INDICATOR SUBSTANCES AND FLOW ANALYSIS 



627 



such as occur in compartment theory problems. 

 In effect, the Laplace transform converts differential 

 equations to algebraic equations which are, of course, 

 easier to solve. The conversion is based upon the 

 group theory concept that if a one-to-one correlation 

 can be established between the members of two 

 groups, and if the rules governing the relationships 

 among operations in the two groups are known, 

 difficulties in performing an operation on one group 

 may be avoided by performing the analogous opera- 

 tion on the second group. 



The most familiar example of this kind of trans- 

 formation is in the use of logarithms to convert the 

 problem of multiplication and division of the real 

 numbers to the problem of addition and subtraction 

 of their corresponding logarithms. 



As with the use of logarithms, the use of the Laplace 

 transform substitutes simple operations for more 

 difficult ones, but does not make it possible to do 

 anything which could not be done by other methods. 

 Also, as an analogy to the use of logarithms, it is 

 possible to use the Laplace transform by looking 

 up the appropriate conversions in a table, without 

 necessarily being able to derive the conversions or 

 even without understanding how the derivations are 

 obtained. 



An understanding of the principles, however, does 

 give one more confidence in the use of the tabulated 

 conversions and may help prevent their misapplica- 

 tion. With this philosophy in mind, the following 

 very brief discussion of the Laplace transform is 

 included. The present treatment largely follows that 

 of HoU et al. (34). 



Definition. The Laplace transform of a function of 

 time, F(t), is symbolized by: L{F{t)], and is defined 

 as the function f{s) given by 



M 



f 



c-''F{t) dt = L\F{t)} 



if f(s) exists. We can assume j^ to be a real number 

 and F(t) to be a continuous real-valued function of 

 the real variable / for / ^ o. It can be shown that 

 some of the transforms are 



F(t). The inverse transform is defined by the nota- 

 tion 



FU) = L-' lf(s) I 



which can be obtained from the equation defining 

 f(s) by handling the operator L as if it were an al- 

 gebraic variable. It can be proved that there is a 

 unique (one-to-one) correspondence between F(t) 

 and f(s) so that a table of Laplace tran.sforms may 

 be used in the opposite direction as a table of inverse 

 Laplace transforms. 



Example. The following example comparing the 

 solution of a differential equation 



dx 

 dr. 



+ 2.\' = e- 



for the initial condition A'(o) = i, with the method 

 of multiplication using logarithms, is a modification 

 of an example in reference (34). 



Original problem: 



3.1416 X 72314 



Look up logs in table 

 Simplified problem: 



497 '5 + -85922 



.•\dd 

 Solution of simplified problem: 



1-35637 



Look up antilog in table 

 Solution to original problem: 

 22.718 



d.Y 



~dt 



+ Q.Y = f-2', .\'(0) 



Look up transform 



sx{s) — I -f- 2Ar(j) = 



s + 2 



Solve for x(,s) 



x{s) = 



.+ 



(S +2y ■ X 4- 2 



Look up inverse transform 

 XiO = e-^'{t + i) 



i|il = -;Llt\ =-(s> o) 



The Laplace transform method may readily be 

 extended to a system of differential equations, and 

 for examples the reader is encouraged to consult 

 texts such as (34) on the subject. Sheppard (59) 

 and Zierler (74) use the Laplace transform in prob- 

 lems of compartment theory and indicator dilution 

 curves. 



L\ri\ = 



(:r > o),-Z.|cj = cL{l\ = - (s > o) 



where c is any constant and L\e^'\ = i/{s — /3), 

 etc. The inverse problem arises when a function of 

 s is given and it is required to find the corresponding 



Notation 



The symbols to be used in the following develop- 

 ment of the analysis of compartmented systems are 

 defined when they first appear. The most frequently 

 used ones are collected here for reference. 



