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 Heat Equation 2 Dimensions

  1. Mar 10, 2016 #1

    RJLiberator

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Show that if v(x,t) and w(y,t) are solutions of the 1-dimensional heat equation (v_t = k*v_xx and w_t = k*w_yy), then u(x,y,t) = v(x,t)w(y,t) satisfies the 2-dimensional heat equation. Can you generalize to 3 dimensions? Is the same result true for solutions of the wave equation?

    2. Relevant equations


    3. The attempt at a solution

    Honestly, I have no idea what I am doing. This is all very interesting and it SEEMS like it should be answered by a "oh, yes, that's obvious just do this" quick few lines.

    But I'm so foreign with my PDE course that this is causing great stress.

    I need a start, if you can tell me what I should look at, or where I should start, please do so. I will attack it then.
     
  2. jcsd
  3. Mar 10, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Step 1: write down the 2D heat equation for a general function u(x,y,t).
    Step 2: plug in the suggested form for u(x,y,t) to see if it "works".
     
  4. Mar 10, 2016 #3

    RJLiberator

    User Avatar
    Gold Member

    I agree with your method and can see that this problem isn't that hard... once you understand the heat equation.

    I am searching now for more information on the 2D heat equation.
     
  5. Mar 10, 2016 #4

    RJLiberator

    User Avatar
    Gold Member

    A good starting point?

    Homogeneous Dirichlet B.C.:

    Screen Shot 2016-03-10 at 9.00.43 PM.png
     
  6. Mar 11, 2016 #5

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Not much of a start I'd say. A better start would be to write down what you are given, then write down what you are to prove. Then you could start working on it...
     
  7. Mar 11, 2016 #6

    RJLiberator

    User Avatar
    Gold Member

    @LCKurtz ,

    I have an extension on this assignment.

    I am going to go through some intense studying tonight and I will report back with my updated tries for this problem and the other PDE problem I posted.

    Thank you for your guidance thus far! Hopefully I come through.
     
  8. Mar 12, 2016 #7

    RJLiberator

    User Avatar
    Gold Member

    @Ray Vickson
    The 2D heat equation I am wondering what this means.

    I have the 1d heat equation as follows:
    u_t =k∇^2u
    u(x,t) = k*∇^2u

    is the 2d heat equation just
    u(x,y,t) = k*(uxx+kuyy)

    ?

    Separation of variable yiels
    T'/(K*T) = X''/X = Y''/Y = -v^2
     
    Last edited: Mar 12, 2016
  9. Mar 12, 2016 #8

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You mean ##u_t = k(u_{xx}+u_{yy})##, but, yes, that is what it means.

    This problem has nothing to do with separation of variables. Start by following my advice in post #5.
     
  10. Mar 12, 2016 #9

    RJLiberator

    User Avatar
    Gold Member

    Okay, what do we know?

    Well, we know the 2 dimension heat equation now:
    u(x,y,t) = k(U_xx+U_yy)
    we know u(x,y,t) = v(x,t)w(y,t) from problem description.

    we know
    V_t = k*V_xx and w_t=k*w_yy

    Do we set:
    v(x,t)w(y,t) = k(U_xx+U_yy)

    The thing that is confusing me is the left hand side, how do I work with a function of this nature.
     
  11. Mar 13, 2016 #10

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    It is very bad form, not to mention confusing, when you use both u and U and v and V to represent the same thing as you have done below. Don't do that.

    I wouldn't call that u(x,y,t) because it suggests it is the same u in the heat equation. Call it something new like$$F(x,y,t) = v(x,t)w(y,t)$$
    The question is whether or not ##F(x,y,t)## satisfies the heat equation ##u_t = k(u_{xx} + u_{yy})##.

    No. You plug ##F(x,y,t)## into the heat equation and see if what you are given makes it work.
     
    Last edited: Mar 14, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: PDE Heat Equation 2 Dimensions
  1. PDE heat equation (Replies: 7)

  2. PDE: heat equation (Replies: 10)

Loading...