarded Systems Exercises 206
- 6.9. Scalar Equations Exercises 210
- 6.10. The Finite Transform Method Exercise 215
- 6.11. Fourier-type Expansions Exercises 224
- Miscellaneous Exercises and Research Problems 226
- Bibliography and Comments 232
-
Chapter 7. The Renewal Equation
234
- 7.1. Introduction 234
- 7.2. Existence and Uniqueness Exercises 235
- 7.3. Further Existence and Uniqueness Theorems 238
- 7.4. Monotonicity and Bounded Variation Exercises 242
- 7.5. The Formal Laplace Transform Solution 244
- 7.6. Exponential Bounds for u(t) 245
- 7.7. Rigorous Solution 246
- 7.8. A Convolution Theorem 247
- 7.10. Use of the Contour Integral Representation 249
- 7.9. Asymptotic Behavior of Solutions 249
- 7.11. ø(t) a Positive Function 250
- 7.12. Shift of the Contour 251
- 7.13. Step Functions 252
- 7.15. A Less Easily Obtained Result 254
- 7.14. An Elementary Result 254
- 7.16. Abelian and Tauberian Results 257
- 7.18. Asymptotic Behavior of Solution of Renewal Equation 258
- 7.17. A Tauberian Theorem of Hardy and Littlewood 258
- 7.19. Discussion 259
- 7.20. A Tauberian Theorem of Ikehara 260
- 7.21. The Tauberian Theorem of Wiener 261
- Miscellaneous Exercises and Research Problems 262
- Bibliography and Comments 273
-
Chapter 8. Systems of Renewal Equations
275
- 8.2. Vector Renewal Equation 275
- 8.1. Introduction 275
- 8.3. Positive Matrices 276
- 8.5. Zero with Largest Real Part 277
- 8.4. Some Consequences 277
- Miscellaneous Exercises and Research Problems 280
- 8.6. Asymptotic Behavior 280
- Bibliography and Comments 281
-
Chapter 9. Asymptotic Behavior of linear Differential-Difference Equations
283
- 9.1. Introduction 283
- 9.2. First Principal Result 284
- 9.3. Preliminaries 285
- 9.4. Discussion 286
- 9.5. ∫∞ |a(t1)|dt1 <∞ 286
- 9.6. The Difficult Part of Theorem 9.1 288
- 9.7. A Lemma 290
- 9.8. Continuation of Proof of Theorem 9.1 292
- 9.9. The Case Where b(t) ≠ 0 293
- 9.12. Asymptotic Series 296
- 9.10. Further Results 296
- 9.11. More Precise Results 296
- 9.13. The Foundations of Asymptotic Series 298
- 9.14. Alternative Formulation 300
- 9.15. Differential and Integral Properties 300
- 9.16. Extension of Definition Exercises 301
- 9.17. First-order Linear Differential Equations 302
- 9.18. Second-order Linear Differential Equations 303
- 9.19. The Case Where a0 = 0 Exercise 304
- 9.20. A Rigorous Derivation of the Asymptotic Expansion 305
- 9.22. A Basic Problem in the Theory of Differential Equations 307
- 9.21. Determination of the Constants Exercises 307
- 9.23. Formal Determination of Coefficients 308
- 9.24. Asymptotic Expansion of Solution 309
- Miscellaneous Exercises and Research Problems 310
- Bibliography and Comments 317
-
Chapter 10. Stability of Solutions of linear Differential-Difference Equations
318
- 10.1. Introduction. 318
- 10.2. Stability Theory for Ordinary Differential Equations 318
- 10.3. The Adjoint Equation 320
- 10.4. The Scalar Linear Differential-Difference Equation 322
- 10.5. The Matrix Equation with Retarded Argument 324
- 10.6. AStability Theorem for Equations with Retarded Argument 326
- 10.7. Equations with Constant Coefficients 328
- 10.8. A Lemma 329
- 10.9. A Stability Theorem for Equations with Constant Coefficients 330
- 10.11. The Scalar Equation of Neutral Type: Integral Representation for a Solution Exercise 331
- 10.10. Boundedness of Solutions of the Unperturbed System 331
- 10.12. The Scalar Equation of Neutral Type: Representation for the Derivative of a Solution 334
- 10.13. Systems of Equations of Neutral Type 338
- 10.14. Stability Theorems for Equations of Neutral Type 342
- 10.15. Stability Theorems for Equations of Neutral Type with Constant Coefficients 345
- Miscellaneous Exercises and Research Problems 348
- Bibliography and Comments 351
-
Chapter 11. Stability Theory and Asymptotic Behavior for Nonlinear Differential-Difference Equations
352
- 11.1. Introduction 352
- 11.2. The Poincaré-Liapunov Theorem 353
- 11.3. Small Perturbations for General Systems 355
- 11.4. Types of Stability 357
- 11.5. Existence Theorem for Nonlinear Differential-Difference Equations 359
- 11.6. Uniqueness 362
- 11.7. Statement of Existence and Uniqueness Theorems Exercises 363
- 11.8. Stability Theorem Exercise 366
- 11.9. Stability Theorem: Second Proof Exercise 368
- 11.10. Asymptotic Behavior of the Solutions. 372
- 11.11. Proof of Theorem 11.5 Exercises 374
- 11.12. Another Stability Theorem 379
- 11.13. Dini-Hukuhara Theorem for Equations with Variable Coefficients 382
- 11.14. Poincaré-Liapunov Theorem for Equations with General Variable Coefficients 385
- 11.15. Asymptotic Behavior for Nonlinear Equations with Almost-constant Coefficients 387
- 11.16. Systems of Nonlinear Equations 390
- 11.17. Liapunov Functions and Functionals 391
- Miscellaneous Exercises and Research Problems 394
- Bibliography and Comments 408
-
Chapter 12. Asymptotic Location of the Zeros of Exponential Polynomials
411
- 12.1. Introduction 411
- 12.2. The Form of det H(s) 412
- 12.3. Zeros of Analytic Functions Exercises 413
- 12.4. Constant Coefficients and Commensurable Exponents 417
- 12.5. Constant Coefficients and General Real Exponents 418
- 12.6. Asymptotically Constant Coefficients 422
- 12.7. Polynomial Coefficients with mj and βj Proportional 424
- 12.8. Polynomial Coefficients 428
- 12.9. Examples Exercise 434
- 12.10. Conditions That All Roots Be of Specified Type Exercises 435
- 12.11. Construction of Contours 438
- 12.12. Order Results for H-1(S) 440
- 12.13. Order Results in the Scalar Case 441
- 12.14. Convergence of Integrals over the Contours Exercises 442
- 12.15. Integrals along Vertical Lines 445
- Miscellaneous Exercises and Research Problems 450
- Bibliography and Comments 457
-
Chapter 13. On Stability Properties of the Zeros of Exponential Polynomials
458
- 13.1. Introduction 458
- 13.2. Exponential Polynomials Exercises 458
- 13.3. Functions of the Form f(z, cos z, sin z) 459
- 13.4. Presence of a Principal Term 460
- 13.5. Zeros of h(z, ez ) 460
- 13.6. The Fundamental Stability Results 461
- 13.7. A Result of Hayes 462
- 13.8. An Important Equation 464
- 13.9. Another Example 468
- Miscellaneous Exercises and Research Problems 470
- Bibliography and Comments 472
- Author Index 475
- Subject Index 478
- Other RAND Books 481