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

In summary, you can flip a rod over at any time and change its initial conditions. Remember to use step functions or generalized functions to solve the problem.
  • #1
mskisoc
2
0
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!
 
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  • #2
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).
 
  • #3
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.
 
  • #4
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.
 
  • #5
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 :)
 

What is the PDE 1-d heat equation?

The PDE 1-d heat equation is a mathematical model that describes the behavior of heat diffusion in a one-dimensional system over time. It is used to calculate the temperature distribution in a material or system based on initial conditions and boundary conditions.

What are the variables involved in the PDE 1-d heat equation?

The variables involved in the PDE 1-d heat equation are time (t) and position (x). The equation also includes the temperature (u) and the thermal diffusivity (k) of the material or system.

What is the physical interpretation of the PDE 1-d heat equation?

The PDE 1-d heat equation represents the flow of heat in a one-dimensional system, where the change in temperature at any point is proportional to the second derivative of temperature with respect to position and the thermal diffusivity.

How is the PDE 1-d heat equation solved?

The PDE 1-d heat equation can be solved using various methods, such as separation of variables, Laplace transform, or numerical methods. The solution depends on the specific boundary conditions and initial conditions of the problem.

What are some real-world applications of the PDE 1-d heat equation?

The PDE 1-d heat equation has various applications in fields such as engineering, physics, and meteorology. Some examples include predicting the temperature distribution in a metal rod during welding, studying the behavior of heat in a building, and modeling the temperature changes in Earth's atmosphere.

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