Question about heat conduction equation.

[DrKareem]

I know how to solve a 1-D heat conduction equation when both ends are kept at 0 temperature(using separation of variables and Fourier series.). In the notes, the prof asked us to solve for one end kept at zero and the other insulated (at x=L) and referred to a non-existent chapter in our textbook for the solution. Can someone give me a few pointers on how to proceed? I don't know what to equate U(L,t) to, and that kinda gets u stuck...

 

[cristo]

Well, firstly, post the equation you're trying to solve!

 

[DrKareem]

I'm sorry, I (falsely) assumed everyone would recognize the equation i had in mind. The equation is

$$\frac{\partial^2 u}{\partial t} = a^2 \frac{\partial^2 u}{\partial x^2}$$

With U(0,t)=0. U(L,t)= ??

 

[cristo]

Firstly, that equation has no function in it! Secondly, are you sure that's the correct expression? I thought the heat equation looked like $$\frac{\partial U}{\partial t}=a^2\frac{\partial ^2U}{\partial x^2}$$ in one dimension?

Anyway, I think the condition that says that at x=L the rod is insulated, means that the flux at x=L is zero; i.e.$$\left. \frac{\partial U}{\partial x}\right|_{x=L}=0$$

 

[DrKareem]

yes, i made those horrible typos sorry again (been up very early having nothing to do but solve). I will try solving it, i think it should be easy. Thanks for the input.

 

[HallsofIvy]

Cristo is right. Saying that one end, x= 0, is held at 0 means, of course, U(0,t)= 0. Saying that the other end, x=L, means that
$$\frac{\partial U}{\partial x}(L, u)= 0$$

 

[DrKareem]

Thanks HallsofIvy. I already solved it, and it was easy as expected (i just didn't know the mathematical interpretation of the word 'insulated' in that context).