Page 18 - Modelling in Transport Phenomena A Conceptual Approach

CONTENTS xv

A Mat hematical Preliminaries 491
A.l THE CYLINDRICAL AND SPHERICAL
COORDINATE SYSTEMS ....................... 491
A.2 MEAN VALUE THEOREM ....................... 493
A.3 SLOPES ON LOG-LOG AND SEMI-LOG
GRAPH PAPERS ............................ 494
A.4 LEIBNITZ’S RULE FOR DIFFERENTIATION OF INTEGRALS . 495
A.5 NUMERICAL DIFFERENTIATION .................. 495
A.5.1 Douglass-Avakian Method .................... 496
A.5.2 Whitaker-Pigford Method .................... 496
A.6 REGRESSION AND CORRELATION ................. 500
A.6.1 Simple Linear Regression .................... 500
A.6.2 Sum of Squared Deviations ................... 500
A.6.3 The Method of Least Squares .................. 501
A.6.4 Correlation Coefficient ...................... 505
A.7 THE ROOT OF AN EQUATION ................... 506
A.7.1 Roots of a Polynomial ...................... 506
A.7.2 Numerical Methods ....................... 510
A.8 METHODS OF INTEGRATION .................... 512
A.8.1 Mean Value Theorem ...................... 512
A.8.2 Graphical Integration ...................... 514
A.8.3 Numerical Integration or Quadrature .............. 514
A.8.4 Numerical Integration When the Integrand is a
Continuous Function ....................... 518
A.9 MATRICES ................................ 522
A.9.1 Fundamental Algebraic Operations ............... 523
A.9.2 Determinants ........................... 525
A.9.3 Types of Matrices ........................ 527
A.9.4 Solution of Simultaneous Algebraic Equations ......... 528

B Solutions of Differential Equations 531
B.l FIRST-ORDER EQUATIONS WITH EXACT SOLUTIONS .... 531
B.l.l Separable Equations ....................... 532
B.1.2 Exact Equations ......................... 532
B . 1.3 Homogeneous Equations ..................... 534
B.1.4 Linear Equations ......................... 535
B.1.5 Bernoulli Equations ....................... 536
B.2 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS .... 537
B.2.1 Special Case of a Second-Order Equation ........... 538
B.2.2 Solution of a Non-Homogenous Differential
Equation ............................. 538
B.2.3 Bessel’s Equation ......................... 539
B.2.4 Numerical Solution of Initial Value Problems ......... 544
B.2.5 Solution of Simultaneous Differential Equations ........ 548