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!

ODE's for 2 space Heat equation

  1. Sep 2, 2010 #1
    1. The problem statement, all variables and given/known data
    The Heat equation in two space is

    [itex]\alpha ^2 \left[\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2} \right]=\frac{\partial u}{\partial t}[/itex]

    Assuming separation solution of the form [itex]u(x,y,t)=F(x)G(y)H(t)[/itex] find ordinary differential equations satisfied by F,G and H.




    2. Relevant equations

    Heat Equation

    3. The attempt at a solution

    Because we can assume [itex]u(x,y,t)=F(x)G(y)H(t)[/itex]

    Is the first step that we require that


    [itex]\alpha^2\frac{F''}{F}+\alpha^2\frac{G''}{G}=\frac{H'}{H}[/itex]

    therefore we need to solve


    [itex]F''+\frac{k}{\alpha^2}F+G''+\frac{k}{\alpha^2}G=0[/itex]

    and

    [itex]H'-kH=0[/itex]


    Is this right so far?
     
  2. jcsd
  3. Sep 2, 2010 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member



    No. F"/F is a function of x, F"/F=f(x). Similarly, G"/G=g(y) and H"/H=h(t)

    α2(f(x)+g(y)) = h(t) holds only when f, g, h are all constant functions: f(x)=K, g(y)=L, h(t)=M

    with the condition that α2(K+L)=M
    Can you proceed from here?

    ehild
     
  4. Sep 2, 2010 #3
    Do I have to try to combine the first functions and set the right side to = 0 ?

    My examples in my notes are all of the form

    [itex]\alpha^2\frac{F''}{F}=\frac{ G'}{G}[/itex]
     
    Last edited: Sep 2, 2010
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook