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