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
There are many books on the use of numerical methods for solving engineering problems and for modeling of engineering artifacts. In addition there are many styles of such presentations ranging from books with a major emphasis on theory to books with an emphasis on applications. The purpose of this book is hopefully to present a somewhat different approach to the use of numerical methods for - gineering applications. Engineering models are in general nonlinear models where the response of some appropriate engineering variable depends in a nonlinear manner on the - plication of some independent parameter. It is certainly true that for many types of engineering models it is sufficient to approximate the real physical world by some linear model. However, when engineering environments are pushed to - treme conditions, nonlinear effects are always encountered. It is also such - treme conditions that are of major importance in determining the reliability or failure limits of engineering systems. Hence it is essential than engineers have a toolbox of modeling techniques that can be used to model nonlinear engineering systems. Such a set of basic numerical methods is the topic of this book. For each subject area treated, nonlinear models are incorporated into the discussion from the very beginning and linear models are simply treated as special cases of more general nonlinear models. This is a basic and fundamental difference in this book from most books on numerical methods.
The emphasis is on nonlinear engineering models and equations as opposed to linear models
Linear models and equations are treated conveniently as just special cases of more general nonlinear models
For each chapter and topic area covered, the theory is discussed and computer code is developed and presented such that the reader can conveniently apply the developed code to practical engineering problems
For each topic covered at least one approach is extensively developed leading to working computer code that can be applied to nonlinear problems
The computer code is developed in a modern scripting language that is very easy to program and understand by the reader