Numerical Solution of Differential Equations - 1st Edition - ISBN: 9780122677809, 9781483262529

Numerical Solution of Differential Equations

1st Edition

Authors: Isaac Fried
Editors: Werner Rheinboldt
eBook ISBN: 9781483262529
Imprint: Academic Press
Published Date: 1st January 1979
Page Count: 278
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Numerical Solution of Differential Equations is a 10-chapter text that provides the numerical solution and practical aspects of differential equations.

After a brief overview of the fundamentals of differential equations, this book goes on presenting the principal useful discretization techniques and their theoretical aspects, along with geometrical and physical examples, mainly from continuum mechanics. Considerable chapters are devoted to the development of the techniques of the numerical solution of differential equations and their analysis. The remaining chapters explore the influential invention in computational mechanics-finite elements. Each chapter emphasizes the relationship among the analytic formulation of the physical event, the discretization techniques applied to it, the algebraic properties of the discrete systems created, and the properties of the digital computer.

This book will be of great value to undergraduate and graduate mathematics and physics students.

Table of Contents


1 Finite Differences

1. Calculus to Algebra to Arithmetic

2. Differentials and Finite Difference Approximations

3. Finite Difference Schemes

4. Accuracy Analysis

5. Higher-Order Schemes


Suggested Further Reading

2 Two-Point Boundary Value Problems

1. Finite Difference Approximation of the Loaded String Equation

2. Incorporation of Boundary Conditions

3. Consistency and Stability: Convergence

4. Higher-Order Consistency

5. Finite Difference Approximation of the Beam Equation

6. Splitting of the Beam Equation into Two String Equations

7. Nonlinear Two-Point Boundary Value Problems


Suggested Further Reading

3 Variational Formulations

1. Energy Error

2. Principle of Minimum Potential Energy

3. More General Boundary Conditions

4. Complementary Variational Principles

5. Euler-Lagrange Equations

6. Total Potential Energy of the Thin Elastic Beam

7. Indefinite Variational Principles

8. A Bound Theorem


Suggested Further Reading

4 Finite Elements

1. The Idea of Ritz

2. Finite Element Basis Functions

3. Finite Element Matrices

4. Assembly of Global Matrices

5. Essential and Natural Boundary Conditions

6. Higher-Order Finite Elements

7. Beam Element

8. Complex Structures


Suggested Further Reading

5 Discretization Accuracy

1. Energy Theorems

2. Energy Rates of Convergence

3. Sharpness of the Energy Error Estimate

4. L2 Error Estimate

5. L∞ Error Estimate

6. Richardson's Extrapolation to the Limit

7. Numerical Integration


Suggested Further Reading

6 Eigenproblems

1. Stability of Columns

2. Vibration of Elastic Systems

3. Finite Difference Approximation

4. Rayleigh's Quotient

5. Finite Element Approximation

6. The Minmax Principle

7. Discretization Accuracy of Eigenvalues

8. Discretization Accuracy of Eigenfunctions

9. Change of Basis: Condensation

10. Numerical Integration: Lumping

11. Nonlinear Eigenproblems


Suggested Further Reading

7 Algebraic Properties of the Global Matrices

1. Eigenvalue Range in Ky = λ My

2. Spectral Norms of K and M

3. Spectral Condition Numbers

4. Irregular Meshes

5. The Influence (Green's) Function

6. Maximum Norms and Condition Numbers

7. Scaling

8. Positive Flexibility Matrices

9. Computational Errors

10. Detection of Computational Errors


Suggested Further Reading

8 Equation of Heat Transfer

1. Nonstationary Heat Transfer in a Rod

2. Finite Difference Approximation

3. Modal Analysis

4. Finite Elements

5. Essential Boundary Conditions

6. Euler's Stepwise Integration in Time

7. Explicit Finite Element Schemes

8. Convergence of Euler's Method

9. Stability

10. Stable Time Step Size Estimate

11. Numerical Example

12. Implicit Unconditionally Stable Schemes

13. Higher-Order Single Step Implicit Schemes

14. Superstable Schemes

15. Multistep Schemes

16. Predictor-Corrector Methods

17. Nonlinear Heat Condition and the Runge-Kutta Method


Suggested Further Reading

9 Equation of Motion

1. Spring-Mass System

2. Single Step Explicit Scheme

3. Conditionally Stable Schemes

5. Modal Decomposition

6. Stability Conditions for Ky + Mÿ = 0

7. Nonlinear Equation of Motion

8. Single Step Unconditionally Stable Implicit Scheme

9. Unconditionally Stable Semiexplicit Schemes

10. Multistep Methods

11. Runge-Kutta-Nyström Method

12. Shooting in Boundary Value Problems


Suggested Further Reading

10 Wave Propagation

1. Standing and Traveling Waves in a String

2. Discretization in Space

3. Spurious Dispersion

4. Effects of (Numerical) Viscosity

5. Higher-Order Elements

6. Spurious Reflection

7. Flexural Waves in a Beam

8. Stiff String


Suggested Further Reading



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© Academic Press 1979
Academic Press
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About the Author

Isaac Fried

About the Editor

Werner Rheinboldt

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