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I need to solve a system of mixed string and wave equations. Omitting some constants it looks like this:

$$u_ {\text {yy}} (y, t) - u_ {\text {tt}} (y, t) = \varphi _{t}(x, y, t)$$ $$\nabla _{\{x,y\}}^{}\varphi (x,y,t) = \varphi _{tt}(x,y,t)$$

With boundary and initial conditions:

$$u(\pm1,t)=0, u_t(y,0)=0,u(y,0)=\cos \left(\frac{\pi y}{2}\right)$$ $$ \varphi _{x}(0, y, t)=u_t(y,t), \varphi _{y}(x, \pm1, t)=0$$

$u(y,0)$ function may be different. I've tried several ways with no success.

    pde = {Derivative[0, 2][u][y, t] == Derivative[2, 0][u][y, t] + Derivative[0, 1, 0][\[CurlyPhi]][0, y, t], 
        Derivative[2, 0, 0][\[CurlyPhi]][x, y, t] + Derivative[0, 2, 0][\[CurlyPhi]][x, y, t] == Derivative[0, 0, 2][\[CurlyPhi]][x, y, t] + 
          NeumannValue[Derivative[0, 1][u][y, t], x == 0] + NeumannValue[0, y == -1 || y == 1]}; 
    bdc = {DirichletCondition[u[y, t] == 0, y == 1], DirichletCondition[u[y, t] == 0, y == -1], 
        DirichletCondition[\[CurlyPhi][x, y, t] == 0, x == 0]}; 
    ic = {u[y, 0] == Cos[(Pi*y)/2], Derivative[0, 1][u][y, 0] == 0}; 
    res = NDSolve[{pde, bdc, ic}, {u, \[CurlyPhi]}, {x, 0, 1}, {y, -1, 1}, {t, 0, 5}]

$$u(y,t) -> u(x,y,t)$$

    pde = {Derivative[0, 0, 2][u][x, y, t] == Derivative[0, 2, 0][u][x, y, t] + Derivative[0, 1, 0][\[CurlyPhi]][0, y, t], 
        Derivative[2, 0, 0][\[CurlyPhi]][x, y, t] + Derivative[0, 2, 0][\[CurlyPhi]][x, y, t] == Derivative[0, 0, 2][\[CurlyPhi]][x, y, t] + 
          NeumannValue[Derivative[0, 0, 1][u][x, y, t], x == 0] + NeumannValue[0, y == -1 || y == 1]}; 
    bdc = {DirichletCondition[u[x, y, t] == 0, y == 1], DirichletCondition[u[x, y, t] == 0, y == -1], 
        DirichletCondition[\[CurlyPhi][x, y, t] == 0, x == 0]}; 
    ic = {u[x, y, 0] == 0, u[x, y, 5] == 0}; 
    res = NDSolve[{pde, bdc, ic}, {u, \[CurlyPhi]}, {x, 0, 1}, {y, -1, 1}, {t, 0, 5}]

Even took the original system:


    pde = {Derivative[0, 2, 0][u][x, y, t] - Derivative[0, 0, 2][u][x, y, t] - \[Rho][0, y, t] == 0, 
        Derivative[0, 0, 1][\[Rho]][x, y, t] + Derivative[1, 0, 0][vx][x, y, t] + Derivative[0, 1, 0][vy][x, y, t] == 0, 
        Derivative[0, 0, 1][vx][x, y, t] - Derivative[1, 0, 0][\[Rho]][x, y, t] == 0, 
        Derivative[0, 0, 1][vy][x, y, t] - Derivative[0, 1, 0][\[Rho]][x, y, t] == 0}; 
    bcs = {u[x, 1, t] == 0, u[x, -1, t] == 0, vx[0, y, t] == Derivative[0, 0, 1][u][x, y, t], vy[0, y, t] == 0, 
        vy[x, -1, t] == 0, vy[x, 1, t] == 0}; 
    ics = {u[x, y, 0] == Cos[(Pi*y)/2], Derivative[0, 0, 1][u][x, y, 0] == 0, \[Rho][x, y, 0] == 0, vx[x, y, 0] == 0, 
        vy[x, y, 0] == 0}; 
    res = NDSolve[{pde, bcs, ics}, {u, \[Rho], vx, vy}, {x, y, t}]; 

In all cases, different errors. I think I have a problem with the formulation of the problem for WM. Maybe I'm going the wrong way? Please, help me find the working solution.

TQ WM11

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One problem is that the equations you have written down do not correspond to the coded equations and I am not sure what the correct input is. Here is one version of what you could mean - I have no idea if this makes any sense:

pde = {
   D[u[x, y, t], {y, 2}] - D[u[x, y, t], {t, 2}] == 
    D[\[CurlyPhi][x, y, t], t],
   Laplacian[\[CurlyPhi][x, y, t], {x, y}] == 
    D[\[CurlyPhi][x, y, t], {t, 2}] + 
     NeumannValue[Derivative[0, 0, 1][u][x, y, t], x == 0]};
bdc = {DirichletCondition[u[x, y, t] == 0, y == 1 || y == -1]};
ic = {u[x, y, 0] == Cos[\[Pi] y/2], 
   Derivative[0, 0, 1][u][x, y, 0] == 0, \[CurlyPhi][x, y, 0] == 0, 
   Derivative[0, 0, 1][\[CurlyPhi]][x, y, 0] == 0};
res = NDSolveValue[{pde, bdc, ic}, {u, \[CurlyPhi]}, {x, 0, 
   1}, {y, -1, 1}, {t, 0, 5}]

You need to think about initial conditions for phi.

GraphicsRow[{Plot3D[
   Evaluate[res[[1]][x, y, 5]], {x, 0, 1}, {y, -1, 1}],
  Plot3D[Evaluate[res[[2]][x, y, 5]], {x, 0, 1}, {y, -1, 1}]}]

enter image description here

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  • $\begingroup$ The code may vary in BC because I tried to simplify it to force to work it. The right BC in the begining of my post. In WM11 this code give me error: CoefficientArrays::poly: [CurlyPhi]$879+[CurlyPhi]$880-[CurlyPhi]$882-NeumannValue[u$881,x==0] is not a polynomial. NDSolveValue::femper: PDE parsing error of {-u$876+u$877-[CurlyPhi]$878,[CurlyPhi]$879+[CurlyPhi]$880-[CurlyPhi]$882-NeumannValue[u$881,x==0]}. Inconsistent equation dimensions. $\endgroup$ – VasilySH Jun 5 at 13:08
  • $\begingroup$ @VasilySH, works fine for me in V11.3 and V12.0 $\endgroup$ – user21 Jun 5 at 13:10
  • $\begingroup$ @VasilySH, try without the NeumannValue to see if this is the only issue. $\endgroup$ – user21 Jun 5 at 13:11
  • $\begingroup$ Without NeumannValue it says: NDSolveValue::fembdnl: The dependent variable in (u^(0,0,1))[0,y,t]==0 in the boundary condition DirichletCondition[(u^(0,0,1))[0,y,t]==0,x==0.] needs to be linear. WM11.0.0 ( Also you put Dirichlet condition instead of Neumann on borders (y == 1 || y == -1) You fix function value instead of derivative. So it's wrong solution. $\endgroup$ – VasilySH Jun 5 at 13:32
  • $\begingroup$ @VasilySH, I updated the equation to use Neumann zero for phi_x. $\endgroup$ – user21 Jun 5 at 13:48

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