Need help! Studing DiffEQ and have a question!

[shikhan]

If given the following Head Conduction problem, what would be your answer?

Given:

u[t] = (1/5) u[xx]
0 < x < 10
t > 0
U(0,t) = u(10, t) = 0
u(x, 0) = 4x

where u[t] is the derivitive of u with respect to t and u[xx] is thesecond derivitive of u with respect to x


I have two possible answers:

One

u(x,t) = sum( b(n) * exp(- (pi^2) (n^2) (1/5) t / 100) * sin (n*pi*x / 10) , from n =1 to infinity)

where b(n) = (1/5) integral( 4x sin(n*pi*x/10), of x from 0 10)

I'm not sure how to solve b(n) in this case (my ti can, but thats no good for me to study by)

my second answer

u(x,t) = sum( (160/ (n * pi)) exp(- (pi^2) (n^2) (1/5) t / 100) * sin( n*pi*x/10), for n = 1, n odd)

Which one do you agree with?

[edit] for spelling

 

[RaynorWolfcastle]

I'm not going to do your homework for you but solve the heat equation in the following manner:

- take the Laplace transform w.r.t. time
- solve ODE for t in the s-domain
- take the Laplace inverse by evaluating a contour integral in the complex plane
- apply IC's and BV's as needed

 

[Legendary]

I can't wait until this crap actually makes sense to me one day. :Q

 

[Nocturnal]

You should work on your grammar. It's THE not TEH.

 

[shikhan]

Originally posted by: RaynorWolfcastle
I'm not going to do your homework for you but solve the heat equation in the following manner:

- take the Laplace transform w.r.t. time
- solve ODE for t in the s-domain
- take the Laplace inverse by evaluating a contour integral in the complex plane
- apply IC's and BVP's as needed

Its not homework (it really isn't ) but it's a problem out of the book i've come across in revewing for my final tomorrow. I know of the normal ways to PDE, but the book has mentioned a specific shortcut for Head Conduction with two cases, one where U(x, 0) = f(x) and one where U(x, 0) = u0

I'm trying to figure out which this falls under since i've gotten the impression that f(x) needs to be sinasodial(no clue how to spell that!)

 

[RaynorWolfcastle]

I don't feel like digging up my DE book to find the heat equation but IIRC the nullspace solution has lin. indep. sinh(x) and cosh(x) portions to the solution.

In the s-domain you should have (again IIRC) something of the form F(x,s) = A(s)sinh(kx) + B(s)cosh(jx) where k and j are constants that you find easily. By taking the Laplace of your IC at X= 0 you get

F(0,s) = A(s) + 0 = 0, A(s) = 0
F(x,s) = B(s)sinh(jx) which simplifies your answer.

 

[shikhan]

Originally posted by: RaynorWolfcastle
I don't feel like digging up my DE book to find the heat equation but IIRC the nullspace solution has lin. indep. sinh(x) and cosh(x) portions to the solution.

In the s-domain you should have (again IIRC) something of the form F(x,s) = A(s)sinh(kx) + B(s)cosh(jx) where k and j are constants that you find easily. By taking the Laplace of your IC at X= 0 you get

F(0,s) = A(s) + 0 = 0, A(s) = 0
F(x,s) = B(s)sinh(jx) which simplifies your answer.

We dont know lapace :/

Its in the next section, but not this one... and frankly, i've heard bad things about laplace

So right now, i'm pounding through it using the 5 step general way to solve pde's

bleh, but i guess i need the practice