1. Not finding help here? Sign up for a free 30min 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!

PDE - Solve heat equation with convection

  1. Aug 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Solve u_t -k u_xx +V u_x=0
    With the initial condition, u(x,0)=f(x)

    Use the transformation y=x-Vt

    2. Relevant equations
    The solution to the equation u_t - k u_xx=0 with the initial condition is
    u(x,t)=1/Sqrt[4[itex]\pi[/itex] kt] [itex]\int[/itex] e^(-(x-y)^2 /4kt)f(y) dy

    3. The attempt at a solution
    I really just need help subbing in the change in variable.

    I think it's something like
    u_y= u_t dt/dy +u_x dx/dy with dx/dy=1/(dy/dx)=1, dt/dy=1/V
    =-1/V u_t +u_x
    But this doesn't put the equation into a useful form...

    the other thing I thought of was
    u_x=u_t dt/dx +u_y dy/dx =0+u_y
    and u_t=u_x dx/dt+u_y dy/dt =-Vu_y
    And then we have that u_t+V u_x=-V u_y +V u_y=0, so the DE is just k u_xx=0; which I'm guessing isn't right either - because then it's just straight integration; (with the constants as functions of y?)...
    Anyway, I'm fairly sure that the change of variables will result in either u_y=u_t+V u_x or possibly some multiple.
     
  2. jcsd
  3. Aug 13, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Ratpigeon! :smile:

    (try using the X2 button just above the Reply box :wink:)
    no, you have two sets of variables: (x,t) and (y,t') with t' = t

    your ut should have uy and ut' parts, not uy and ux parts :wink:
     
  4. Aug 13, 2012 #3
    So is s=x+Vt
    Then is it ux= uy +us, uxx=uyy+2uys+uss
    ut=-Vuy+Vus and the total equaton is 2Vuy+uyy+2uys+uss?
     
  5. Aug 13, 2012 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    what on earth is "s" ? :confused:

    you have two sets of variables: (x,t) and (y,t') with y = x - Vt and t' = t

    your ut should have uy and ut' parts, not uy and ux parts :wink:
     
  6. Aug 13, 2012 #5
    Okay so i define h(x-Vt,t')=u(x,t) and get
    ht=ut-Vuy and hy=ux for y=x-Vt
    So:
    ht-khyy=ut-kuxx+Vux=0
    And the initial conditon u(x,0)=f(x) becomes h(x-Vt,0)=f(x-Vt); and since t=0 this is just h(x,0)=f(x)
    and then i have an integral of...

    exp(-(x-Vt-y)^2/4kt) f(y)dy (with the square root scalar out the front) and thats the solution for h and hence also u?
     
  7. Aug 13, 2012 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    sorry, this is too difficult to read :redface:

    but anyway wouldn't it be easier to find ux and ut, since they're actually mentioned in the question?
     
  8. Aug 13, 2012 #7
    Im about 90% sure i got it. thankyou so much for your help :) i just wasnt getting the variable change until you explained it... :S
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: PDE - Solve heat equation with convection
  1. PDE heat equation (Replies: 7)

  2. PDE: heat equation (Replies: 10)

Loading...