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Solve this PDEs system?

Posted 7 months ago
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Hello everybody,

I would like to solve numerically this system of PDEs with the following initial conditions: PDEs System

Here is my attempt to do it in Mathematica:

clear[\[Phi], \[Chi]];
sol = NDSolve[{
    -2*D[\[Phi][x, t], t, t] + 
      2 D[\[Phi][x, t], x, 
        x] + (Exp (\[Phi][x, t])/
          2)*((D[\[Chi][x, t], t])^2 - (D[\[Chi][x, t], x])^2) - 
      Exp[\[Phi][x, t]]*(Exp[\[Phi][x, t]] - 1) == 0,
    -D[\[Chi][x, t], t, t] + D[\[Chi][x, t], x, x] - 
      D[\[Phi][x, t], t]*D[\[Chi][x, t], t] +  
      D[\[Phi][x, t], x]*D[\[Chi][x, t], x] == 0 ,
      \[Phi][x, 0]  == Exp[-x^2],
     D[\[Phi][0, t], t] == 0,
    \[Chi][x, 0]  == Exp[-x^2],
    D[\[Chi][0, t], t] == 0
    },
   {\[Phi], \[Chi]}, {x, 0, 10}, {t, 0, 10}];
 Plot3D[\[Phi][x, t] /. sol[[1]], {x, 0, 10}, {t, 0, 10}] 

Unfortunately it gives me an unexpected error.

I have reread the code many times but I do not find the error.

Can someone help me please?

As it stands (fifth line) (Exp (\[Phi][x,t])/ it is an syntax error. But even that fixed it runs into

clear[\[Phi], \[Chi]];
sol = NDSolve[{-2*D[\[Phi][x, t], t, t] + 
      2 D[\[Phi][x, t], x, 
        x] + (Exp[\[Phi][x, t]]/
         2)*((D[\[Chi][x, t], t])^2 - (D[\[Chi][x, t], x])^2) - 
      Exp[\[Phi][x, t]]*(Exp[\[Phi][x, t]] - 1) == 
     0, -D[\[Chi][x, t], t, t] + D[\[Chi][x, t], x, x] - 
      D[\[Phi][x, t], t]*D[\[Chi][x, t], t] + 
      D[\[Phi][x, t], x]*D[\[Chi][x, t], x] == 0, \[Phi][x, 0] == 
     Exp[-x^2], D[\[Phi][0, t], t] == 0, \[Chi][x, 0] == Exp[-x^2], 
    D[\[Chi][0, t], t] == 0}, {\[Phi], \[Chi]}, {x, 0, 10}, {t, 0, 
    10}];


CoefficientArrays::poly: -E^\[Phi] (-1+E^\[Phi])-2 \[Phi]$4455+2 \[Phi]$4458+1/2 E^\[Phi] (\[Chi]$4456^2-\[Chi]$4457^2) is not a polynomial. >>

NDSolve::femper: PDE parsing error of {-E^\[Phi] (-1+E^\[Phi])-2 \[Phi]$4455+2 \[Phi]$4458+1/2 E^\[Phi] (\[Chi]$4456^2-\[Chi]$4457^2),-\[Phi]$4459 \[Chi]$4456+\[Phi]$4461 \[Chi]$4457-\[Chi]$4460+\[Chi]$4462}. Inconsistent equation dimensions. >>

if your modelling is meaningful and correct, then the first thing (CoefficientArrays::poly) is interesting (see also ref/message/General/poly in Mathematica Help) and can possibly treated by options, i.e. use another solver, else think about inconsistent equation dimensions.

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