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Numerical solution of ordinary differential equations / Kendall E. Atkinson, Weimin Han, David Stewart.
Math/Physics/Astronomy Library QA372 .A85 2009
Available
- Format:
- Book
- Author/Creator:
- Atkinson, Kendall E.
- Series:
- Pure and applied mathematics (John Wiley & Sons : Unnumbered)
- Pure and applied mathematics
- Language:
- English
- Subjects (All):
- Differential equations--Numerical solutions.
- Differential equations.
- Physical Description:
- xii, 252 pages : illustrations ; 25 cm.
- Place of Publication:
- Hoboken, N.J. : Wiley, [2009]
- Contents:
- 1 Theory of differential equations: An introduction 3
- 1.1 General solvability theory 7
- 1.2 Stability of the initial value problem 8
- 1.3 Direction fields 11
- Problems 13
- 2 Euler's method 15
- 2.1 Definition of Euler's method 16
- 2.2 Error analysis of Euler's method 21
- 2.3 Asymptotic error analysis 26
- 2.3.1 Richardson extrapolation 28
- 2.4 Numerical stability 29
- 2.4.1 Rounding error accumulation 30
- Problems 32
- 3 Systems of differential equations 37
- 3.1 Higher-order differential equations 39
- 3.2 Numerical methods for systems 42
- Problems 46
- 4 The backward Euler method and the trapezoidal method 49
- 4.1 The backward Euler method 51
- 4.2 The trapezoidal method 56
- Problems 62
- 5 Taylor and Runge-Kutta methods 67
- 5.1 Taylor methods 68
- 5.2 Runge-Kutta methods 70
- 5.2.1 A general framework for explicit Runge-Kutta methods 73
- 5.3 Convergence, stability, and asymptotic error 75
- 5.3.1 Error prediction and control 78
- 5.4 Runge-Kutta-Fehlberg methods 80
- 5.5 MATLAB codes 82
- 5.6 Implicit Runge-Kutta methods 86
- 5.6.1 Two-point collocation methods 87
- Problems 89
- 6 Multistep methods 95
- 6.1 Adams-Bashforth methods 96
- 6.2 Adams-Moulton methods 101
- 6.3 Computer codes 104
- 6.3.1 Matlab Ode codes 105
- Problems 106
- 7 General error analysis for multistep methods 111
- 7.1 Truncation error 112
- 7.2 Convergence 115
- 7.3 A general error analysis 117
- 7.3.1 Stability theory 118
- 7.3.2 Convergence theory 122
- 7.3.3 Relative stability and week stability 122
- Problems 123
- 8 Stiff differential equations 127
- 8.1 The method of lines for a parabolic equation 131
- 8.1.1 MATLAB programs for the method of lines 135
- 8.2 Backward differentiation formulas 140
- 8.3 Stability regions for multistep methods 141
- 8.4 Additional sources of difficulty 143
- 8.4.1 A-stability and L-stability 143
- 8.4.2 Time-varying problems and stability 145
- 8.5 Solving the finite-difference method 145
- 8.6 Computer codes 146
- Problems 147
- 9 Implicit RK methods for stiff differential equations 149
- 9.1 Families of implicit Runge-Kutta methods 149
- 9.2 Stability of Runge-Kutta methods 154
- 9.3 Order reduction 156
- 9.4 Runge-Kutta methods for stiff equations in practice 160
- Problems 161
- 10 Differential algebraic equations 163
- 10.1 Initial conditions and drift 165
- 10.2 DAEs as stiff differential equations 168
- 10.3 Numerical issues: higher index problems 169
- 10.4 Backward differentiation methods for DAEs 173
- 10.4.1 Index 1 problems 173
- 10.4.2 Index 2 problems 174
- 10.5 Runge-Kutta methods for DAEs 175
- 10.5.1 Index 1 problems 176
- 10.5.2 Index 2 problems 179
- 10.6 Index three problems from mechanics 181
- 10.6.1 Runge-Kutta methods for mechanical index 3 systems 183
- 10.7 Higher index DAEs 184
- Problems 185
- 11 Two-point boundary value problems 187
- 11.1 A finite-difference method 188
- 11.1.1 Convergence 190
- 11.1.2 A numerical example 190
- 11.1.3 Boundary conditions involving the derivative 194
- 11.2 Nonlinear two-point boundary value problems 195
- 11.2.1 Finite difference methods 197
- 11.2.2 Shooting methods 201
- 11.2.3 Collocation methods 204
- 11.2.4 Other methods and problems 206
- Problems 206
- 12 Volterra integral equations 211
- 12.1 Solvability theory 212
- 12.1.1 Special equations 214
- 12.2 Numerical methods 215
- 12.2.1 The trapezoidal method 216
- 12.2.2 Error for the trapezoidal method 217
- 12.2.3 General schema for numerical methods 219
- 12.3 Numerical methods: Theory 223
- 12.3.1 Numerical stability 225
- 12.3.2 Practical numerical stability 227
- Problems 231.
- Notes:
- Includes bibliographical references (pages 245-249) and index.
- ISBN:
- 9780470042946
- 047004294X
- OCLC:
- 243960534
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