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Chapter 1 THE MATHEMATICAL STATEMENT OF THE PROBLEM 1

1.1 INTRODUCTION 1

1.2 REPRESENTATION OF THE PROBLEM 1

1.3 SOLVENT EXTRACTION IN TWO STAGES 3

1.4 SOLVENT EXTRACTION IN N STAGES 4

1.5 SIMPLE WATER STILL WITH PREHEATED FEED 6

1.6 UNSTEADY STATE OPERATION 8

1.7 SALT ACCUMULATION IN A STIRRED TANK 11

1.8 RADIAL HEAT TRANSFER THROUGH A CYLINDRICAL CONDUCTOR 14

1.9 HEATING A CLOSED KETTLE 16

1.10 DEPENDENT AND INDEPENDENT VARIABLES,PARAMETERS 17

1.11 BOUNDARY CONDITIONS 18

1.12 SIGN CONVENTIONS 19

1.13 SUMMARY OF THE METHOD OF FORMULATION 21

Chapter 2 ORDINARY DIFFERENTIAL EQUATIONS 23

2.1 INTRODUCTION 23

2.2 ORDER AND DEGREE 23

2.3 FIRST ORDER DIFFERENTIAL EQUATIONS 24

2.4 SECOND ORDER DIFFERENTIAL EQUATIONS 33

2.5 LINEAR DIFFERENTIAL EQUATIONS 41

2.6 SIMULTANEOUS DIFFERENTIAL EQUATIONS 66

2.7 CONCLUSIONS 72

Chapter 3 SOLUTION BY SERIES 74

3.1 INTRODUCTION 74

3.2 INFINITE SERIES 74

3.3 POWER SERIES 79

3.4 SIMPLE SERIES SOLUTIONS 86

3.5 METHOD OF FROBENIUS 90

3.6 BESSEL'S EQUATION 106

3.7 PROPERTIES OF BESSEL FUNCTIONS 113

Chapter 4 COMPLEX ALGEBRA 117

4.1 INTRODUCTION 117

4.2 THE COMPLEX NUMBER 117

4.3 THE ARGAND DIAGRAM 118

4.4 PRINCIPAL VALUES 119

4.5 ALGEBRAIC OPERATIONS ON THE ARGAND DIAGRAM 120

4.6 CONJUGATE NUMBERS 123

4.7 DE MOIVRE'S THEOREM 124

4.8 THE nTH ROOTS OF UNITY 125

4.9 COMPLEX NUMBER SERIES 126

4.10 TRIGONOMETRICAL—EXPONENTIAL IDENTITIES 128

4.11 THE COMPLEX VARIABLE 128

4.12 DERIVATIVES OF A COMPLEX VARIABLE 130

4.13 ANALYTIC FUNCTIONS 131

4.14 SINGULARITIES 132

4.15 INTEGRATION OF FUNCTIONS OF COMPLEX VARIABLES,AND CAUCHY'S THEOREM 137

4.16 LAURENT'S EXPANSION AND THE THEORY OF RESIDUES 142

Chapter 5 FUNCTIONS AND DEFINITE INTEGRALS 149

5.1 INTRODUCTION 149

5.2 THE ERROR FUNCTION 149

5.3 THE GAMMA FUNCTION 151

5.4 THE BETA FUNCTION 154

5.5 OTHER TABULATED FUNCTIONS WHICH ARE DEFINED BY INTEGRALS 157

5.6 EVALUATION OF DEFINITE INTEGRALS 159

Chapter 6 THE LAPLACE TRANSFORMATION 163

6.1 INTRODUCTION 163

6.2 THE LAPLACE TRANSFORM 163

6.3 THE INVERSE TRANSFORMATION 167

6.4 PROPERTIES OF THE LAPLACE TRANSFORMATION 170

6.5 THE STEP FUNCTIONS 174

6.6 CONVOLUTION 179

6.7 FURTHER ELEMENTARY METHODS OF INVERSION 180

6.8 INVERSION OF THE LAPLACE TRANSFORM BY CONTOUR INTEGRATION 182

6.9 APPLICATION OF THE LAPLACE TRANSFORM TO AUTOMATIC CONTROL THEORY 188

Chapter 7 VECTOR ANALYSIS 199

7.1 INTRODUCTION 199

7.2 TENSORS 200

7.3 ADDITION AND SUBTRACTION OF VECTORS 203

7.4 MULTIPLICATION OF VECTORS 210

7.5 DIFFERENTIATION OF VECTORS 216

7.6 HAMILTON'S OPERATOR,？ 218

7.7 INTEGRATION OF VECTORS AND SCALARS 222

7.8 STANDARD IDENTITIES 227

7.9 CURVILINEAR COORDINATE SYSTEMS 228

