Preface xi

**1 Introduction 1**

1.1 What is a Differential Equation? 1

1.2 Examples of an Ordinary and a Partial Differential Equation, 2

1.3 Numerical Analysis, a Necessity for Scientific Computing, 5

1.4 Outline of the Contents of this Book, 8

Bibliography, 10

**2 Ordinary Differential Equations 11**

2.1 Problem Classification, 11

2.2 Linear Systems of ODEs with Constant Coefficients, 16

2.3 Some Stability Concepts for ODEs, 19

2.3.1 Stability for a Solution Trajectory of an ODE System, 20

2.3.2 Stability for Critical Points of ODE Systems, 23

2.4 Some ODE models in Science and Engineering, 26

2.4.1 Newton’s Second Law, 26

2.4.2 Hamilton’s Equations, 27

2.4.3 Electrical Networks, 27

2.4.4 Chemical Kinetics, 28

2.4.5 Control Theory, 29

2.4.6 Compartment Models, 29

2.5 Some Examples from Applications, 30

Bibliography, 36

**3 Numerical Methods for Initial Value Problems 37**

3.1 Graphical Representation of Solutions, 38

3.2 Basic Principles of Numerical Approximation of ODEs, 40

3.3 Numerical Solution of IVPs with Euler’s method, 41

3.3.1 Euler’s Explicit Method: Accuracy, 43

3.3.2 Euler’s Explicit Method: Improving the Accuracy, 46

3.3.3 Euler’s Explicit Method: Stability, 48

3.3.4 Euler’s Implicit Method, 53

3.3.5 The Trapezoidal Method, 55

3.4 Higher Order Methods for the IVP, 56

3.4.1 Runge–Kutta Methods, 56

3.4.2 Linear Multistep Methods, 60

3.5 Special Methods for Special Problems, 62

3.5.1 Preserving Linear and Quadratic Invariants, 62

3.5.2 Preserving Positivity of the Numerical Solution, 64

3.5.3 Methods for Newton’s Equations of Motion, 64

3.6 The Variational Equation and Parameter Fitting in IVPs, 66

Bibliography, 69

**4 Numerical Methods for Boundary Value Problems 71**

4.1 Applications, 73

4.2 Difference Methods for BVPs, 78

4.2.1 A Model Problem for BVPs, Dirichlet’s BCs, 79

4.2.2 A Model Problem for BVPs, Mixed BCs, 83

4.2.3 Accuracy, 86

4.2.4 Spurious Solutions, 87

4.2.5 Linear Two-Point BVPs, 89

4.2.6 Nonlinear Two-Point BVPs, 91

4.2.7 The Shooting Method, 92

4.3 Ansatz Methods for BVPs, 94

4.3.1 Starting with the ODE Formulation, 95

4.3.2 Starting with the Weak Formulation, 96

4.3.3 The Finite Element Method, 100

Bibliography, 103

**5 Partial Differential Equations 105**

5.1 Classical PDE Problems, 106

5.2 Differential Operators Used for PDEs, 110

5.3 Some PDEs in Science and Engineering, 114

5.3.1 Navier–Stokes Equations for Incompressible Flow, 114

5.3.2 Euler’s Equations for Compressible Flow, 115

5.3.3 The Convection–Diffusion–Reaction Equations, 116

5.3.4 The Heat Equation, 117

5.3.5 The Diffusion Equation, 117

5.3.6 Maxwell’s Equations for the Electromagnetic Field, 117

5.3.7 Acoustic Waves, 118

5.3.8 Schrödinger’s Equation in Quantum Mechanics, 119

5.3.9 Navier’s Equations in Structural Mechanics, 119

5.3.10 Black–Scholes Equation in Financial Mathematics, 120

5.4 Initial and Boundary Conditions for PDEs, 121

5.5 Numerical Solution of PDEs, Some General Comments, 121

Bibliography, 122

**6 Numerical Methods for Parabolic Partial Differential Equations 123**

6.1 Applications, 125

6.2 An Introductory Example of Discretization, 127

6.3 The Method of Lines for Parabolic PDEs, 130

6.3.1 Solving the Test Problem with MoL, 130

6.3.2 Various Types of Boundary Conditions, 134

6.3.3 An Example of the Use of MoL for a Mixed Boundary Condition, 135

6.4 Generalizations of the Heat Equation, 136

6.4.1 The Heat Equation with Variable Conductivity, 136

6.4.2 The Convection – Diffusion – Reaction PDE, 138

6.4.3 The General Nonlinear Parabolic PDE, 138

