Compartment Graphs and Linear Systems - Environmental Modeling - page 323

(Jordan and Smith 1977 ; Walter and Contreras 1999 ). The expressions with
the exponential function need further explanation, as the arguments Ct and - Cs
are matrices. The exponential value of a square matrix is defined by the infinite
series:

k C k t k
k! þ :::

C 2 t 2
2

exp ð CtÞ¼I Ct þ

! þ ::: þð

1 Þ (18.9)

which is the analogue to the infinite series of the exponential function for single
numbers. The powers of C in the higher order terms are results of matrix multipli-
cation. The result of the operation is also a matrix. The so computed matrix is
different from the element wise evaluation of the infinite series or the exponential
function. In MATLAB can be programmed easily, as the

exponential function of a matrix is available. In m-code it is written using the matrix
exponential call expm (compare Sidebar 18.2):

the expression exp ð CtÞ

®

is reserved for the elementwise evaluation of
the exponential function. The vector c in formula ( 18.8 ) contains unknown
parameters, which have to be determined from boundary or initial conditions.
The computation of the general solution ( 18.8 ) is quite complex. It includes the
evaluation of an integral, which usually needs special analysis. We keep it simple
by assuming constant f i . For constant f i

The exp command in MATLAB

®

the solution of ( 18.6 ) can be written as:

C 1 f i

c ðtÞ¼ exp ð CtÞc (18.10)

The result can be derived easiest by the application of the integration formula for
the exponential:

ð

t

C 1 exp exp ð CsÞds ¼ ð CtÞ

0

which is common knowledge for single values, but also holds for matrices. C 1
is the inverse matrix of C . Thus, the formula ( 18.10 ) is applicable only for
regular matrices (for which an inverse exists). The first term in ( 18.10 ) is the
general solution of the problem @c=@t ¼ Cc , which mathematicians call
homogeneous . The second term in ( 18.10 ) is a particular solution of the differential
( 18.6 ), representing the equilibrium solution with

@ c =@t ¼ 0, which can be verified

easily.