# Numerical Methods for Engineers and Scientists

Second Edition Revised and Expanded By Joe D. Hoffman

**Contents of Numerical Methods for Engineers and Scientists**

Preface

Chapter 0.

Introduction

Objectives and Approach

Organization of the Book

Examples

Programs

Problems

Significant Digits, Precision, Accuracy, Errors, and NumbeRr epresentation

Software Packages and Libraries

The Taylor Series and the Taylor Polynomial

Part I

Basic Tools of Numerical Analysis

Systems of Linear Algebraic Equations

Eigenproblems

Roots of Nonlinear Equations

Polynomial Approximation and Interpolation

Numerical Differentiation and Difference Formulas

Numerical Integration

Summary

Chapter 1. Systems of Linear Algebraic Equations

1.1. Introduction

1.2. Properties of Matrices and Determinants

1.3. Direct Elimination Methods

1.4. LU Factorization

1.5. Tridiagonal Systems of Equations

1.6. Pitfalls of Elimination Methods

1.7. Iterative Methods

1.8. Programs

1.9. Summary

Exercise Problems

Chapter 2. Eigenproblems

2.1. Introduction

2.2. Mathematical Characteristics of Eigenproblems

2.3. The Power Method

2.4. The Direct Method

2.5. The QR Method

2.6. Eigenvectors

2.7. Other Methods

2.8. Programs

2.9. Summary ,

Exercise Problem

Chapter 3. Nonlinear Equations

3.1. Introduction

3.2. General Features of Root Finding

3.3. Closed Domain (Bracketing) Methods

3.4. Open Domain Methods

3.5. Polynomials

3.6. Pitfalls of Root Finding Methods and Other Methods of Root Finding

3.7. Systems of Nonlinear Equations

3.8. Programs

3.9. Summary

Exercise Problem.

Chapter 4. Polynomial Approximation and Interpolation

4.1. Introduction

4.2. Properties of Polynomials

4.3. Direct Fit Polynomials

4.4. Lagrange Polynomials

4.5. Divided Difference Tables and Divided Difference Polynomials

4.6. Difference Tables and Difference Polynomials

4.7. Inverse Interpolation

4.8. Multivariate Approximation

4.9. Cubic Splines

4.10. Least Squares Approximation

4.11. Programs

4.12. Summary

Exercise Problems

Chapter 5. Numerical Differentiation and Difference Formulas

5.1. Introduction

5.2. Unequally Spaced Data

5.3. Equally Spaced Data

5.4. Taylor Series Approach

5.5. Difference Formulas

5.6. Error Estimation and Extrapolation

5.7. Programs

5.8. Summary

Exercise Problems

Chapter 6. Numerical Integration

6.1. Introduction

6.2. Direct Fit Polynomials

6.3. Newton-Cotes Formulas

6.4. Extrapolation and RombergI ntegration

6.5. Adaptive Integration

6.6. Gaussian Quadrature

6.7. Multiple Integrals

6.8. Programs

6.9. Summary

Exercise Problems

Part II.

Ordinary Differential Equations

Introduction

General Features of Ordinary Differential Equations

Classification of Ordinary Differential Equations

Classification of Physical Problems

Initial-Value Ordinary Differential Equations

Boundary-ValueO rdinary Differential Equations

Summary

Chapter 7.

One-DimensionaIln itial-Value Ordinary Differential Equations

Introduction

General Features of Initial-Value ODEs

The Taylor Series Method

The Finite Difference Method

The First-Order Euler Methods

Consistency, Order, Stability, and Convergence

Single-Point Methods

Extrapolation Methods

Multipoint Methods

Summary of Methods and Results

Nonlinear Implicit Finite Difference Equations

Higher-Order Ordinary Differential Equations

Systems of First-Order Ordinary Differential Equations

Stiff Ordinary Differential Equations

Programs

Summary

Exercise Problems

Chapter 8.

One-Dimensional Boundary-Value Ordinary Differential Equations

Introduction

General Features of Boundary-Value ODEs

The Shooting (initial-Value) Method

The Equilibrium (Boundary-Value) Method

Derivative (and Other) Boundary Conditions

Higher-Order Equilibrium Methods

The Equilibrium Method for Nonlinear Boundary-Value Problems

The Equilibrium Method on Nonuniform Grids

Eigenproblems

Programs

Summary

Exercise Problems

Part III.

Partial Differential Equations

Introduction

General Features of Partial Differential Equations

Classification of Partial Differential Equations

Classification of Physical Problems

Elliptic Partial Differential Equations

Parabolic Partial Differential Equations

Hyperbolic Partial Differential Equations

The Convection-Diffusion Equation

Initial Values and Boundary Conditions

Well-Posed Problems

Summary

Chapter 9.

Elliptic Partial Differential Equations

Introduction

General Features of Elliptic PDEs

The Finite Difference Method

Finite Difference Solution of the Laplace Equation

Consistency, Order, and Convergence

Iterative Methods of Solution

Derivative Boundary Conditions

Finite Difference Solution of the Poisson Equation

Higher-Order Methods

Nonrectangular Domains

Nonlinear Equations and Three-Dimensional Problems

The Control Volume Method

Programs

Summary

Exercise Problems

Chapter 10.

Parabolic Partial Differential Equations

Introduction

General Features of Parabolic PDEs

The Finite Difference Method

The Forward-Time Centered-Space (FTCS) Method

Consistency, Order, Stability, and Convergence

The Richardson and DuFort-Frankel Methods

Implicit Methods

Derivative Boundary Conditions

Nonlinear Equations and Multidimensional Problems

The Convection-Diffusion Equation

Asymptotic Steady State Solution to Propagation Problems

Programs

Summary

Exercise Problems

Chapter 11.

Hyperbolic Partial Differential Equations

Introduction

General Features of Hyperbolic PDEs

The Finite Difference Method

The Forward-Time Centered-Space (FTCS) Method and the Lax Method

Lax-Wendroff Type Methods

Upwind Methods

The Backward-Time Centered-Space (BTCS) Method

Nonlinear Equations and Multidimensional Problems

The Wave Equation

Programs

Summary

Exercise Problems

Chapter 12

Finite Element Method

Introduction

The Rayleigh-Ritz, Collocation, and Galerkin Methods

The Finite Element Method for Boundary Value Problems

The Finite Element Method for the Laplace (Poisson) Equation

The Finite Element Method for the Diffusion Equation

Programs

Summary

Exercise Problems

References

Answers to Selected Problems

Index

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