Numerical Methods for Initial Value Problems in Ordinary Differential Equations
- 1st Edition - May 10, 2014
- Author: Simeon Ola Fatunla
- Editors: Werner Rheinboldt, Daniel Siewiorek
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 2 - 3 8 6 8 - 5
- eBook ISBN:9 7 8 - 1 - 4 8 3 2 - 6 9 2 6 - 9
Numerical Method for Initial Value Problems in Ordinary Differential Equations deals with numerical treatment of special differential equations: stiff, stiff oscillatory, singular,… Read more
Purchase options
Institutional subscription on ScienceDirect
Request a sales quoteNumerical Method for Initial Value Problems in Ordinary Differential Equations deals with numerical treatment of special differential equations: stiff, stiff oscillatory, singular, and discontinuous initial value problems, characterized by large Lipschitz constants. The book reviews the difference operators, the theory of interpolation, first integral mean value theorem, and numerical integration algorithms. The text explains the theory of one-step methods, the Euler scheme, the inverse Euler scheme, and also Richardson's extrapolation. The book discusses the general theory of Runge-Kutta processes, including the error estimation, and stepsize selection of the R-K process. The text evaluates the different linear multistep methods such as the explicit linear multistep methods (Adams-Bashforth, 1883), the implicit linear multistep methods (Adams-Moulton scheme, 1926), and the general theory of linear multistep methods. The book also reviews the existing stiff codes based on the implicit/semi-implicit, singly/diagonally implicit Runge-Kutta schemes, the backward differentiation formulas, the second derivative formulas, as well as the related extrapolation processes. The text is intended for undergraduates in mathematics, computer science, or engineering courses, andfor postgraduate students or researchers in related disciplines.
Preface
1 Preliminaries
1.1 The Difference Operators
1.2 Theory of Interpolation
1.3 Finite Difference Equations
1.4 Linear Systems with Constant Coefficients
1.5 Distribution of Roots of Polynomials
1.6 First Integral Mean Value Theorem
1.7 Common Norms in ODEs
2 Numerical Integration Algorithms
2.1 Introduction
2.2 Existence of Solution, Numerical Approach
2.3 Special IVPs
2.4 Error Propagation, Stability and Convergence of Discretization Methods
3 Theory of One-Step Methods
3.1 General Theory of One-Step Methods
3.2 The Euler Scheme, the Inverse Euler Schem and Richardson's Extrapolation
3.3 The Convergence of Euler's Scheme
3.4 The Trapezoidal Scheme
4 Runge-Kutta Processes
4.1 General Theory of Runge-Kutta Processes
4.2 The Explicit Two-Stage Process
4.3 Convergence and Stability of Two-Stage Explicit R-K Scheme
4.4 Matrix Representation of the R-K Processes
4.5 Error Estimation and Stepsize Selection in R-K Processes
4.6 Implicit and Semi-Implicit R-K Processes
4.7 Rosenbrock Methods
5 Linear Multistep Methods
5.1 Starting Procedure
5.2 Explicit Linear Multistep Methods
5.3 Implicit Linear Multistep Methods
5.4 Implementation of the Predictor-Corrector Formulas
5.5 General Theory of Linear Multistep Methods
5.6 Automatic Implementation of the Adams Scheme
6 Numerical Treatment of Singular/Discontinuous Initial Value Problems
6.1 Introduction
6.2 Non-Polynomial Methods
6.3 The Inverse Polynomial Methods
6.4 Local Error Estimates in Automatic Codes for Discontinuous Systems
7 Extrapolation Processes and Singularities
7.1 Introduction
7.2 Generation of the Zero-th Column of Extrapolation Table
7.3 Polynomial and Rational Extrapolation
7.4 Convergence and Stability Properties of Extrapolation Processes
7.5 Practical Implementation of Extrapolation Processes
8 Stiff Initial Value Problems
8.1 The Concept of Stiffness
8.2 Stiff and Nonstiff Algorithms
8.3 Solution of Nonlinear Equations and Estimation of Jacobians
8.4 Region of Absolute Stability
8.5 Stability Criteria for Stiff Methods
8.6 Stronger Stability Properties of IRK Processes
8.7 One-Leg Multistep Methods
9 Stiff Algorithms
9.1 What are Stiff Algorithms
9.2 Efficient Implementation of Implicit Runge-Kutta Methods
9.3 The Backward Differentiation Formula
9.4 Second Derivative Formulas
9.5 Extrapolation Processes for Stiff Systems
9.6 Mono-Implicit Runge-Kutta Methods
10 Second Order Differential Equations
10.1 Introduction
10.2 Linear Multistep Methods and the Concept of P-Stability
10.3 Derivation of P-Stable Formulas
10.4 One-Leg Multistep Methods for Second Order IVPs
10.5 Multiderivative Methods
11 Recent Developments in Ode Solvers
References
- No. of pages: 308
- Language: English
- Edition: 1
- Published: May 10, 2014
- Imprint: Academic Press
- Paperback ISBN: 9781483238685
- eBook ISBN: 9781483269269
DS
Daniel Siewiorek
Affiliations and expertise
Carnegie-Mellon UniversityRead Numerical Methods for Initial Value Problems in Ordinary Differential Equations on ScienceDirect