1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Nonhomog heat equation that's piecewise

  1. Mar 6, 2016 #1
    1. The problem statement, all variables and given/known data

    $$u_{t}=u_{xx}+f(x) \\ u(0,t)=50 \\ u(\pi , t)=0 \\ u(x,0)=g(x)$$
    $$0<x<\pi \\ t>0$$
    50 & 0<x<\frac{\pi}{2} \\
    0 & \frac{\pi}{2}\leq x< \pi
    0 & 0<x<\frac{\pi}{2} \\
    50 & \frac{\pi}{2}\leq x< \pi

    So what I tried to do here is use the principle of superposition to split this problem up into two different problems ##m(x,t),n(x,t)##.

    $$m_t=m_{xx} \\ m(0,t)=50 \\ m(\pi,t)=0 \\ m(x,0) = g(x)$$
    $$n_t=n_{xx}+f(x) \\ n(0,t)=0 \\ n(\pi,t)=0 \\ n(x,0) =0$$

    I know how solve the first PDE easily, but the second one is giving me some trouble. I know that you are supposed to do a change of variables and then solve it that way, but how do you take care of the piecewise function ##f(x)## when you are transforming back from the change of variables? Will you just have a piecewise solution in the end?

  2. jcsd
  3. Mar 6, 2016 #2
    The heat equation has the smoothing property which smooths out any discontinuities in the data, so no the solution after t=0 should not be piece wise
  4. Mar 6, 2016 #3
    So when you transform the second with a change of variables you get something like $$v_t=v_{xx} \\ v(0,t)=0 \\ v(\pi, t)=0 \\ v(x,0)=\int \int f(x) dx - Ax - B$$. Solving for this problem $v$ isn't too difficult, but when you transform back, by doing $$u(x,t) = v(x,t)-\int \int f(x) dx + Ax + B$$. Then we see that ##f(x)## is piecewise, so wouldn't that make the whole solution piecewise too?
  5. Mar 6, 2016 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    As Brian T said, any discontinuities get smoothed out, but, yes, you are correct that the function would still be expressed as a piecewise formula.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted