Numerical Methods for Engineers and Scientists

Amos Gilat, Ph.D., is Professor of Mechanical Engineering at The Ohio State University. Dr. Gillat’s main research interests are in plasticity, specifically, in developing experimental techniques for testing materials over a wide range of strain rates and temperatures and in investigating constitutive relations for viscoplasticity. Dr. Gilat's research has been supported by the National Science Foundation, NASA, Federal Aviation Administration, Department of Defense, and various industries. Vish Subramaniam, Ph.D., is a Professor of Mechanical Engineering & Chemical Physics at The Ohio State University. Dr. Subramaniam’s main research interests are in cancer detection and imaging, plasma and laser physics and processes, particularly those that involve non-equilibirum phenomena. Dr. Subramaniam's research is both experimental and computational, and has been supported by The Department of Defense, National Science Foundation, and numerous industries.

• Preface iiiChapter 1 Introduction 11.1 Background 11.2 Representation of Numbers on a Computer 41.3 Errors in Numerical Solutions 101.4 Computers and Programming 151.5 Problems 18Chapter 2 Mathematical Background 232.1 Background 232.2 Concepts from Pre-Calculus and Calculus 242.3 Vectors 282.4 Matrices and Linear Algebra 322.5 Ordinary Differential Equations (ODE) 412.6 Functions of Two or More Independent Variables 442.7 Taylor Series Expansion of Functions 472.8 Inner Product and Orthogonality 502.9 Problems 51Chapter 3 Solving Nonlinear Equations 573.1 Background 573.2 Estimation of Errors in Numerical Solutions 593.3 Bisection Method 613.4 Regula Falsi Method 643.5 Newton’s Method 663.6 Secant Method 713.7 Fixed-Point Iteration Method 743.8 Use of MATLAB Built-In Functions for Solving Nonlinear Equations 773.9 Equations with Multiple Solutions 793.10 Systems of Nonlinear Equations 813.11 Problems 88Chapter 4 Solving a System of Linear Equations 994.1 Background 994.2 Gauss Elimination Method 1024.3 Gauss Elimination with Pivoting 1124.4 Gauss–Jordan Elimination Method 1154.5 LU Decomposition Method 1184.6 Inverse of a Matrix 1284.7 Iterative Methods 1324.8 Use of MATLAB Built-In Functions for Solving a System of Linear Equations 1364.9 Tridiagonal Systems of Equations 1414.10 Error, Residual, Norms, and Condition Number 1464.11 Ill-Conditioned Systems 1514.12 Problems 155Chapter 5 Eigenvalues and Eigenvectors 1655.1 Background 1655.2 The Characteristic Equation 1675.3 The Basic Power Method 1675.4 The Inverse Power Method 1725.5 The Shifted Power Method 1735.6 The QR Factorization and Iteration Method 1745.7 Use of MATLAB Built-In Functions for Determining Eigenvalues andEigenvectors 1845.8 Problems 186Chapter 6 Curve Fitting and Interpolation 1936.1 Background 1936.2 Curve Fitting with a Linear Equation 1956.3 Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form 2016.4 Curve Fitting with Quadratic and Higher-Order Polynomials 2056.5 Interpolation Using a Single Polynomial 2106.6 Piecewise (Spline) Interpolation 2236.7 Use of MATLAB Built-In Functions for Curve Fitting and Interpolation 2366.8 Curve Fitting with a Linear Combination of Nonlinear Functions 2386.9 Problems 241Chapter 7 Fourier Methods 2517.1 Background 2517.2 Approximating a Square Wave by a Series of sine functions 2547.3 General (Infinite) Fourier Series 2577.4 Complex Form of the Fourier Series 2627.5 The Discrete Fourier Series and Discrete Fourier transform 2637.6 Complex Discrete Fourier Transform 2697.7 Power (Energy) Spectrum 2727.8 Aliasing and Nyquist Frequency 2737.9 Alternative Forms of the Discrete Fourier Transform 2787.10 Use of MATLAB Built-In Functions for Calculating Discrete Fourier Transform 2797.11 Leakage and Windowing 2847.12 Bandwidth and Filters 2867.13 The Fast Fourier Transform (FFT) 2887.14 Problems 298Chapter 8 Numerical Differentiation 3038.1 Background 3038.2 Finite Difference Approximation of the Derivative 3058.3 Finite Difference Formulas Using Taylor Series Expansion 3108.4 Summary of Finite Difference Formulas for Numerical Differentiation 3178.5 Differentiation Formulas Using Lagrange Polynomials 3198.6 Differentiation Using Curve Fitting 3208.7 Use of MATLAB Built-In Functions for Numerical Differentiation 3208.8 Richardson’s Extrapolation 3228.9 Error in Numerical Differentiation 3258.10 Numerical Partial Differentiation 3278.11 Problems 330Chapter 9 Numerical Integration 3419.1 Background 3419.1.1 Overview of Approaches in Numerical Integration 3429.2 Rectangle and Midpoint Methods 3449.3 Trapezoidal Method 3469.4 Simpson’s Methods 3509.5 Gauss Quadrature 3559.6 Evaluation of Multiple Integrals 3609.7 Use of MATLAB Built-In Functions for Integration 3629.8 Estimation of Error in Numerical Integration 3649.9 Richardson’s Extrapolation 3669.10 Romberg Integration 3699.11 Improper Integrals 3729.12 Problems 374Chapter 10 Ordinary Differential Equations: Initial-ValueProblems 38510.1 Background 38510.2 Euler’s Methods 39010.3 Modified Euler’s Method 40110.4 Midpoint Method 40410.5 Runge–Kutta Methods 40510.6 Multistep Methods 41710.6.1 Adams–Bashforth Method 41810.6.2 Adams–Moulton Method 41910.7 Predictor–Corrector Methods 42010.8 System of First-Order Ordinary Differential Equations 42210.9 Solving a Higher-Order Initial Value Problem 43210.10 Use of MATLAB Built-In Functions for Solving Initial-Value Problems 43710.11 Local Truncation Error in Second-Order Range–Kutta Method 44710.12 Step Size for Desired Accuracy 44810.13 Stability 45210.14 Stiff Ordinary Differential Equations 45410.15 Problems 457Chapter 11 Ordinary Differential Equations: Boundary-ValueProblems 47111.1 Background 47111.2 The Shooting Method 47411.3 Finite Difference Method 48211.4 Use of MATLAB Built-In Functions for Solving Boundary Value Problems 49211.5 Error and Stability in Numerical Solution of Boundary Value Problems 49711.6 Problems 499Appendix A Introductory MATLAB 509A.1 Background 509A.2 Starting with MATLAB 509A.3 Arrays 514A.4 Mathematical Operations with Arrays 519A.5 Script Files 524A.6 Plotting 526A.7 User-Defined Functions and Function Files 528A.8 Anonymous Functions 530A.9 Function functions 532A.10 Subfunctions 535A.11 Programming in MATLAB 537A.11.1 Relational and Logical Operators 537A.11.2 Conditional Statements, if-else Structures 538A.11.3 Loops 541A.12 Problems 542Appendix B MATLAB Programs 547Appendix C Derivation of the Real Discrete Fourier Transform 551C.1 Orthogonality of Sines and Cosines for Discrete Points 551C.2 Determination of the Real DFT 553Index 555