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How to solve two coupled pde's 
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#1
Nov912, 08:14 PM

P: 3

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. 


#2
Nov912, 09:51 PM

HW Helper
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#3
Nov1012, 05:21 AM

P: 3

(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. 


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