# Numerical Methods for Engineers and Scientists

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## Book Description

Emphasizing the finite difference approach for solving differential equations, the second edition of Numerical Methods for Engineers and Scientists presents a methodology for systematically constructing individual computer programs. Providing easy access to accurate solutions to complex scientific and engineering problems, each chapter begins with objectives, a discussion of a representative application, and an outline of special features, summing up with a list of tasks students should be able to complete after reading the chapter- perfect for use as a study guide or for review. The AIAA Journal calls the book "…a good, solid instructional text on the basic tools of numerical analysis."

## 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

## Reviews

"…a good, solid instructional text on the basic tools of numerical analysis."

-AIAA Journal