A concise introduction to numerical methodsand the mathematical framework neededto understand their performanceNumerical Solution of Ordinary Differential Equations presents a complete and easy-to-follow introduction to classical topics in the numerical solution of ordinary differential equations. The book's approach not only explains the presented mathematics, but also helps readers understand how these numerical methods are used to solve real-world problems.Unifying perspectives are provided throughout the text, bringing together and categorizing different types of problems in order to help readers comprehend the applications of ordinary differential equations. In addition, the authors' collective academic experience ensures a coherent and accessible discussion of key topics, including:Euler's methodTaylor and Runge-Kutta methodsGeneral error analysis for multi-step methodsStiff differential equationsDifferential algebraic equationsTwo-point boundary value problemsVolterra integral equationsEach chapter features problem sets that enable readers to test and build their knowledge of the presented methods, and a related Web site features MATLAB® programs that facilitate the exploration of numerical methods in greater depth. Detailed references outline additional literature on both analytical and numerical aspects of ordinary differential equations for further exploration of individual topics.Numerical Solution of Ordinary Differential Equations is an excellent textbook for courses on the numerical solution of differential equations at the upper-undergraduate and beginning graduate levels. It also serves as a valuable reference for researchers in the fields of mathematics and engineering.
Author(s): Kendall Atkinson, Weimin Han, David E. Stewart
Series: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
Edition: 1
Publisher: Wiley
Year: 2009
Language: English
Pages: 272
Tags: Математика;Вычислительная математика;
Numerical Solution of Ordinary Differential Equations......Page 7
CONTENTS......Page 13
Introduction......Page 17
1 Theory of differential equations: An introduction......Page 19
1.1 General solvability theory......Page 23
1.2 Stability of the initial value problem......Page 24
1.3 Direction fields......Page 27
Problems......Page 29
2 Euler's method......Page 31
2.1 Definition of Euler's method......Page 32
2.2 Error analysis of Euler's method......Page 37
2.3 Asymptotic error analysis......Page 42
2.3.1 Richardson extrapolation......Page 44
2.4 Numerical stability......Page 45
2.4.1 Rounding error accumulation......Page 46
Problems......Page 48
3 Systems of differential equations......Page 53
3.1 Higher-order differential equations......Page 55
3.2 Numerical methods for systems......Page 58
Problems......Page 62
4 The backward Euler method and the trapezoidal method......Page 65
4.1 The backward Euler method......Page 67
4.2 The trapezoidal method......Page 72
Problems......Page 78
5 Taylor and Runge–Kutta methods......Page 83
5.1 Taylor methods......Page 84
5.2 Runge–Kutta methods......Page 86
5.2.1 A general framework for explicit Runge–Kutta methods......Page 89
5.3 Convergence, stability, and asymptotic error......Page 91
5.3.1 Error prediction and control......Page 94
5.4 Runge–Kutta–Fehlberg methods......Page 96
5.5 MATLAB codes......Page 98
5.6 Implicit Runge–Kutta methods......Page 102
5.6.1 Two-point collocation methods......Page 103
Problems......Page 105
6 Multistep methods......Page 111
6.1 Adams–Bashforth methods......Page 112
6.2 Adams–Moulton methods......Page 117
6.3 Computer codes......Page 120
6.3.1 MATLAB ODE codes......Page 121
Problems......Page 122
7 General error analysis for multistep methods......Page 127
7.1 Truncation error......Page 128
7.2 Convergence......Page 131
7.3 A general error analysis......Page 133
7.3.1 Stability theory......Page 134
7.3.3 Relative stability and weak stability......Page 138
Problems......Page 139
8 Stiff differential equations......Page 143
8.1 The method of lines for a parabolic equation......Page 147
8.1.1 MATLAB programs for the method of lines......Page 151
8.2 Backward differentiation formulas......Page 156
8.3 Stability regions for multistep methods......Page 157
8.4.1 A-stability and L-stability......Page 159
8.5 Solving the finite-difference method......Page 161
8.6 Computer codes......Page 162
Problems......Page 163
9.1 Families of implicit Runge–Kutta methods......Page 165
9.2 Stability of Runge–Kutta methods......Page 170
9.3 Order reduction......Page 172
9.4 Runge–Kutta methods for stiff equations in practice......Page 176
Problems......Page 177
10 Differential algebraic equations......Page 179
10.1 Initial conditions and drift......Page 181
10.2 DAEs as stiff differential equations......Page 184
10.3 Numerical issues: higher index problems......Page 185
10.4.1 Index 1 problems......Page 189
10.4.2 Index 2 problems......Page 190
10.5 Runge–Kutta methods for DAEs......Page 191
10.5.1 Index 1 problems......Page 192
10.5.2 Index 2 problems......Page 195
10.6 Index three problems from mechanics......Page 197
10.6.1 Runge–Kutta methods for mechanical index 3 systems......Page 199
10.7 Higher index DAEs......Page 200
Problems......Page 201
11 Two-point boundary value problems......Page 203
11.1 A finite-difference method......Page 204
11.1.2 A numerical example......Page 206
11.1.3 Boundary conditions involving the derivative......Page 210
11.2 Nonlinear two-point boundary value problems......Page 211
11.2.1 Finite difference methods......Page 213
11.2.2 Shooting methods......Page 217
11.2.3 Collocation methods......Page 220
Problems......Page 222
12 Volterra integral equations......Page 227
12.1 Solvability theory......Page 228
12.1.1 Special equations......Page 230
12.2 Numerical methods......Page 231
12.2.1 The trapezoidal method......Page 232
12.2.2 Error for the trapezoidal method......Page 233
12.2.3 General schema for numerical methods......Page 235
12.3 Numerical methods: Theory......Page 239
12.3.1 Numerical stability......Page 241
12.3.2 Practical numerical stability......Page 243
Problems......Page 247
Appendix A. Taylor's Theorem......Page 251
Appendix B. Polynomial interpolation......Page 257
References......Page 261
Index......Page 266
