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Reduction of 2nd order PDE to a first order equations system

  1. Sep 30, 2015 #1
    I want to convert this linear second order general form PDE to two equations:
    ##ϕ_{xx}+bϕ_{xy}+cϕ_{yy}+dϕ_x+eϕ_y+fϕ=g(x,y)##

    Converted equations:

    ##a_1 u_x+b_1 u_y+c_1 v_x+d_1 v_y=f_1##

    ##a_2 u_x+b_2 u_y+c_2 v_x+d_2 v_y=f_2##

    I want to find parametric values of ##a_1 ....f_2##

    How can I do it?

    Hint:

    ##ϕ_{xx}+ϕ_{yy}=0##

    ##u=ϕ_x , v=ϕ_y##

    ##ϕ_{xx}=u_x,ϕ_{yy}=ϕ_y##

    ##u_x+v_y=0##

    ##u_y-v_x=0##
     
    Last edited: Sep 30, 2015
  2. jcsd
  3. Oct 5, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Oct 5, 2015 #3

    andrewkirk

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    1. The converted equations don't look right, as they do not incorporate the function ##g##.

    2. What is the justification for the first line of the Hint: ##\phi_{xx}+\phi_{yy}=0##? It is not derivable from the given equation.
     
  5. Oct 7, 2015 #4
    No that hint is a typical example
     
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