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