Presents standard numerical approaches for solving common mathematical problems in engineering using Python. Covers the most common numerical calculations used by engineering students Covers Numerical Differentiation and Integration, Initial Value Problems, Boundary Value Problems, and Partial Differential Equations Focuses on open ended, real world problems that require students to write a short report/memo as part of the solution process Includes an electronic download of the Python codes presented in the book

Table of Contents:
Preface xi About the Companion Website xv 1 Problem Solving in Engineering 1 1.1 Equation Identification and Categorization 4 1.1.1 Algebraic versus Differential Equations 4 1.1.2 Linear versus Nonlinear Equations 5 1.1.3 Ordinary versus Partial Differential Equations 6 1.1.4 Interpolation versus Regression 8 Problems 10 Additional Resources 11 References 11 2 Programming with Python 12 2.1 Why Python? 12 2.1.1 Compiled versus Interpreted Computer Languages 13 2.1.2 A Note on Python Versions 14 2.2 Getting Python 15 2.2.1 Installation of Python 17 2.2.2 Alternative to Installation: SageMathCloud 18 2.3 Python Variables and Operators 19 2.3.1 Updating Variables 21 2.3.2 Containers 23 2.4 External Libraries 25 2.4.1 Finding Documentation 27 Problems 28 Additional Resources 29 References 30 3 Programming Basics 31 3.1 Comparators and Conditionals 31 3.2 Iterators and Loops 34 3.2.1 Indentation Style 39 3.3 Functions 39 3.3.1 Pizza Example 43 3.3.2 Print Function 44 3.4 Debugging or Fixing Errors 45 3.5 Top 10+ Python Error Messages 45 Problems 47 Additional Resources 49 References 49 4 External Libraries for Engineering 51 4.1 Numpy Library 51 4.1.1 Array and Vector Creation 51 4.1.2 Array Operations 55 4.1.3 Getting Helping with Numpy 55 4.1.4 Numpy Mathematical Functions 56 4.1.5 Random Vectors with Numpy 57 4.1.6 Sorting and Searching 57 4.1.7 Polynomials 58 4.1.8 Loading and Saving Arrays 59 4.2 Matplotlib Library 60 4.3 Application: Gillespie Algorithm 63 Problems 66 Additional Resources 68 References 68 5 Symbolic Mathematics 70 5.1 Introduction 70 5.2 Symbolic Mathematics Packages 71 5.3 An Introduction to SymPy 72 5.3.1 Multiple Equations 75 5.4 Factoring and Expanding Functions 76 5.4.1 Equilibrium Kinetics Example 77 5.4.2 Partial Fraction Decomposition 78 5.5 Derivatives and Integrals 78 5.5.1 Reaction Example 79 5.5.2 Symbolic Integration 80 5.5.3 Reactor Sizing Example 80 5.6 Cryptography 81 Problems 83 References 86 6 Linear Systems 87 6.1 Example Problem 88 6.2 A Direct Solution Method 91 6.2.1 Distillation Example 95 6.2.2 Blood Flow Network Example 95 6.2.3 Computational Cost 98 6.3 Iterative Solution Methods 100 6.3.1 Vector Norms 100 6.3.2 Jacobi Iteration 100 6.3.3 Gauss–Seidel Iteration 103 6.3.4 Relaxation Methods 105 6.3.5 Convergence of Iterative Methods 105 Problems 107 References 112 7 Regression 113 7.1 Motivation 113 7.2 Fitting Vapor Pressure Data 114 7.3 Linear Regression 115 7.3.1 Alternative Derivation of the Normal Equations 118 7.4 Nonlinear Regression 119 7.4.1 Lunar Disintegration 122 7.5 Multivariable Regression 126 7.5.1 Machine Learning 127 Problems 129 References 134 8 Nonlinear Equations 135 8.1 Introduction 135 8.2 Bisection Method 137 8.3 Newton’s Method 140 8.4 Broyden’s Method 143 8.5 Multiple Nonlinear Equations 146 8.5.1 The Point Inside a Square 149 Problems 151 9 Statistics 156 9.1 Introduction 156 9.2 Reading Data from a File 156 9.2.1 Numpy Library 157 9.2.2 CVS Library 159 9.2.3 Pandas 159 9.2.4 Parsing an Array 162 9.3 Statistical Analysis 162 9.4 Advanced Linear Regression 164 9.5 U.S. Electrical Rates Example 168 Problems 172 References 175 10 Numerical Differentiation and Integration 176 10.1 Introduction 176 10.2 Numerical Differentiation 176 10.2.1 First Derivative Approximation 177 10.2.2 Second Derivative Approximation 180 10.2.3 Scipy Derivative Approximation 181 10.3 Numerical Integration 183 10.3.1 Trapezoid Rule 185 10.3.2 Numerical Integration Using Scipy 186 10.3.3 Error Function 187 Problems 190 Reference 192 11 Initial Value Problems 193 11.1 Introduction 193 11.2 Biochemical Reactors 193 11.3 Forward Euler 195 11.4 Modified Euler Method 198 11.5 Systems of Equations 199 11.5.1 The Lorenz System and Chaotic Solutions 200 11.5.2 Second-Order Initial Value Problems 203 11.6 Stiff Differential Equations 203 Problems 206 References 210 12 Boundary Value Problems 211 12.1 Introduction 211 12.2 Shooting Method 212 12.3 Finite Difference Method 216 12.3.1 Reactions in Spherical Catalysts 220 Problems 224 Reference 226 13 Partial Differential Equations 227 13.1 Finite Difference Method for Steady-State PDEs 227 13.1.1 Setup 228 13.1.2 Matrix Assembly 230 13.1.3 Solving and Plotting 232 13.2 Convection 233 13.3 Finite Difference Method for Transient PDEs 236 Problems 241 Reference 244 14 Finite Element Method 245 14.1 A Warning 245 14.2 Why FEM? 246 14.3 Laplace’s Equation 246 14.3.1 The Mesh 246 14.3.2 Discretization 247 14.3.3 Wait! Why Are We Doing This? 248 14.3.4 FEniCS Implementation 248 14.4 Pattern Formation 249 Additional Resources 253 References 254 Index 255