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How to solve two coupled pde's

  1. Nov 9, 2012 #1
    I'm new, hi all.

    I have two coupled equations, one of which is continuity. Basically, my problem comes down to the following system:

    (1) [itex]u=f(v)[/itex] (similarly, [itex]v=g(u)[/itex]. Here, u and v are the components of a vector field, ie u=u(x,y) and v=v(x,y).

    (2) Continuity: [itex]\nabla \cdot \textbf{u} = 0[/itex] or [itex]u_{x}+v_{y}=0[/itex]

    From here, I can find the following expressions

    [itex]u_{x} = -g_{y} \left( u \right)[/itex]
    [itex]v_{y} = -f_{x} \left( v \right)[/itex]

    Which I think leaves an equation of the form [itex]G \left( u,u_{x},u_{y} \right)=0[/itex] and [itex]F \left( v,v_{x},v_{y} \right)=0[/itex]

    It seems to me my original problem has two variables and I have two equations. I think this should be solvable, but I don't know how. Any help please? Thanks in advance!

    --edit-- p.s. I'm looking for a numerical (discrete) solution.
    Last edited: Nov 9, 2012
  2. jcsd
  3. Nov 9, 2012 #2


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    If u = f(v) and v = g(u), don't you just have a (possibly nonlinear) system of equations to solve for v and u? What is the continuity condition (2) for? Do you get more than one solution by solving (1) and you need (2) to choose a solution of interest?
  4. Nov 10, 2012 #3
    Actually I have only one relation for [itex]u[/itex] and [itex]v[/itex] that I can write it as [itex]u(v)[/itex] or [itex]v(u)[/itex]. Sorry for the confusion. Otherwise you would be right. To clarify my equations:

    (1) A relation for [itex]u[/itex] and [itex]v[/itex] (if I have [itex]u[/itex], I have [itex]v[/itex] and vice versa)
    (2) A relation for [itex]u_{x}[/itex] and [itex]v_{y}[/itex] (if I have [itex]u_{x}[/itex], I have [itex]v_{y}[/itex] and vice versa

    --edit-- Which then, after some rewriting, leads to two ODE's [itex]G[/itex] and [itex]F[/itex] as stated before. I just don't know how to solve those.
    Last edited: Nov 10, 2012
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