Numerical Methods for Nonlinear Engineering Models

Preface; 1 Introduction to Nonlinear Engineering Problems and Models; 1.1 Science and Engineering; 1.2 The Engineering Method; 1.3 Some General Features of Engineering Models; 1.4 Linear and Nonlinear; 1.5 A Brief Look Ahead; 2 Numerical Fundamentals and Computer Programming; 2.1 Computer Programming Languages; 2.2 Lua as a Programming Language; 2.3 Data Representation and Associated Limitations; 2.4 Language Extensibility; 2.5 Some Language Enhancement Functions; 2.6 Software Coding Practices; 2.7 Summary; 3 Roots of Nonlinear Equations; 3.1 Successive Substitutions or Fixed Point Iteration; 3.2 Newton's Method or Newton-Ralphson Method; 3.3 Halley's Iteration Method; 3.4 Other Solution Methods; 3.5 Some Final Considerations for Finding Roots of Functions; 3.6 Summary; 4 Solving Sets of Equations: Linear and Nonlinear; 4.1 The Solution of Sets of Linear Equations; 4.2 Solution of Sets of Nonlinear Equations; 4.3 Some Examples of Sets of Equations; 4.4 Polynomial Equations and Roots of Polynomial Equations; 4.5 Matrix Equations and Matrix Operations; 4.6 Matrix Eigenvalue Problems; 4.7 Summary; 5 Numerical Derivatives and Numerical Integration; 5.1 Fundamentals of Numerical Derivatives; 5.2 Maximum and Minimum Problems; 5.3 Numerical Partial Derivatives and Min/Max Applications; 5.4 Fundamentals of Numerical Integration; 5.5 Integrals with Singularities and Infinite Limits; 5.6 Summary; 6 Interpolation; 6.1 Introduction to Interpolation - Linear Interpolation; 6.2 Interpolation using Local Cubic (LCB) Functions; 6.3 Interpolation using Cubic Spline Functions (CSP); 6.4 Interpolation Examples with Known Functions; 6.5 Interpolation Examples with Unknown Functions; 6.6 Summary; 7 Curve Fitting and Data Plotting; 7.1 Introduction; 7.2 Linear Least Squares Data Fitting; 7.3 General Least Squares Fitting with Linear Coefficients; 7.4 The Fourier Series Method; 7.5 Nonlinear Least Squares Curve Fitting; 7.6 Data Fitting and Plotting with Known Functional Foms;7.7 General Data Fitting and Plotting; 7.8 Rational Function Approximations to Implicit Functions; 7.9 Weighting Factors; 7.10 Summary; 8 Statistical Methods and Basic Statistical Functions; 8.1 Introduction; 8.2 Basic Statistical Properties and Functions; 8.3 Distributions and More Distributions; 8.4 Analysis of Mean and Variance; 8.5 Comparing Distribution Functions - The Kolmogorov-Smirnov Test; 8.6 Monte Carlo Simulations and Confidence Limits; 8.7 Non-Gaussian Distributions and Reliability Modeling; 8.8 Summary; 9 Data Models and Parameter Estimation; 9.1 Introduction; 9.2 Goodness of Data Fit and the 6-Plot Approach; 9.3 Confidence Limits on Estimated Parameters and MC Analysis; 9.4 Examples of Single Variable Data Fitting and Parameter Estimation; 9.5 Data Fitting and Parameter Estimation with Weighting Factors; 9.6 Data Fitting and Parameter Estimation with Transcendental Functions; 9.7 Data Fitting and Parameter Estimation with Piecewise Model Equations; 9.8 Data Fitting and Parameter Estimation with Multiple Independent Parameters; 9.9 Response Surface Modeling and Parameter Estimation; 9.10 Summary; 10 Differential Equations: Initial Value Problems; 10.1 Introduction to the Numerical Solution of Differential Equations; 10.2 Systems of Differential Equations; 10.3 Exploring Stability and Accuracy Issues with Simple Examples; 10.4 Development of a General Differential Equation Algorithm; 10.5 Variable Time Step Solutions; 10.6 A More Detailed Look at Accuracy Issues with the TP Algorithm; 10.7 Runge-Kutta Algorithms; 10.8 An Adaptive Step Size Algorithm; 10.9 Comparison with MATLAB Differential Equation Routines; 10.10 Direct Solution of Second Order Differential Equations; 10.11 Differential-Algebraic Systems of Equations; 10.12 Examples of Initial Value Problems; 10.13 Summary; 11 Differential Equations: Boundary Value Problems; 11.1 Introduction to Boundary Value Problems in One Independent Variable; 11.2 Shooting(ST) Methods and Boundary Value