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Solve the following partial differential equation?

Posted 5 months ago
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Consider the following code:

In[1]:= Subscript[a, 11] = 1.0

Out[1]= 1.

In[2]:= Subscript[a, 12] = 0.0015

Out[2]= 0.0015

In[3]:= Subscript[a, 13] = 0.0055

Out[3]= 0.0055

In[4]:= Subscript[a, 21] = 0.5

Out[4]= 0.5

In[5]:= Subscript[a, 22] = 0.00025

Out[5]= 0.00025

In[6]:= Subscript[a, 23] = 0.0055

Out[6]= 0.0055

In[7]:= Subscript[d, 1] = 0.23

Out[7]= 0.23

In[8]:= Subscript[d, 2] = 0.45

Out[8]= 0.45

In[9]:= j = 
 NDSolveValue[{D[u[x, \[Psi]], \[Psi], \[Psi]] == 0, 
   u[x, -1] == u[x, 1] == u[-1, \[Psi]] == u[1, \[Psi]] == 0}, 
  u, {x, -1, 1}, {\[Psi], -1, 1}]

Out[9]= InterpolatingFunction[{{-1., 1.}, {-1., 1.}}, <>]

In[10]:= k = 
 NDSolveValue[{D[u[y, \[Psi]], \[Psi], \[Psi]] == 0, 
   u[y, -1] == u[y, 1] == u[-1, \[Psi]] == u[1, \[Psi]] == 0}, 
  u, {y, -1, 1}, {\[Psi], -1, 1}]

Out[10]= InterpolatingFunction[{{-1., 1.}, {-1., 1.}}, <>]

In[12]:= Equationp = 
 NDSolve[{x'[t] == 
    Subscript[a, 11] x[t] - Subscript[a, 12] x[t]^2 - 
     Subscript[a, 13] x[t] y[t] + Subscript[d, 1] j[t], 
   y'[t] == -Subscript[a, 21] y[t] - Subscript[a, 22] y[t]^2 + 
     Subscript[a, 23]  x[t]  y[t] + Subscript[d, 2] k[t], x[0] == 10, 
   y[0] == 5, \[Psi][0] == 5, D[u[x, \[Psi]], \[Psi]] == 10, 
   D[u[y, \[Psi]], \[Psi]] == 5}, {x, y, \[Psi]}, {\[Psi], 0, 
   2 \[Pi]}, {t, 0, 25}]

During evaluation of In[12]:= NDSolve::dsfun: [Psi] cannot be used as a function.

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3 Replies

Is Psi an independent variable or an unknown function? What are j[t] and u[x,Psi]?

Usually in a diff'eq there is only one unknown function being looked for, if it is a "book problem".

I have issues reading the equations: it would be useful to me to see the 2D equations (not function wrapper version).

I ask if the poster knows if the solution involves Laplace, separation of variables (and if orthagonal or fouriers are useful).

I often find people are trying to solve diff'eq that have been solved 1,000,000 times over (the function being found has already been found, and can be looked up in a table). Is this equation one that can be located in a table of solutions?

Can I ask why Psi is used both as an Function[] parameter to D and as a Function[] parameter to NDSolve? I'm unsure but I sense problems there. D is Function[] parameter to NDSolve, and the "scope of Psi" is unclear (to me, perhaps not Mathematica). I don't know if NDSolve has a firm "HoldAll" on all it's Function[] parameter (when Psi will become in what context). i would avoid that usage because not all Mathematica functions Hold arguments the same: some Functions intently evaluate parameters before "beginning solving", while others Hold parameters to prevent it. Obviously this must be known to know if Psi will become "used 2x / used for both Functions" (which won't work).

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