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E-grāmata: Numerical Solution of Ordinary Differential Equations [Wiley Online]

(University of Iowa), (University of Iowa), (University of Iowa)
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Numerical Solution of Ordinary Differential EquationsPalielināt
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A concise introduction to numerical methodsand the mathematical framework neededto understand their performance Numerical 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 method



Taylor and Runge-Kutta methods



General error analysis for multi-step methods



Stiff differential equations



Differential algebraic equations



Two-point boundary value problems



Volterra integral equations





Each 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.
Introduction 1
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 weak 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
Appendix A. Taylor's Theorem 235
Appendix B. Polynomial interpolation 241
References 245
Index 250
Kendall E. Atkinson, PhD, is Professor Emeritus in the Departments of Mathematics and Computer Science at the University of Iowa. He has authored books and journal articles in his areas of research interest, which include the numerical solution of integral equations and boundary integral equation methods. Weimin Han, PhD, is Professor in the Department of Mathematics at the University of Iowa, where he is also Director of the interdisciplinary PhD Program in Applied Mathematical and Computational Science. Dr. Han currently focuses his research on the numerical solution of partial differential equations. David E. Stewart, PhD, is Professor and Associate Chair in the Department of Mathematics at the University of Iowa, where he is also the departmental Director of Undergraduate Studies. Dr. Stewart's research interests include numerical analysis, computational models of mechanics, scientific computing, and optimization.
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