Differential Equations As Models In Science And Engineering

Differential Equations As Models In Science And Engineering

by Gregory Richard Baker

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Overview

Differential Equations As Models In Science And Engineering by Gregory Richard Baker

This textbook develops a coherent view of differential equations by progressing through a series of typical examples in science and engineering that arise as mathematical models. All steps of the modeling process are covered: formulation of a mathematical model; the development and use of mathematical concepts that lead to constructive solutions; validation of the solutions; and consideration of the consequences. The volume engages students in thinking mathematically, while emphasizing the power and relevance of mathematics in science and engineering. There are just a few guidelines that bring coherence to the construction of solutions as the book progresses through ordinary to partial differential equations using examples from mixing, electric circuits, chemical reactions and transport processes, among others. The development of differential equations as mathematical models and the construction of their solution is placed center stage in this volume.

Product Details

ISBN-13: 9789814656962
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 09/22/2016
Pages: 392
Product dimensions: 6.80(w) x 9.60(h) x 0.90(d)

Table of Contents

Preface vii

A Note to the Student xv

1 Linear Ordinary Differential Equations 1

1.1 Growth and decay 1

1.1.1 Bacterial growth 2

1.1.2 From discrete to continuous 6

1.1.3 Conservation of quantity 8

1.1.4 Simple electric circuits 11

1.1.5 Abstract, viewpoint 13

14.6 Exercises 17

1.1.7 Additional deliberations 21

1.2 Forcing effects 21

1.2.1 Constant inflow as an input 22

1.2.2 Periodic inflow as an input 27

1.2.3 Discontinuous inflow as an input 31

1.2.4 Abstract viewpoint 34

1.2.5 Exercises 38

1.2.6 General forcing terms 43

1.3 Coefficients with time variation 46

1.3.1 Variable river flow 46

1.3.2 Abstract viewpoint 49

1.3.3 Exercises 53

1.3.4 Summary 54

1.3.5 Flowchart: Linear, first-order differential equations 56

1.4 Second-order equations: Growth and decay 56

1.4.1 A simple chain of chemical reactions 57

1.4.2 Chained chemical reactions with forcing 63

1.4.3 Abstract viewpoint 65

1.4.4 Exercises 70

1.4.5 Time-varying coefficients in the differential equation 73

1.5 Second-order equations: Oscillations 78

1.5.1 Exponentials with complex arguments 78

1.5.2 Application to differential equations 81

1.5.3 The LCR circuit 83

1.5.4 Abstract viewpoint 89

1.5.5 Exercises 91

1.5.6 Higher-order differential equations 94

1.6 Forcing terms: Resonances 95

1.6.1 LOR circuit with constant applied voltage 96

1.6.2 LCR circuit with alternating applied voltage 97

1.6.3 Complex solutions 101

1.6.4 Resonance 102

1.6.5 Abstract view 104

1.6.6 Exercises 107

1.6.7 General forcing terms 111

1.6.8 Summary 115

1.6.9 Flowchart: Linear, second-order differential equations 117

2 Periodic Behavior 119

2.1 Periodic functions 119

2.1.1 Mathematical expression 119

2.1.2 Periodic forcing of an LCR circuit 122

2.1.3 Abstract view 125

2.1.4 Exercises 129

2.1.5 Periodic complex functions 131

2.2 The Fourier series 132

2.2.1 Construction of a Fourier series 134

2.2.2 Calculation of the Fourier coefficients 140

2.2.3 Abstract view 143

2.2.4 Exercises 152

2.2.5 Complex Fourier coefficients 153

2.3 Symmetry in the Fourier series 155

2.3.1 Even and odd functions 155

2.3.2 Fourier coefficients for even and odd periodic functions 157

2.3.3 Abstract view 160

2.3.4 Exercises 161

2.3.5 Other symmetries 163

3 Boundary Value Problems 167

3.1 Spatially varying steady states 167

3.1.1 Steady state transport 168

3.1.2 Diffusion 170

3.1.3 Abstract view 176

3.1.4 Exercises 178

3.2 Bifurcation 180

3.2.1 Column buckling 180

3.2.2 Abstract viewpoint 184

3.2.3 Summary 186

3.2.4 Exercises 188

3.3 Forcing effects 190

3.3.1 Cooling fins 191

3.3.2 Heated plate; homogeneous boundary conditions 194

3.3.3 Heated plate; inhomogeneous boundary conditions 196

3.3.4 Abstract viewpoint 199

3.3.5 Exercises 203

3.3.6 Flowchart: Linear, second-order boundary-value problems 205

4 Linear Partial Differential Equations 207

4.1 Diffusion: Part I 207

4.1.1 Transport equations 208

4.1.2 Initial and boundary conditions 210

4.1.3 Exponential solutions 213

4.1.4 Separation of variables 214

4.1.5 Abstract view 224

4.3.1 Exercises 226

4.2 Diffusion: Part II 229

4.2.1 Inhomogeneous boundary conditions 229

4.2.2 Homogeneous solution 231

4.2.3 Time-dependent boundary conditions 235

4.2.4 Abstract view 238

4.2.5 Exercises 241

4.3 Propagation 244

4.3.1 Transmission line 245

4.3.2 Initial and boundary conditions 246

4.3.3 The homogeneous solution 247

4.3.4 Inhomogeneous boundary conditions 254

4.3.5 No resistance 257

4.3.6 Abstract view 258

4.3.7 Exercises 261

4.3.8 Inhomogeneous partial differential equations 263

4.4 Laplace's equation 264

4.4.1 Heat transport in two dimensions 265

4.4.2 Steady state solution as a particular solution 268

4.4.3 Electrostatic potential 275

4.4.4 Abstract view 279

4.4.5 Exercises 281

4.4.6 Heat transport in two dimensions - continued 283

5 Systems of Differential Equations 287

5.1 First-order equations 287

5.1.1 Population dynamics 287

5.1.2 Abstract view 295

5.1.3 Exercises 299

5.2 Homogeneous linear equations 303

5.2.1 Basic concepts 304

5.2.2 Chemical reactions 305

5.2.3 The LCR circuit 310

5.2.4 Abstract viewpoint 313

5.2.5 Exercises 318

5.2.6 Higher dimensional systems 320

5.3 Inhomogeneous linear equations 322

5.3.1 LCR circuit with constant applied voltage 323

5.3.2 LCR circuit with alternating applied voltage 324

5.3.3 Stability 325

5.3.4 Abstract view 327

5.3.5 Exercises 330

5.3.6 General forcing term 331

5.4 Nonlinear autonomous equations 333

5.4.1 Predator-prey model 333

5.4.2 Abstract view 340

5.4.3 Exercises 343

Appendix A The Exponential Function 345

A.1 A review of its properties 345

Appendix B The Taylor Series 347

B.1 A derivation of the Taylor series 347

B.2 Accuracy of the truncated Taylor series 349

B.3 Standard examples 351

Appendix C Systems of Linear Equations 353

C.1 Algebraic equations 353

C.2 Gaussian elimination 354

C.3 Matrix form 355

C.4 Eigenvalues and eigenvectors 356

Appendix D Complex Variables 359

D.1 Basic properties 359

D.2 Connections with trigonometric functions 363

Index 365

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