This text emphasizes the general area of non-linear engineering models with an in-depth discussion of computer algorithms for solving non-linear problems coupled with appropriate examples of such non-linear engineering problems. The present book is a balance of numerical theory applied to non-linear or linear engineering models with many examples taken for typical engineering type problems. It is is for the engineer who wants to learn the basics of the theory but who wishes to rapidly get to solving real non-linear engineering type problems with modern desktop computers.

Table of Contents

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 Problems(BVP); 11.3 Accuracy of the Shooting Method for Boundary Value Problems; 11.4 Eigenvalue and Eigenfunction Problems in Differential Equations; 11.5 Finite Difference Methods and Boundary Value Problems; 11.6 Boundary Value Problems with Coupled Second Order Differential Equations; 11.7 Other Selected Examples of Boundary Value Problems; 11.8 Estimating Parameters of Differential Equations; 11.9 Summary; 12 Partial Differential Equations: Finite Difference Approaches; 12.1 Introduction to Single Partial Differential Equations; 12.2 Introduction to Boundary Conditions; 12.3 Introduction to the Finite Difference Method; 12.4 Coupled Systems of Partial Differential Equations; 12.5 Exploring the Accuracy of the Finite Difference Method For Partial Differential Equations; 12.6 Some Examples of Partial Differential Equations of the Initial Value, Boundary Value Type; 12.7 Boundary Value Problems (BVP) in Two Dimensions; 12.8 Two Dimensional BVPs with One Time Dimension; 12.9 Some Selected Examples of Two Dimensional BVPs; 12.10 Summary; 13 Partial Differential Equations: The Finite Element Method; 13.1 An Introduction to the Finite Element Method; 13.2 Selecting and Determining Finite Elements; 13.3 Shape Functions and Natural Coordinates; 13.4 Formulation of Finite Element Equations; 13.5 Interpolation Functions and Integral Evaluation; 13.6 Boundary Conditions with the Finite Element Method; 13.7 The Complete Finite Element Method; 13.8 Exploring the Accuracy of the Finite Element Method; 13.9 Two Dimensional BVPs with One Time Dimension; 13.10 Selected Examples of PDEs with the Finite Element Method; 13.11 Some Final Considerations for the FE Method; 13.12 Summary; Appendix A: A Brief Summary of the Lua Programming Language; Appendix B: Software Installation; Subject Index.