7.10 THE EQUATIONS OF FLUID FLOW 231

7.11 TRANSPORT OF HEAT,MASS,AND MOMENTUM 236

Chapter 8 PARTIAL DIFFERENTIATION AND PARTIAL DIFFERENTIAL EQUATIONS 238

8.1 INTRODUCTION 238

8.2 INTERPRETATION OF PARTIAL DERIVATIVES 239

8.3 FORMULATING PARTIAL DIFFERENTIAL EQUATIONS 245

8.4 BOUNDARY CONDITIONS 252

8.5 PARTICULAR SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 259

8.6 ORTHOGONAL FUNCTIONS 269

8.7 METHOD OF SEPARATION OF VARIABLES 272

8.8 THE LAPLACE TRANSFORM METHOD 290

8.9 OTHER TRANSFORMS 302

8.10 CONCLUSIONS 306

Chapter 9 FINITE DIFFERENCES 307

9.1 INTRODUCTION 307

9.2 THE DIFFERENCE OPERATOR,△ 307

9.3 OTHER DIFFERENCE OPERATORS 311

9.4 INTERPOLATION 315

9.5 FINITE DIFFERENCE EQUATIONS 321

9.6 LINEAR FINITE DIFFERENCE EQUATIONS 322

9.7 NON-LINEAR FINITE DIFFERENCE EQUATIONS 331

9.8 DIFFERENTIAL-DIFFERENCE EQUATIONS 338

Chapter 10 TREATMENT OF EXPERIMENTAL RESULTS 349

10.1 INTRODUCTION 349

10.2 GRAPH PAPER 349

10.3 THEORETICAL PROPERTIES 354

10.4 CONTOUR PLOTS 355

10.5 PROPAGATION OF ERRORS 356

10.6 CURVE FITTING 360

10.7 NUMERICAL INTEGRATION 369

Chapter 11 NUMERICAL METHODS 380

11.1 INTRODUCTION 380

11.2 FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS 380

11.3 HIGHER ORDER DIFFERENTIAL EQUATIONS(INITIAL VALUE TYPE) 385

11.4 HIGHER ORDER DIFFERENTIAL EQUATIONS(BOUNDARY VALUE TYPE) 388

11.5 ALGEBRAIC EQUATIONS 397

11.6 DIFFERENCE-DIFFERENTIAL EQUATIONS 406

11.7 PARTIAL DIFFERENTIAL EQUATIONS 409

Chapter 12 MATRICES 437

12.1 INTRODUCTION 437

12.2 THE MATRIX 438

12.3 MATRIX ALGEBRA 439

12.4 DETERMINANTS OF SQUARE MATRICES AND MATRIX PRODUCTS 443

12.5 THE TRANSPOSE OF A MATRIX 443

12.6 ADJOINT MATRICES 444

12.7 RECIPROCAL OF A SQUARE MATRIX 444

12.8 THE RANK AND DEGENERACY OF A MATRIX 446

12.9 THE SUB-MATRIX 448

12.10 SOLUTION OF LINEAR ALGEBRAIC EQUATIONS 448

12.11 MATRIX SERIES 449

12.12 DIFFERENTIATION AND INTEGRATION OF MATRICES 451

12.13 LAMBDA-MATRICES 452

12.14 THE CHARACTERISTIC EQUATION 454

12.15 SYLVESTER'S THEOREM 457

12.16 TRANSFORMATION OF MATRICES 459

12.17 QUADRATIC FORM 461

12.18 APPLICATION TO THE SOLUTION OF DIFFERENTIAL EQUATIONS 463

12.19 SOLUTIONS OF SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS 465

12.20 CONCLUSIONS 472

Chapter 13 OPTIMIZATION 473

13.1 INTRODUCTION 473

13.2 TYPES OF OPTIMIZATION 474

13.3 ANALYTICAL PROCEDURES 475

13.4 THE METHOD OF STEEPEST ASCENT 483

13.5 THE SEQUENTIAL SIMPLEX METHOD 485

13.6 DYNAMIC PROGRAMMING 486

Chapter 14 COMPUTERS 492

14.1 INTRODUCTION 492

14.2 PASSIVE ANALOGUE COMPUTERS 493

14.3 ACTIVE ANALOGUE COMPUTERS 496

14.4 DIGITAL COMPUTERS 505

14.5 COMPARISON OF THE USES OF ANALOGUE AND DIGITAL COMPUTERS 509

PROBLEMS 511

APPENDIX 532

SUBJECT INDEX 543