2nd Edition

Numerical Methods for Engineers and Scientists

    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."

    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

    Biography

    Hoffman, Joe D.; Hoffman, Joe D.; Frankel, Steven

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