0
$\begingroup$

I'm having the following issue that can not find a solution , by using this boundary condition 

 u[0, t] == 1,
 v[0, t] == u[1, t],
 w[0, t] == v[1, t],
 z[0, t] == w[1, t]

that Mathematica does not understand, someone please help me!

s = NDSolve[
  {
   D[u[x, t], t] == -D[u[x, t], x] - 1 v[x, t] w[x, t] z[x, t],
   D[v[x, t], t] == -D[v[x, t], x] - 2 u[x, t] w[x, t] z[x, t],
   D[w[x, t], t] == -D[w[x, t], x] - 3 u[x, t] v[x, t] z[x, t],
   D[z[x, t], t] == -D[z[x, t], x] - 4 u[x, t] v[x, t] w[x, t],

   u[x, 0] == 1,
   v[x, 0] == 2,
   w[x, 0] == 3,
   z[x, 0] == 4,

   u[0, t] == 1,
   v[0, t] == u[1, t],
   w[0, t] == v[1, t],
   z[0, t] == w[1, t]
   },

  {u[x, t], v[x, t], w[x, t], z[x, t]},

  {x, 0, 1}, {t, 0, 1}]

I'm getting this error message :

NDSolve::bcedge: Boundary condition v[0,t]==u[1,t] is not specified on a single edge of the boundary of the computational domain. >>

I have already presented a similar problem here, but the reasoning suggested is not a good solution for my case.

What strategies do you know to treat this problem ? I appreciate any help...

Improve this question
$\endgroup$
4
  • $\begingroup$ not sure why the downvote. Anyway NDSolve is not so general that it can handle any sort of bc you can dream up, even if mathematically valid. In this case the pde's look to be fully decoupled, so solve first for u, then use that solution as a bc as you solve for v and so on. Alternately formulate as a single unknown on domain x,0,4 $\endgroup$
    – george2079
    Commented Apr 27, 2016 at 12:28
  • $\begingroup$ Thank you. But I need to solve all equations at once. $\endgroup$
    – Jeiveison Gobério Maia
    Commented Apr 27, 2016 at 18:02
  • $\begingroup$ I'm afraid you will need to work out your own finite difference scheme. $\endgroup$
    – george2079
    Commented Apr 27, 2016 at 20:43
  • $\begingroup$ How you could do this, any suggestions? $\endgroup$
    – Jeiveison Gobério Maia
    Commented Apr 27, 2016 at 20:46

2 Answers 2

Reset to default
1
$\begingroup$

As suggested by the Documentation Center, you can remove excess (last 3) conditions.

s = NDSolve[{D[u[x, t], t] == -D[u[x, t], x] - u[x, t], 
    D[v[x, t], t] == -D[v[x, t], x] - v[x, t], 
    D[w[x, t], t] == -D[w[x, t], x] - w[x, t], 
    D[z[x, t], t] == -D[z[x, t], x] - z[x, t],
    u[x, 0] == 1, v[x, 0] == 1, w[x, 0] == 1,
    z[x, 0] == 1, u[0, t] == 1},
  {u[x, t], v[x, t], w[x, t], z[x, t]}, {x, 0, 1}, {t, 0, 1}];

Plot3D[Evaluate[u[x, t] /. s[[1, 1]]], {x, 0, 1}, {t, 0, 1}]
Plot3D[Evaluate[v[x, t] /. s[[1, 2]]], {x, 0, 1}, {t, 0, 1}]
Plot3D[Evaluate[w[x, t] /. s[[1, 3]]], {x, 0, 1}, {t, 0, 1}]
Plot3D[Evaluate[z[x, t] /. s[[1, 4]]], {x, 0, 1}, {t, 0, 1}]

Will give you the solutions.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thus the boundary conditions for the other variables are wrong because the ultilizados values ​​in the solution does not match the proposed problem . $\endgroup$
    – Jeiveison Gobério Maia
    Commented Apr 27, 2016 at 18:02
0
$\begingroup$

A method to solve the original decoupled system :

s = NDSolve[{
   D[u[x, t], t] == -D[u[x, t], x] - u[x, t],
   u[x, 0] == 1,
   u[0, t] == 1}, {u[x, t]}, {x, 0, 4}, {t, 0, 1}]
v[x_, t_] = (u[x, t] /. s[[1, 1]]) /. x -> x + 1;
w[x_, t_] = (u[x, t] /. s[[1, 1]]) /. x -> x + 2;
z[x_, t_] = (u[x, t] /. s[[1, 1]]) /. x -> x + 3;
Row[{
  ContourPlot[u[x, t] /. s[[1, 1]], {x, 0, 1}, {t, 0, 1}],
  ContourPlot[v[x, t], {x, 0, 1}, {t, 0, 1}],
  ContourPlot[w[x, t], {x, 0, 1}, {t, 0, 1}],
  ContourPlot[z[x, t], {x, 0, 1}, {t, 0, 1}]}]

enter image description here

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you. But I need to solve all equations at once. The example it is simple , but the truth is that each variable has its own distinct equation among others. $\endgroup$
    – Jeiveison Gobério Maia
    Commented Apr 27, 2016 at 18:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.