6.5 Ansatz Methods for the Model Equation, 139

Bibliography, 140

**7 Numerical Methods for Elliptic Partial Differential Equations 143**

7.1 Applications, 145

7.2 The Finite Difference Method, 150

7.3 Discretization of a Problem with Different BCs, 154

7.4 Ansatz Methods for Elliptic PDEs, 156

7.4.1 Starting with the PDE Formulation, 156

7.4.2 Starting with the Weak Formulation, 158

7.4.3 The Finite Element Method, 159

Bibliography, 164

**8 Numerical Methods for Hyperbolic PDEs 165**

8.1 Applications, 171

8.2 Numerical Solution of Hyperbolic PDEs, 174

8.2.1 The Upwind Method (FTBS), 175

8.2.2 The FTFS Method, 177

8.2.3 The FTCS Method, 178

8.2.4 The Lax–Friedrichs Method, 178

8.2.5 The Leap-Frog Method, 179

8.2.6 The Lax–Wendroff Method, 179

8.2.7 Numerical Method for the Wave Equation, 181

8.3 The Finite Volume Method, 183

8.4 Some Examples of Stability Analysis for Hyperbolic PDEs, 185

Bibliography, 187

**9 Mathematical Modeling with Differential Equations 189**

9.1 Nature Laws, 190

9.2 Constitutive Equations, 192

9.2.1 Equations in Heat Transfer Problems, 192

9.2.2 Equations in Mass Diffusion Problems, 193

9.2.3 Equations in Mechanical Moment Diffusion Problems, 193

9.2.4 Equations in Elastic Solid Mechanics Problems, 194

9.2.5 Equations in Chemical Reaction Engineering Problems, 194

9.2.6 Equations in Electrical Engineering Problems, 195

9.3 Conservative Equations, 195

9.3.1 Some Examples of Lumped Models, 196

9.3.2 Some Examples of Distributed Models, 197

9.4 Scaling of Differential Equations to Dimensionless Form, 201

Bibliography, 204

**10 Applied Projects on Differential Equations 205**

Project 1 Signal propagation in a long electrical conductor, 205

Project 2 Flow in a cylindrical pipe, 206

Project 3 Soliton waves, 208

Project 4 Wave scattering in a waveguide, 209

Project 5 Metal block with heat sourse and thermometer, 210

Project 6 Deformation of a circular metal plate, 211

Project 7 Cooling of a chrystal glass, 212

Project 8 Rotating fluid in a cylinder, 212

**Appendix A Some Numerical and Mathematical Tools 215**

A.1 Newton’s Method for Systems of Nonlinear Algebraic Equations, 215

A.1.1 Quadratic Systems, 215

A.1.2 Overdetermined Systems, 218

A.2 Some Facts about Linear Difference Equations, 219

A.3 Derivation of Difference Approximations, 223

Bibliography, 225

A.4 The Interpretations of Grad, Div, and Curl, 225

A.5 Numerical Solution of Algebraic Systems of Equations, 229

A.5.1 Direct Methods, 229

A.5.2 Iterative Methods for Linear Systems of Equations, 233

A.6 Some Results for Fourier Transforms, 237

Bibliography, 239

**Appendix B Software for Scientific Computing 241**

B.1 MATLAB, 242

B.1.1 Chapter 3: IVPs, 242

B.1.2 Chapter 4: BVPs, 244

B.1.3 Chapter 6: Parabolic PDEs, 245

B.1.4 Chapter 7: Elliptic PDEs, 246

B.1.5 Chapter 8: Hyperbolic PDEs, 246

B.2 COMSOL MULTIPHYSICS, 247

Bibliography and Resources, 249

**Appendix C Computer Exercises to Support the Chapters 251**

C.1 Computer Lab 1 Supporting Chapter 2, 251

C.1.1 ODE Systems of LCC Type and Stability, 251

C.2 Computer Lab 2 Supporting Chapter 3, 254

C.2.1 Numerical Solution of Initial Value Problems, 254

C.3 Computer Lab 3 Supporting Chapter 4, 257

C.3.1 Numerical Solution of a Boundary Value Problem, 257

C.4 Computer Lab 4 Supporting Chapter 6, 258

C.4.1 Partial Differential Equation of Parabolic Type, 258

C.5 Computer Lab 5 Supporting Chapter 7, 261

C.5.1 Numerical Solution of Elliptic PDE Problems, 261

C.6 Computer Lab 6 Supporting Chapter 8, 263

C.6.1 Numerical Experiments with the Hyperbolic Model PDE

Problem, 263

Index 265