Information Technology Reference
In-Depth Information
The necessary condition
G
0
.a/ D 0
for a minimum leads to the equation
t D5
X
.2:555 10
9
e
at
d
t
/2:555 10
9
te
at
D 0;
(9.20)
t D1
which must be solved to determine an optimal value for
a
, see Exercise
9.1
.
In this case, the standard output least squares form is
"
1
2
#
t D
X
.r.t I a/ d
t
/
2
min
a
t D1
subject to the constraints
r
0
.t / D ar.t/
for
t>0;
r.0/ D 2:555 10
9
:
9.2
The Backward Diffusion Equation
The estimation of constant parameters in differential equations often leads to one or
more algebraic equations that must be solved numerically. What about non-constant
parameters? As you might have guessed, this is a far more subtle issue and can
involve equations that are very difficult to solve. This topic, in its full complexity,
is certainly far beyond the scope of this topic. Nevertheless, in order to provide
the reader with some basic understanding of the matter, we will consider a classic
example, commonly referred to as the
backward diffusion equation
, in some detail.
Assume that a substance in an industrial process must have a prescribed temper-
ature distribution, say, g(x), at time
T
in the future. Furthermore, the substance must
be introduced/implanted into the process at time
t D 0
. (This could typically be the
case in various molding processes or in steel casting). What should the temperature
distribution
f.x/
at time
t D 0
be in order to ensure that the temperature is
g.x/
at
time
T
?
For the sake of simplicity, let us consider a medium with a constant diffusion
coefficient
k.x/ D 1
for all
x
, occupying the unit interval. In mathematical terms,
we may formulate our challenge as follows: Determine the initial condition
f D
f.x/
such that the solution
u
D
u
.x; t I f/
of
u
t
D
u
xx
for
x 2 .0; 1/; t > 0;
(9.21)
u
.0; t / D
u
.1; t / D 0
for
t>0;
(9.22)
u
.x; 0/ D f.x/
for
x 2 .0; 1/;
(9.23)
