Solving Inhomogeneous PDE for Equilibrium Temperature Distribution

[Lacero]

Hi all,

Homework Statement

Determine the equilibrium temperature distribution (if it exists). For what values of B, are there solutions.

Homework Equations

a) Ut = Uxx + 1, U(x,0) = f(x), Ux(0,t) = 1, U(L,t) = B

b) Ut = Uxx + X - B, U(x,0) = f(x), Ux(0,t) = 0, U(L,t) = 0

The Attempt at a Solution

a) Assume solution U(x,t) = V(x,t) + K

=> Ut = Vt
=> Uxx = Vxx -- Sub into Ut = Uxx + 1

Vt = Vxx + 1

Now to get homogeneous boundary conditions,

U(0,t) = V(0,t) + K = 1 ?
but K = 1 => V(0,t) = 1?

// Trouble at the BC

b)

Assume solution U(x,t) = V(x,t) + W(x)

=> Ut = Vt
=> Uxx = Vxx + W'' -- Sub into Ut = Uxx + Q(x)

Vt = Vxx + W'' + Q

Now, let W'' + Q = 0 or W'' = -Q
=> Vt = Vxx

V(0,t) = 0
V(L,t) = 0

U(x,0) = V(x,0) + W(x) = f(x)
=> V(x,0) = f(x) - W(x)

Now Solve transient solution, v(x,t)
. .
. .
. .
V(x,t) = (2/L)sum(n=1 to inf)(int((f(x)-W(x))sin(npi/L)xdx)... etc

Now Solve steady state solution, W" = -Q B - X
W" + X + B = 0
. .
. .
W(x) = Scos(zx) + Tsin(zx) + Yp... etc

Now U(x,t) = W(x) + V(x,t)

Am I right at all?

 

[mighty2000]

Hi there,

it is important to approach problems like this with a clear and logical thought process. In this case, it seems that you are on the right track with your solution attempts, but there are a few areas that could use some clarification and further exploration.

Firstly, in both cases a) and b), it is important to clearly define the boundary conditions. In a), it seems that the boundary condition at x=0 is U(0,t) = 1, but in b) it is Ux(0,t) = 0. These are two different conditions and will result in different solutions.

Additionally, when solving for the steady state solution in b), it is important to consider the boundary conditions at x=0 and x=L. In your solution attempt, you only considered one of these boundary conditions (W(0) = 0), but the other one (W(L) = 0) should also be taken into account.

Overall, your approach seems to be on the right track, but it would be helpful to clearly define the boundary conditions and to consider all of them when solving for the steady state solution. Keep up the good work!