Numerical Methods for Initial Value Problems in Ordinary Differential Equations

Numerical Methods for Initial Value Problems in Ordinary Differential Equations

1st Edition - August 28, 1988

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  • Author: Simeon Ola Fatunla
  • eBook ISBN: 9781483269269

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Numerical 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.

Table of Contents

  • 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


Product details

  • No. of pages: 308
  • Language: English
  • Copyright: © Academic Press 1988
  • Published: August 28, 1988
  • Imprint: Academic Press
  • eBook ISBN: 9781483269269

About the Author

Simeon Ola Fatunla

About the Editors

Werner Rheinboldt

Daniel Siewiorek

Affiliations and Expertise

Carnegie-Mellon University

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