Question

我正在尝试使用NDsolve来解决与ginzburg landau相关的以下非线性耦合PDE方程。我是Mathematica的新手。我收到以下错误。我正在做的错误是什么？

pde = {D[u[t, x, y], t] == 
D[u[t, x, y], {x, x}] + 
 D[u[t, x, y], {y, 
   y}] - (1/u[t, x, y])^3*(D[v[t, x, y], y]^2 + 
    D[v[t, x, y], x]^2) - u[t, x, y] + u[t, x, y]^3, 
   D[v[t, x, y], t] == 
D[v[t, x, y], {x, x}] + D[v[t, x, y], {y, y}] - 
 v[t, x, y]*u[t, x, y] + 
       (2/u[t, x, y])*(D[u[t, x, y], x]*D[v[t, x, y], x] - 
    D[u[t, x, y], y]*D[v[t, x, y], y])};bc = {u[0, x, y] == 0, v[0, x, y]== 0, u[t, 5, y] == 1, u[t, x, 5] == 1, D[v[t, 0, y], x] == 0, D[v[t, x, 0], y] == 0};
NDSolve[{pde, bc}, {u, v}, {x, 0, 5}, {y, 0, 5}, {t, 0, 2}]

'Error: NDSolve::deqn: Equation or list of equations expected instead of True in the first argument {{(u^(1,0,0))[t,x,y]==-u[t,x,y]+u[t,x,y]^3+(u^(0,0,y))[t,x,y]-((<<1>>^(<<3>>))[<<3>>]^2+(<<1>>^(<<3>>))[<<3>>]^2)/u[t,x,y]^3+(u^(0,x,0))[t,x,y],(v^(1,0,0))[t,x,y]==-u[t,x,y] v[t,x,y]+(v^(0,0,y))[t,x,y]+(2 (-(<<1>>^(<<3>>))[<<3>>] (<<1>>^(<<3>>))[<<3>>]+(<<1>>^(<<3>>))[<<3>>] (<<1>>^(<<3>>))[<<3>>]))/u[t,x,y]+(v^(0,x,0))[t,x,y]},{u[0,x,y]==0,v[0,x,y]==0,u[t,5,y]==1,u[t,x,5]==1,True,True}}.

NDSolve[{{Derivative[1, 0, 0][u][t, x, y] == -u[t, x, y] + 
     u[t, x, y]^3 + Derivative[0, 0, y][u][t, x, y] - 
              (Derivative[0, 0, 1][v][t, x, y]^2 + 
     Derivative[0, 1, 0][v][t, x, y]^2)/u[t, x, y]^3 + 
     Derivative[0, x, 0][u][t, x, y], 
       Derivative[1, 0, 0][v][t, x, y] == (-u[t, x, y])*v[t, x, y] + 
     Derivative[0, 0, y][v][t, x, y] + 
           (2*((-Derivative[0, 0, 1][u][t, x, y])*
           Derivative[0, 0, 1][v][t, x, y] + 
          Derivative[0, 1, 0][u][t, x, y]*
           Derivative[0, 1, 0][v][t, x, y]))/u[t, x, y] + 
           Derivative[0, x, 0][v][t, x, y]}, {u[0, x, y] == 0, 
   v[0, x, y] == 0, u[t, 5, y] == 1, u[t, x, 5] == 1, True, 
   True}}, {u, v}, {x, 0, 5}, {y, 0, 5}, {t, 0, 2}]

Answer 1

如果查看bc的值，您会看到

bc = {u[0, x, y] == 0, v[0, x, y] == 0, u[t, 5, y] == 1, 
u[t, x, 5] == 1, D[v[t, 0, y], x] == 0, D[v[t, x, 0], y] == 0}

给你

{u[0, x, y] == 0, v[0, x, y] == 0, u[t, 5, y] == 1, u[t, x, 5] == 1, True, True}

这是关于True的错误消息的来源。

你正在做的是区分表达式相对于x，但表达式中没有x，因此结果为零。并且0 == 0始终为True。与y同样。因此，让我们改变你试图告诉它边界条件的方式。

bc = {u[0, x, y] == 0, v[0, x, y] == 0, u[t, 5, y] == 1, u[t, x, 5] == 1,
Derivative[0, 1, 0][v][t, 0, y] == 0, Derivative[0, 0, 1][v][t, x, 0] == 0}

或

bc = {u[0, x, y] == 0, v[0, x, y] == 0, u[t, 5, y] == 1, u[t, x, 5] == 1,
D[v[t, x, y], x] == 0/.x->0, D[v[t, x, y], y] == 0/.y->0}

我认为其中任何一个都应该能满足您的需求。

一旦你修复了那些，那么你会得到一个关于导数阶和非负整数的不同错误。

我相信您可以通过将pde从{x,x}和{y,y}更改为{x,2}和{y,2}来解决此问题

pde = {D[u[t, x, y], t] == D[u[t, x, y], {x, 2}] + D[u[t, x, y], {y, 2}] -
  (1/u[t, x, y])^3*(D[v[t, x, y], y]^2 + D[v[t, x, y], x]^2) - u[t, x, y] +
  u[t, x, y]^3,
  D[v[t, x, y], t] == D[v[t, x, y], {x, 2}] + D[v[t, x, y], {y, 2}] - v[t, x, y]*
  u[t, x, y] + (2/u[t, x, y])*(D[u[t, x, y], x]*D[v[t, x, y], x] - D[u[t, x, y], y]*
  D[v[t, x, y], y])};

这使得该错误消失了。

一旦你修复了它并尝试了NDSolve，那么你的分母中的零就会开始咬你。

修复这些看起来不仅仅是了解MMA语法。这可能需要了解您的问题，看看您是否可以消除这些零分母。

Mathematica中的非线性PDE求解

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