HW 1, MAE 200B.
Problem 2
by Nasser Abbasi
UCI, Winter 2006.
Question
Solution
This is a heat conduction PDE with 3 spatial dimensions and one in time. Hence
the PDE is
Use method of separation of variables to solve.
Assume
Where the function
depends on the spatial dimensions only and the function
depends on time only. Substitute eq(2) into (1) we get
or
Hence as we reasoned in the first problem, since the LHS of eq(3) depends on
time only and the RHS depends on the spatial coordinates only, then we way
that each side of eq. (3) must be equal to a constant, say
,
hence eq (3) can be written as
from which we obtain 2 equations to solve
and
The second of the above equations,
or
is called the Helmholtz equation. To solve, we again assume that
where
is a function of
only, and
is a function of
only and
is a function of
only. And now we substitite back in the Helmholtz equation and we get
,
and divide by
we obtain
In eq(4), We apply separation of variables again to obtain 3 ODE's each for
and
(4) can be written as
or
hence we see that the LHS depends on
only and the right hand side does not depend on
and both are equal to each others, then they must be both equal to the same
constant, say
,
hence we obtain that
and
From
we get
or
and
now since
is a constant, call it
hence the ODE is
whose solution is
where
for some positive constant
Now that we obtained solution for
we go back and obtain solutions for
and
.
From equation (5) we have
or
hence since the LHS depend on
only and the RHS depends on
only and they both equal, then they must be equal to some constant, say
hence we obtain that
and
Looking at the
equation, we obtain
or
which has solution
where
And from the equation
we obtain the solution as
or
whos solution is
where
So, to summarise we have now 4 ODE's to solve, which are