Paul's Online Math Notes
Differential Equations (Notes) / Partial Differential Equations (Notes) / Heat Equation with Non-Zero Temperature Boundaries   [Notes]
Differential Equations - Notes

Internet Explorer 10 & 11 Users : If you have been using Internet Explorer 10 or 11 to view the site (or did at one point anyway) then you know that the equations were not properly placed on the pages unless you put IE into "Compatibility Mode". I beleive that I have partially figured out a way around that and have implimented the "fix" in the Algebra notes (not the practice/assignment problems yet). It's not perfect as some equations that are "inline" (i.e. equations that are in sentences as opposed to those on lines by themselves) are now shifted upwards or downwards slightly but it is better than it was.

If you wish to test this out please make sure the IE is not in Compatibility Mode and give it a test run in the Algebra notes. If you run into any problems please let me know. If things go well over the next week or two then I'll push the fix the full site. I'll also continue to see if I can get the inline equations to display properly.
Boundary Value Problems & Fourier Series Previous Chapter  
Solving the Heat Equation Previous Section   Next Section Laplace's Equation

 Heat Equation with Non-Zero Temperature Boundaries

In this section we want to expand one of the cases from the previous section a little bit.  In the previous section we look at the following heat problem.

 

 

 

Now, there is nothing inherently wrong with this problem, but the fact that we’re fixing the temperature on both ends at zero is a little unrealistic.  The other two problems we looked at, insulated boundaries and the thin ring, are a little more realistic problems, but this one just isn’t all that realistic so we’d like to extend it a little.

 

What we’d like to do in this section is instead look at the following problem.

(1)

 

In this case we’ll allow the boundaries to be any fixed temperature,  or .  The problem here is that separation of variables will no longer work on this problem because the boundary conditions are no longer homogeneous.  Recall that separation of variables will only work if both the partial differential equation and the boundary conditions are linear and homogeneous.  So, we’re going to need to deal with the boundary conditions in some way before we actually try and solve this. 

 

Luckily for us there is an easy way to deal with them.  Let’s consider this problem a little bit.  There are no sources to add/subtract heat energy anywhere in the bar.  Also our boundary conditions are fixed temperatures and so can’t change with time and we aren’t prescribing a heat flux on the boundaries to continually add/subtract heat energy.  So, what this all means is that there will not ever be any forcing of heat energy into or out of the bar and so while some heat energy may well naturally flow into our out of the bar at the end points as the temperature changes eventually the temperature distribution in the bar should stabilize out and no longer depend on time.

 

Or, in other words it makes some sense that we should expect that as  our temperature distribution,  should behave as,

 

 

where  is called the equilibrium temperature.  Note as well that is should still satisfy the heat equation and boundary conditions.  It won’t satisfy the initial condition however because it is the temperature distribution as  whereas the initial condition is at .  So, the equilibrium temperature distribution should satisfy,

(2)

 

This is a really easy 2nd order ordinary differential equation to solve.  If we integrate twice we get,

 

 

and applying the boundary conditions (we’ll leave this to you to verify) gives us,

 

 

 

Okay, just what does this have to do with solving the problem given by (1) above?  We’ll let’s define the function,

 

(3)

where  is the solution to (1) and  is the equilibrium temperature for (1)

 

Now let’s rewrite this as,

 

 

 

and let’s take some derivatives.

 

 

 

In both of these derivatives we used the fact that  is the equilibrium temperature and so is independent of time t and must satisfy the differential equation in (2).

 

What this tells us is that both  and  must satisfy the same partial differential equation.  Let’s see what the initial conditions and boundary conditions would need to be for .

 

 

 

So, the initial condition just gets potentially messier, but the boundary conditions are now homogeneous!  The partial differential equation that  must satisfy is,

 

 

 

We saw how to solve this in the previous section and so we the solution is,

 

 

 

where the coefficients are given by,

 

 

 

The solution to (1) is then,

 

 

 

and the coefficients are given above.

Solving the Heat Equation Previous Section   Next Section Laplace's Equation
Boundary Value Problems & Fourier Series Previous Chapter  

Differential Equations (Notes) / Partial Differential Equations (Notes) / Heat Equation with Non-Zero Temperature Boundaries    [Notes]

© 2003 - 2016 Paul Dawkins