1. ## PDE double checking, unsure of answer

Hi,

I've done this question but I'm unsure if I used the boundary conditions right. The question is:

Find u(x,t) given u_xx=0 0<x<1, t>0 with boundary conditions u(0,t) = sin(t), u(1,t) = 0.5 t>0

To solve this question I found u = cx + f(t) where f(t) is any differentiable equation of t by integrating twice. I then used the boundary conditions to find c in terms of t. Is this correct?

Thank you!

2. What equation are you solving?

3. Originally Posted by ThePerfectHacker
What equation are you solving?
oh gosh sorry! the equation is u_xx=0

4. Originally Posted by ThePerfectHacker
If u_xx = 0 then u_x = k + f(y) for any function f(y). That means u = kx + f(t) for any function f(t).

The boundary condition say u(0,t)= sin t and u(1,t) = .5, thus, k*0 + f(t) = sin t and so f(t) = sin t. Thus, u(x,t) = kx + sin t using the second boundary condition we have u(1,t) = k + sin t = .5 for t>0, which seems impossible no constant k will make this possible because t varies.
Yea, thats pretty much what I found, I ended up with a constant in terms of t, which is why I was unsure if I was using the boundary conditions correct. (First time solving this type of equation and I don't have any examples to follow from) But thank you anyways!

5. I thought that maybe I was sepposed to use the boundary conditions to find a value of t...meaning that in f(t) t had to be something certain...or am I just going way off track?

6. Originally Posted by ThePerfectHacker
If u_xx = 0 then u_x = k + f(y) for any function f(y). That means u = kx + f(t) for any function f(t).

The boundary condition say u(0,t)= sin t and u(1,t) = .5, thus, k*0 + f(t) = sin t and so f(t) = sin t. Thus, u(x,t) = kx + sin t using the second boundary condition we have u(1,t) = k + sin t = .5 for t>0, which seems impossible no constant k will make this possible because t varies.
$\displaystyle u_{xx} = 0$.

Therefore $\displaystyle u_x = f(t)$.

Therefore $\displaystyle u(x, t) = x f(t) + g(t)$.

B.C. $\displaystyle u(0,t)= \sin t: \,$ $\displaystyle \sin t = g(t).$

Therefore $\displaystyle u(x, t) = x f(t) + \sin t$.

B.C. $\displaystyle u(1,t) = 0.5: \,$ $\displaystyle 0.5 = f(t) + \sin t \therefore f(t) = 0.5 - \sin t.$

Therefore $\displaystyle u(x, t) = x (0.5 - \sin t) + \sin t = \frac{x}{2} - x \sin t + \sin t.$

The checking (satisfies the P.D.E., satisfies the given B.C.'s) I leave to angels sympthony.

Mind you, this is only how a *ahem* physicist would do it ......