Numerical Methods for Engineers and Scientists

Table of Contents

Introduction
. Objectives and Approach
. Organization of the Book
. Examples
. Programs
. Problems
. Significant Digits, Precision, Accuracy, Errors, and Number Representation
. Software Packages and Libraries
. The Taylor Series and the Taylor Polynomial

BASIC TOOLS OF NUMERICAL ANALYSIS
. Systems of Linear Algebraic Equations
. Eigenproblems
. Nonlinear Equations
. Polynomial Approximation and Interpolation
. Numerical Differentiation and Difference Formulas
. Numerical Integration

Systems of Linear Algebraic Equations
. Introduction
. Properties of Matrices and Determinants
. Direct Elimination Methods
. LU Factorization
. Tridiagonal Systems of Equations
. Pitfalls of Elimination Methods
. Iterative Methods
. Programs
. Summary
. Exercise Problems
Eigenproblems
. Introduction
. Mathematical Characteristics of Eigenproblems
. The Power Method
. The Direct Method
. The QR Method
. Eigenvectors
. Other Methods
. Programs Summary
. Exercise Problems
Nonlinear Equations
. Introduction
. General Features of Root Finding
. Closed Domain (Bracketing) Methods
. Open Domain Methods
. Polynomials
. Pitfalls of Root Finding Methods and Other Methods of Root Finding
. Systems of Nonlinear Equations
. Programs
. Summary
. Exercise Problems
Polynomial Approximation and Interpolation
. Introduction
. Properties of Polynomials
. Direct Fit Polynomials
. Lagrange Polynomials
. Divided Difference Tables and Divided Difference Polynomials
. Difference Tables and Difference Polynomials
. Inverse Interpolation
. Multivariate Approximation
. Cubic Splines
. Least Squares Approximation
. Programs
. Summary
. Exercise Problems
Numerical Differentiation and Difference Formulas
. Introduction
. Unequally Spaced Data
. Equally Spaced Data
. Taylor Series Approach
. Difference Formulas
. Error Estimation and Extrapolation
. Programs
. Summary
. Exercise Problems
Numerical Integration
. Introduction
. Direct Fit Polynomials
. Newton-Cotes Formulas
. Extrapolation and Romberg Integration
. Adaptive Integration
. Gaussian Quadrature
. Multiple Integrals
. Programs
. Summary
. Exercise Problems

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-Value Ordinary Differential Equations
. Summary

One-Dimensional Initial-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
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

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 Equaitons
. The Convection-Diffusion Equation
. Initial Values and Boundary Conditions
. Well-Posed Problems
. Summary

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
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
Hyperbolic Partial Differential Equations
. Introduction
. General Features of Hyperbolic PDEs
. The Finite Difference Method
. The Forward-Time Centered-Space (FTCS) Methods 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
The 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