# Fourier Series Solution of 1-D Heat Flow

1. Mar 30, 2013

### Curtis15

1. The problem statement, all variables and given/known data

Length of rod = 1

Initial Conditions: u(x,0)=sin(πx)

Boundary conditions: u(0,t)=0 and u(1,t)=5.

Alright I am supposed to find the temperature at all times, but I am curious about the setup of the problem itself.

When x = 1, the boundary condition says that u = 5.

When t = 0, the initial condition says that u = sin(x∏).

So u(1,0) is supposed to equal what exactly? The boundary says it should be 5, but the initial condition says that sin(∏) = 0, so what would the answer be. I feel like this is contradictory but people are saying that it isn't and I am an idiot.

I have asked this somewhere else and got responses just saying this was a stupid question.

Thanks for any help

2. Mar 30, 2013

### micromass

Staff Emeritus

A possible way to interpret the problem is that the $u(1,t)=5$ condition only holds for large $t$ and not for all $t$.

3. Mar 30, 2013

### Curtis15

You have no idea how much I appreciate this response. Thank you very much!

4. Mar 30, 2013

### LCKurtz

@curtis15: You didn't specify for what $t$ your boundary conditions $u(0,t)=0,\ u(1,t)=5$ apply. A reasonable interpretation would be for $t>0$. Or you could think of the bar having the initial temperature distribution $u(x,0) = \sin(\pi x)$ suddenly inserted into a situation with those boundary conditions. I don't think there is anything contradictory here and working the problem should be straightforward.

[Edit, added later:] Another reasonable interpretation is to assume the initial condition $u(x,0) = \sin(\pi x)$ holds for $0 < x < 1$. That works even better intuitively.

Last edited: Mar 30, 2013