Theory of Difference Equations: Numerical Methods and Applications

Theory of Difference Equations: Numerical Methods and Applications, Second Edition......Page 1
Preface......Page 3
Contents......Page 6
1.1 Discrete Calculus......Page 10
Table of Contents......Page 0
1.2 Summation and Negative Powers of A......Page 13
1.2.1 Equations reducible to simple form......Page 15
1.3 Factorial Powers and Stirling Numbers......Page 18
1.4 Bernoulli Numbers and Polynomials......Page 20
1.5 Matrix Form......Page 22
1.5.1 Pascal matrix and combinatorics......Page 23
1.5.2 Pascal matrix and Bernoulli polynomials......Page 28
1.5.3 Pascal matrix and Bernstein polynomials......Page 29
1.5.4 Pascal matrix and Stirling numbers......Page 31
1.6 Comparison Principle......Page 33
1.7 Problems and Remarks......Page 38
1.8 Notes......Page 43
2.1 Preliminaries......Page 44
2.2 Fundamental Theory......Page 47
2.2.1 Adjoint and transposed equations......Page 51
2.3 The Method of Variation of Constants......Page 52
2.4 Linear Equations with Constant Coefficients......Page 53
2.5 Use of Operators A and E......Page 59
2.6 Method of Generating Functions......Page 62
2.7 Stability of Solutions......Page 68
2.8 Absolute Stability......Page 70
2.9 Boundary Value Problems......Page 76
2.10 Problems and Remarks......Page 78
2.11 Notes......Page 81
3.1 Basic Theory......Page 83
3.2 Method of Variation of Constants......Page 87
3.4 Systems Representing High-Order Equations......Page 90
3.4.1 One-sided Green\'s functions......Page 93
3.5 Poincare Theorem......Page 96
3.6 Periodic Solutions......Page 98
3.7 Boundary Value Problems......Page 103
3.8 Problems......Page 107
3.9 Notes......Page 109
4.1 Stability Notions......Page 111
4.2 The Linear Case......Page 115
4.3 Autonomous Linear Systems......Page 116
4.4 Linear Equations with Periodic Coefficients......Page 118
4.5 Use of the Comparison Principle......Page 120
4.6 Variation of Constants......Page 124
4.7 Stability by First Approximation......Page 127
4.8 Liapunov Functions......Page 129
4.9 Domain of Asymptotic Stability......Page 137
4.10 Converse Theorems......Page 140
4.11 Total and Practical Stability......Page 145
4.12 Problems......Page 148
4.13 Notes......Page 150
5.1 Initial Value Problems......Page 151
5.2 Boundary Values Problems......Page 154
5.2.1 Invert ibility of tridiagonal matrices......Page 157
5.2.2 Sufficient conditions for well-conditioning......Page 162
5.3 Cyclic Reduction......Page 165
5.3.1 The case of Toeplitz tridiagonal matrices......Page 167
5.5 Notes......Page 171
6.1 Iterative Methods......Page 173
6.2 Local Results......Page 175
6.3 Semilocal Results......Page 178
6.3.1 Newton-Kantorovich-like theorems......Page 180
6.3.2 Effect of perturbations......Page 187
6.4 Miller\'s, Olver\'s, and Clenshaw\'s Algorithms......Page 188
6.5 Boundary Value Problems......Page 192
6.6 Monotone Iterative Methods......Page 193
6.7 Monotone Approximations......Page 196
6.8 Problems......Page 200
6.9 Notes......Page 202
7.0 Introduction......Page 203
7.1 Linear Multistep Methods......Page 204
7.2 Finite Interval......Page 206
7.3 Infinite Interval......Page 209
7.4 Nonlinear Case......Page 211
7.5 Other Techniques......Page 213
7.6 The Method of Lines......Page 214
7.7 Spectrum of a Family of Matrices......Page 216
7.8 Problems......Page 219
7.9 Notes......Page 220
8.1 Linear Models for Population Dynamics......Page 222
8.2 The Logistic Equation......Page 226
8.3 Distillation of a Binary Liquid......Page 228
8.4 Models from Economics......Page 231
8.5 Models of Traffic in Channels......Page 234
8.6 Problems......Page 238
8.7 Notes......Page 239
9.1 Combinations of Means......Page 240
9.2 Arithmetic-Geometric (Borchard)......Page 242
9.2.1 Arithmetic-geometric mean II......Page 244
9.3 The Weierstrass Method......Page 245
9.4 Difference Equations and Prime Numbers......Page 246
9.6 Notes......Page 248
A.I Introduction......Page 249
A.2 Properties of Component Matrices......Page 252
A.3 Particular Matrices......Page 254
A.4 Sequence of Matrices......Page 257
A.5 Jordan Canonical Form......Page 258
A.6 Norms of Matrices and Related Topics......Page 260
A.7 Nonnegative Matrices......Page 262
B.I The Schur Criteria......Page 265
C.I Definitions......Page 268
C.2 Properties of Tn(z) and Un(z)......Page 269
D.I Chapter 1......Page 271
D.2 Chapter 2......Page 274
D.3 Chapter 3......Page 276
D.4 Chapter 4......Page 279
D.6 Chapter 6......Page 281
D.7 Chapter 7......Page 283
D.8 Chapter 8......Page 284
D.9 Chapter 9......Page 285
Bibliography......Page 287