This Second Edition of the go-to reference combines the classical analysis and modern applications of applied mathematics for chemical engineers. The book introduces traditional techniques for solving ordinary differential equations (ODEs), adding new material on approximate solution methods such as perturbation techniques and elementary numerical solutions. It also includes analytical methods to deal with important classes of finite-difference equations. The last half discusses numerical solution techniques and partial differential equations (PDEs). The reader will then be equipped to apply mathematics in the formulation of problems in chemical engineering. Like the first edition, there are many examples provided as homework and worked examples.

Preface to the Second Edition xi

**Part I. 1**

**1. Formulation of Physicochemical Problems 3**

1.1 Introduction 3

1.2 Illustration of the Formulation Process (Cooling of Fluids) 3

1.3 Combining Rate and Equilibrium Concepts (Packed Bed Adsorber) 7

1.4 Boundary Conditions and Sign Conventions 8

1.5 Models with Many Variables: Vectors and Matrices 10

1.6 Matrix Definition 10

1.7 Types of Matrices 11

1.8 Matrix Algebra 12

1.9 Useful Row Operations 13

1.10 Direct Elimination Methods 14

1.11 Iterative Methods 18

1.12 Summary of the Model Building Process 19

1.13 Model Hierarchy and its Importance in Analysis 19

Problems 25

**2. Solution Techniques for Models Yielding Ordinary Differential Equations 31**

2.1 Geometric Basis and Functionality 31

2.2 Classification of ODE 32

2.3 First-Order Equations 32

2.4 Solution Methods for Second-Order Nonlinear Equations 37

2.5 Linear Equations of Higher Order 42

2.6 Coupled Simultaneous ODE 55

2.7 Eigenproblems 59

2.8 Coupled Linear Differential Equations 59

2.9 Summary of Solution Methods for ODE 60

Problems 60

References 73

**3. Series Solution Methods and Special Functions 75**

3.1 Introduction to Series Methods 75

3.2 Properties of Infinite Series 76

3.3 Method of Frobenius 77

3.4 Summary of the Frobenius Method 85

3.5 Special Functions 86

Problems 93

References 95

**4. Integral Functions 97**

4.1 Introduction 97

4.2 The Error Function 97

4.3 The Gamma and Beta Functions 98

4.4 The Elliptic Integrals 99

4.5 The Exponential and Trigonometric Integrals 101

Problems 102

References 104

**5. Staged-Process Models: The Calculus of Finite Differences 105**

5.1 Introduction 105

5.2 Solution Methods for Linear Finite Difference Equations 106

5.3 Particular Solution Methods 109

5.4 Nonlinear Equations (Riccati Equations) 111

Problems 112

References 115

**6. Approximate Solution Methods for ODE: Perturbation Methods 117**

6.1 Perturbation Methods 117

6.2 The Basic Concepts 120

6.3 The Method of Matched Asymptotic Expansion 122

6.4 Matched Asymptotic Expansions for Coupled Equations 125

Problems 128

References 136

**Part II. 137**

**7. Numerical Solution Methods (Initial Value Problems) 139**

7.1 Introduction 139

7.2 Type of Method 142

7.3 Stability 142

7.4 Stiffness 147

7.5 Interpolation and Quadrature 149

7.6 Explicit Integration Methods 150

7.7 Implicit Integration Methods 152

7.8 Predictor-Corrector Methods and Runge-Kutta Methods 152

7.9 Runge-Kutta Methods 153

7.10 Extrapolation 155

7.11 Step Size Control 155

7.12 Higher Order Integration Methods 156

Problems 156

References 159

**8. Approximate Methods for Boundary Value Problems: Weighted Residuals 161**

8.1 The Method of Weighted Residuals 161

8.2 Jacobi Polynomials 179

8.3 Lagrange Interpolation Polynomials 172

8.4 Orthogonal Collocation Method 172

8.5 Linear Boundary Value Problem: Dirichlet Boundary Condition 175

8.6 Linear Boundary Value Problem: Robin Boundary Condition 177

8.7 Nonlinear Boundary Value Problem: Dirichlet Boundary Condition 179

8.8 One-Point Collocation 181

8.9 Summary of Collocation Methods 182

8.10 Concluding Remarks 183

Problems 184

References 192

**9. Introduction to Complex Variables and Laplace Transforms 193**

9.1 Introduction 193

9.2 Elements of Complex Variables 193

9.3 Elementary Functions of Complex Variables 194

9.4 Multivalued Functions 195

9.5 Continuity Properties for Complex Variables: Analyticity 196

9.6 Integration: Cauchy’s Theorem 198

9.7 Cauchy’s Theory of Residues 201

9.8 Inversion of Laplace Transforms by Contour Integration 202

9.9 Laplace Transformations: Building Blocks 204

9.10 Practical Inversion Methods 209

9.11 Applications of Laplace Transforms for Solutions of ODE 211

9.12 Inversion Theory for Multivalued Functions: the Second Bromwich Path 215

9.13 Numerical Inversion Techniques 218

Problems 221

References 225

**10. Solution Techniques for Models Producing PDEs 227**

10.1 Introduction 227

10.2 Particular Solutions for PDES 231

10.3 Combination of Variables Method 233

10.4 Separation of Variables Method 238

10.5 Orthogonal Functions and Sturm-Liouville Conditions 241

10.6 Inhomogeneous Equations 245

10.7 Applications of Laplace Transforms for Solutions of PDES 248

Problems 254

References 271

**11. Transform Methods for Linear PDEs 273**

11.1 Introduction 273

11.2 Transforms in Finite Domain: Sturm-Liouville Transforms 273

11.3 Generalized Sturm-Liouville Integral Transforms 289

Problems 297

References 301

**12. Approximate and Numerical Solution Methods for PDEs 303**

12.1 Polynomial Approximation 303

12.2 Singular Perturbation 310

12.3 Finite Difference 315

12.4 Orthogonal Collocation for Solving PDEs 324

12.5 Orthogonal Collocation on Finite Elements 330

Problems 335

References 342

Appendix A. Review of Methods for Nonlinear Algebraic Equations 343

Appendix B. Derivation of the Fourier-Mellin Inversion Theorem 351

Appendix C. Table of Laplace Transforms 357

Appendix D. Numerical Integration 363

References 372

Appendix E. Nomenclature 373

Postface 377

Index 379