practise problems for solving first order PDEs using method of characteristics
by Nasser Abbasi
Solve
Solution
Seek solution where
constant,hence
Compare to (1) we see that
or
and
,
but since
then
,
and this has solution
but
hence
Now at
,
the solution is
but this solution is valid any where on this characteristic line and not just
when
.
hence
But
from (2), hence
Solve
Solution
Seek solution where
constant,hence
Compare to (1) we see that
or
and
,
but since
then
,
and this has solution
but
hence
Hence
At
,
Now we are told the solution at
is
,
or
but
this solution is valid any where on this characteristic line and not just when
.
hence
Replace the value of
obtained in (2) we obtain
Hence
Solve
Solution
Seek solution where
constant,hence
Compare to (1) we see that
or
and
,
but since
then
hence we need to solve
but
hence
Hence
At
,
Now we are told the solution at
is
,
or
but
this solution is valid any where on this characteristic line and not just when
.
hence
Replace the value of
obtained in (2) we obtain
Hence
To avoid a solution
which blow up, we need
,
hence
for
example,
and
will not give a valid solution. so all region in
plane in which
is not a valid region to apply this solution at.
The solution breaks down along this line in the
plane
To see it in 3D, here is the
solution that includes the above line, and we see that the solution below the
line and the above the line are not continuous across it. ( I think there is a
name to this phenomena that I remember reading about sometime, may be related
to shockwaves but do not now know how this would happen in reality)
Solution
Nonhomogeneous pde first order.
(TO DO)