About the Author xvii
Preface xix
Outline xxi
About the Companion Website xxv

1 Modeling in Chemical and Subsurface Energy Resources Engineering 1
1.1 Introduction 1
1.2 Motivation 2
1.3 Development of Mathematical Models 3
1.4 Modeling Simplification 4
1.5 Organization of the Book and Guidance for Readers 4
1.6 Summary 5

2 Fundamentals of Mathematical Modeling 7
2.1 Introduction 7
2.2 Basics 7
2.3 Fundamental Laws 8
2.4 Coordinate Systems for Heat, Energy, and Momentum Balances 10
2.5 Lumped Versus Distributed Models 12
2.6 Additional Problems 22

3 Review of Ordinary Differential Equations 25
3.1 Introduction 25
3.2 Definitions 25
3.3 Integrating Factor Method 27
3.4 Bernoulli's ODE 28
3.5 Riccati's ODE 30
3.6 Second-order ODEs 31
3.7 Laplace Transform 35
3.8 Cauchy–Euler Equation 47
3.9 Higher-order ODEs 48
3.10 Additional Problems 49

4 Boundary Value Problems 51
4.1 Introduction 51
4.2 Regular Sturm–Liouville Problem 51
4.3 Singular BVPs 52
4.4 Periodic BVPs 53Contents ix
4.5 Orthogonality Property 53
4.6 Boundary Conditions 54
4.7 Determination of Eigenvalues and Eigenfunctions for Various Boundary Conditions 55
4.8 A General Note on the Sign of Eigenvalues 61
4.9 Evaluation of the Norm for ;;′′ + λ;; = 0 Subject to Various Boundary Conditions 63
4.10 Additional Problems 67

5 Separation of Variables 69
5.1 Classification of Partial Differential Equations 69
5.2 Scope 71
5.3 Separation of Variables for Homogeneous PDEs With Homogeneous Boundary Conditions 71
5.4 Separation of Variables for Homogeneous PDEs With Nonhomogeneous Boundary Conditions 84
5.5 Separation of Variables for Nonhomogeneous PDEs 107
5.6 Application of Separation of Variables to Multidimensional Diffusion-type Problems 115
5.7 Additional Problems 117

6 Eigenfunction Expansion Method 119
6.1 Introduction 119
6.2 Scope 119
6.3 Application of the Method of Eigenfunction Expansion 120
6.4 Additional Problems 139

7 Advection–Diffusion–Reaction Problems 141
7.1 Introduction 141
7.2 A Note on Boundary Conditions for ADR-type Problems 141
7.3 BVPs of ADR 142
7.4 Application of Separation of Variables to ADR Problems 145
7.5 Solution of ADR Problems Using Transformation 163
7.6 Additional Problems 164

8 Steady-state Problems 167
8.1 Introduction 167
8.2 Solution of the Laplace Equation 168
8.3 Solution of the Poisson Equation 188
8.4 Laplace Equation in Cylindrical Coordinates 193
8.5 Additional Problems 196

9 Diffusion Problems Involving Bessel Functions 199
9.1 Introduction 199
9.2 Bessel Functions 199
9.3 Airy ODE 203
9.4 The General form of Diffusion (Conduction) in Cylindrical Polar Coordinates 204
9.5 Useful Formulas 238
9.6 Additional Problems 240

10 Diffusion and Conduction in a Sphere 243
10.1 Introduction 243
10.2 The General Form of Diffusion (Conduction) in Spherical Coordinates 243
10.3 Additional Problems 253

11 Wave Equation 255
11.1 Introduction 255
11.2 Wave Equation and Boundary Conditions 255
11.3 Application of the Separation of Variables to the Wave Equation 257
11.4 Additional Problems 267

12 Laplace Transform 269
12.1 Introduction 269
12.2 Preliminaries 269
12.3 Applications of the Laplace Transform 270
12.4 Inverse Laplace Transform Using the Residue Theorem 312
12.5 Numerical Laplace Inversion Methods 331
12.6 Additional Problems 334

13 Application of Duhamel's Theorem 337
13.1 Introduction 337
13.2 Duhamel's Theorem 337
13.3 Application to Partial Differential Equations with Time-dependent Nonhomogeneity 338
13.4 Application of Duhamel's Theorem With a Jump Boundary Condition 361
13.5 Notes 380
13.6 Additional Problems 380

14 Fourier Transform 381
14.1 Introduction 381
14.2 Fourier Transform for Infinite Domains 381
14.3 Fourier Transform for Semi-infinite Domains 386
14.4 Fourier Transform for Finite Domains 399
14.5 Additional Problems 413

15 Green's Function Method 415
15.1 Introduction 415
15.2 Solution of Nonhomogeneous Time-dependent Diffusion-type Problems 415
15.3 Application of Green's Function Technique 418
15.4 Additional Problems 457

16 Integral Transform Method 459
16.1 Introduction 459
16.2 Integral Transform 459
16.3 Application of the Integral Transform 465
16.4 Additional Problems 484

17 Similarity Method 485
17.1 Introduction 485
17.2 Determination of the Similarity Variable 485
17.3 Useful Scaling Transformation for Nonlinear PDEs 505
17.4 Additional Problems 508

References for Further Reading 508
Table of Laplace Transforms 509
Table of Useful Series Equivalents 513
Index 519