TIME-DEPENDENT AND NON-LINEAR PROBLEMS 167
formulation (equation (6.54)) could then be used with these modifi-
cations and the inversions over the parameter κ performed in the
usual way.
6.8 TIME INTEGRATION USING TIME STEPPING
If the time dependence in the governing equations cannot be removed
using a transformation we have to integrate in time a hyperbolic or
parabolic system.
The usual way of solving the problem is to use a time stepping type
technique, where the problem is solved at each time interval. The
evolution of the time-dependent terms is generally carried out in some
finite difference way. Values at the interior of the region at any time
may be calculated using the normal boundary element method.
Consider an equation of the type,
&(u) = du/dt
(6.73)
on the regions defined in Figure 6.4 (for simplicity we assume that
u
is
a potential type function, although the same arguments apply to the
u
=
u
t
Γι "
=
"
Δ
ί
Figure 6.4 Time-dependent problem using time stepping technique
168 TIME-DEPENDENT AND NON-LINEAR PROBLEMS
equations of
elasticity).
To solve the problem we require some initial
values given over the whole of
Ω
at time t = 0, denoted by
Ω(0).
Say
u(x,0) = w
o
(6.74)
We may also have time-dependent boundary conditions, i.e.
u =
Ti
0
onii at time 0 ,, _
c
.
u = u
t
on
Γ
l
at time t
and
du/dn = q
0
on Γ
2
at time 0
(f
.
η
„
du/dn = q
t
on Γ
2
at time t
If
we
wish to find u at a later time At (we can call this value of
u,
u
At
),
where At is sufficiently small, we can write,
*(„
Α
)
Ä
ΪΪ4Ζ*»
(6
.77)
or
dii?(ii
df
)-ii
Jf
= -II
0
(6.78)
If
we
can find the fundamental solution to this equation, i.e.
u%,
then
we can use the boundary element method to solve for u at time At.
Example 6.3
Consider the equation of heat conduction in two dimensions, i.e.
V
2
u = du/et (a)
with
u = u
0
given at t = 0 in Ω (b)
u = u
t
on Γ
ι
at time t (c)
du/dn = q
t
on Γ
2
at time t (d)
Then equation (a) can be written as,
zlf At
*
2
<«
A
>-=£--2
(e)
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