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I am trying to solve a set of coupled partial differential equations, with defined boundary conditions using mathematica. Here are the equations and the boundary conditions.

$0=\frac{\partial}{\partial y} \left(\frac{1}{2} \frac{\partial v(y,t)}{\partial y} \left(\alpha \cos ^2(\theta (y,t))+\beta \sin ^2(\theta (y,t))\right)+\frac{\partial \theta (y,t)}{\partial t} \sin ^2(\theta (y,t))\right)$

$\frac{\partial \theta (y,t)}{\partial t}=\text{EE}^2 \sin (\theta (y,t)) \cos (\theta (y,t))-\frac{1}{2} \frac{\partial v(y,t)}{\partial y}+\frac{\partial^2 \theta (y,t)}{\partial y^2}$

$v(0,t)=0$

$v(1,t)=0$

$\theta(0,t)=0$

$\theta(1,t)=\frac{\pi}{2}$

Here is my attempt at solving this in mathematica:

EE = 100; A = 10^6; \[Alpha] = -.63; \[Beta] = -2.7;
Eqns = {D[\[Theta][y, t], {t}] == 
   EE^2*Sin[\[Theta][y, t]]*Cos[\[Theta][y, t]] - 
     (1/2)*D[v[y, t], {y}] + D[D[\[Theta][y, t], {y}], {y}], 
 0 == D[(1/2)*D[v[y, t], {y}]*(\[Alpha]*Cos[\[Theta][y, t]]^2 + 
            \[Beta]*Sin[\[Theta][y, t]]^2) + 
 D[\[Theta][y, t], {t}]*
         Sin[\[Theta][y, t]]^2, {y}]}
y0 = {v[0, t] == 0, \[Theta][0, t] == 0}
NDSolve[{Eqns, y0, y1}, {v[y, t], \[Theta][y, t]}, {y, 0, 1}, {t, 0, 10}]

And I get this error:

NDSolve::ivone: Boundary values may only be specified for one independent variable. Initial values may only be specified at one value of the other independent variable. >>

What's odd, is that it seems that I have only specified boundary values for one independent variable, $y$. I also understand that NDSolve is designed for use with Initial Value Problems, and not Boundary Value Problems. I've consulted this Wolfram Resource which presents some methods for solving BVPs with NDSolve, but I have been unable to figure out how to put those methods to use in attacking this problem. Does anyone know of a good way of solving this type of BVP with Mathematica, or do I need to turn to an iterative approach too integrating solving this in Matlab? I know that there are some expensive packages for solving BVPs with Mathematica, but I'm looking to do something out of the box, or with free extensions. Thanks in advance for any help you might have.

share|improve this question
    
I get NDSolve::deqn: Equation or list of equations expected instead of y1 in the first argument please test your code on a fresh Mathematica session –  belisarius Mar 20 at 21:41
    
Is y1 a extra variable? Or did you define it? –  joojaa Apr 4 at 12:27

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