Solving the PDE 1-d Heat Equation for a Flipped Rod

[mskisoc]

regarding 1-d Head Equations on rods. I am aware of how to long a rod with length x=0 to x=L. and initial conditions of u(0,t)=0 degrees and u(L,t)=100 degrees. But how does the problem change if before t=0 the rod at x=0 was at 100 degrees and x=L was at 0 degrees. So at time=0 the rod was flipped over. Any help setting this up would be great!

 

[LCKurtz]

In the first case your initial condition is u(x,0) is a straight line between (0,0) and (L,100) and in the second case it goes from (0,100) to (L,0).

 

[mskisoc]

So, how would you set up to solve the problem if the two cases were combined. For example the rod is sitting in a certain set of initial conditions and then the rod if flipped 180 degrees so now it is in a different set of initial conditions and then I am interested in finding out what the temperature distribution would be after that one flip occurred.

 

[LCKurtz]

OK, let's get the terminology straight; I think I misunderstood you at first. The conditions on u(0,t) and u(L,t) are boundary conditions, not initial conditions. The initial condition u(x,0) = f(x), which needs to be specified to have a well posed problem. It is the temperature at t = 0 along the rod and you don't get to change it.

So I guess you want to let it run until some time t₀ > 0 and then change things. What you can't change is the initial conditions. You can change the temperatures at the ends. Is that what you are trying to describe? If so, you could work the problem in two parts. Use the first solution up until t₀, then use u(x,t₀) as the initial condition and solve again with the new boundary conditions.

 

[elibj123]

You can also write your boundary conditions with step functions, then generalized functions will enter your equation ( a delta function). I think it will be much more interesting and perhaps even faster :)