Transport Solutions - Environmental Modeling - page 75

The exponential function is reached simply by exp.
To mention is also the error function and its complement, the complemen-
tary error function, defined by

ð

x

2
p

ðt 2

p

erf ðxÞ¼ exp Þdt

0

1

2
p ðt 2

erfcðxÞ¼ 1 erf ðxÞ¼ p exp Þdt

x

erfcxðxÞ¼e x 2

erfcðxÞ (4.2)

which are called using MATLAB by erf , erfc and erfcx . There are also

the inverse error functions, which are reached by the keywords er nv and
erfcinv .
Another integral of the exponential function, the so-called exponential
integral or well function , is defined as

®

1

exp ðtÞ

EðxÞ¼ dt

(4.3)

t

x

and is reached by expint. Other special functions, which one may meet, are
implemented in MATLAB ® , like the Bessel functions, and the modified
Bessel functions, the Airy functions, the Beta-function, the incomplete
Beta-function, the Gamma-Function, the incomplete Gamma-function:

e.g. when the component is not present at the initial simulation time and is
introduced into the system by inflow of concentration c in at position x ¼

0. The
second boundary condition concerns the behavior of the system at a point that is
infinitely far away from the inflow boundary, where the concentration remains
constant. For the application of the formula in practice follows that the outflow
boundary is far away during all time instants of interest. When the front approaches
the outflow boundary the solution of Ogata-Banks is not valid anymore.
The generalisation of the formula of Ogata & Banks for a non-zero initial
condition cðx ¼ 0 ; tÞ¼c 0 is given by:

þ

D x erfc

c in c 0
2

x vt
2

v x þ vt

2

cðx; tÞ¼c 0 þ erfc p exp p (4.4)

Dt Dt