Abstract:
Advection-diffusion equation and its related analyt
ical solutions have gained wide
applications in different areas. Compared with nume
rical solutions, the analytical
solutions benefit from some advantages. As such, ma
ny analytical solutions have been
presented for the advection-diffusion equation. The
difference between these solutions is
mainly in the type of boundary conditions, e.g. tim
e patterns of the sources. Almost all the
existing analytical solutions to this equation invo
lve simple boundary conditions. Most
practical problems, however, involve complex bounda
ry conditions where it is very
difficult and sometimes impossible to find the corr
esponding analytical solutions. In this
research, first, an analytical solution of advectio
n-diffusion equation was initially derived
for a point source with a linear pulse time pattern
involving constant-parameters
condition (constant velocity and diffusion coeffici
ent). Hence, using the superposition
principle, the derived solution can be extended for
an arbitrary time pattern involving
several point sources. The given analytical solutio
n was verified using four hypothetical
test problems for a stream. Three of these test pro
blems have analytical solutions given by
previous researchers while the last one involves a
complicated case of several point
sources, which can only be numerically solved. The
results show that the proposed
analytical solution can provide an accurate estimat
ion of the concentration; hence it is
suitable for other such applications, as verifying
the transport codes. Moreover, it can be
applied in applications that involve optimization p
rocess where estimation of the solution
in a finite number of points (e.g. as an objective
function) is required. The limitations of
the proposed solution are that it is valid only for
constant-parameters condition, and is
not computationally efficient for problems involvin
g either a high temporal or a high
spatial resolution.
Keywords:
Advection-diffusion equation, Analytical solution,
Laplace transformation, Point
source, Solute transport.
