Separation of Variables for Solving PDEs

[joriarty]

See attached image for the question and my working. Hopefully you can read it OK, I had to resize it to fit to the allowed dimensions.

I'm unsure how to proceed or if I have done something wrong previously - the initial and boundary conditions are tripping me up. The boundary conditions in red are given, I have written down the initial conditions (in the blue writing) by interpreting the question.

In past experience I should be able to end up with a Fourier series to solve, but I'm not sure how to get there! Could I please have a hint? Thanks :)

 

[AlephZero]

You got to

[tex]u(L,0) = C_1C_3\sin(\lambda L/ 2 \sqrt k) = 0[/tex]

Clearly [itex]C_1 = 0[/itex] or [itex]C_3 = 0[/itex] are not interesting solutions, so

[tex]\sin(\lambda L/ 2 \sqrt k) = 0[/tex]

[tex]\lambda L/ 2 \sqrt k = n\pi[/tex]

for n = 1, 2, ...

The question says the temperature profile is triangular at time 0, so you know the temperature for all values of x when t = 0, not just for x = 0, L/2, and L.

That's where the Fourier series solution comes from.

 

[joriarty]

Thanks for your reply. I'm not familiar with the term "triangular" - I presume this simply means that the temperature gradient between 0 < x < L/2 is uniform? Which would make it easy enough to get the temperature for any x at t = 0.

Also, I agree that having C1 or C3 equal to zero isn't very interesting, however I don't see how C1C3 can be anything but zero, as u(0,0) = C1C3 = 0.

 

[phyzguy]

u(0,0) isn't equal to C1*C3, it's equal to C1*C3*sin(0). Since sin(0)=0, you can't conclude anything about C1 and C3 from this boundary condition.

 

[joriarty]

Shouldn't that be u(0,0) = C1*C2*sin(0) + C1*C3 = 0 ? C2 and C3 could be anything but I think this restricts C1 to being zero.

 

[phyzguy]

Let's try again. I had read AlephZero's post, which is different than your derivation. First of all, you have defined too many constants - there are really only two independent constants. You should absorb C1 into C2 and C3 by defining two new constants C2' = C1*C2 and C3' = C1*C3. Then you only have two constants to solve for. But for now let's stick with the constants as you have defined them. As you say, the boundary condition at (0,0) tells you that C1*C3=0. If C1=0, then the whole solution is zero everywhere, which isn't the solution that you are looking for, so we discard this option. Any linear system always has this trivial solution (zero everywhere), but it isn't of any value. So we have C3 =0. Then the other condition at (L,0) tells us that C1*C2*Sin(Lambda*L/2*Sqrt(K))=0-C2'*Sin(Lambda*L/(2*Sqrt(K))), so as AlephZero says, you conclude that Lambda*L/(2*Sqrt(K)) = n*Pi. Again you discard the option that C2'=0 since it is just the trivial solution (zero everywhere). Does this make sense?

 

[joriarty]

Got it! Thanks :)