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The internal documentation on NDEigenvalues is a bit unclear. I want to solve two coupled eigenvalue equations in 2D for two coupled eigenfunctions. The problem is of the form:

LinOp1[ u[x,y], v[x,y] ] ==  eval u[x,y]

LinOp2[ u[x,y], v[x,y] ] ==  eval v[x,y]

where LinOp1 and LinOp2 are linear combinations of derivatives of the functions u[x,y] and v[x,y]. The documentation implies that this is possible, because it discusses coupled systems with a syntax giving the list of linear differential operators and a list of functions. But it's not clear in the documentation how Mathematica "knows" the eigenfunction on the RHS for the first operator is u[x,y], and the second one is v[x,y]. Is it just a matter of the order in the list of arguments? Can anyone point me to a sample of a coupled system solved with NDEigenvalues?

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  • $\begingroup$ Yes, it is given by the order of the second argument. An example from the documentation is the last problem under "Properties & Relations": NDEigenvalues[{D[u[t, x], t] == v[t, x], D[v[t, x], t] == Laplacian[u[t, x], {x}]}, {u[t, x], v[t, x]}, t, {x, 0, Pi}, 6]. $\endgroup$ – Marius Ladegård Meyer Aug 4 '17 at 7:55
  • $\begingroup$ @MariusLadegårdMeyer, I took the liberty of using your example in my answer. I hope that is OK? $\endgroup$ – user21 Aug 4 '17 at 11:13
  • $\begingroup$ @user21 Sure, no problem :) $\endgroup$ – Marius Ladegård Meyer Aug 4 '17 at 22:35
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This

NDEigenvalues[{D[u[t, x], t] == v[t, x], 
  D[v[t, x], t] == Laplacian[u[t, x], {x}]}, {u[t, x], 
  v[t, x]}, t, {x, 0, Pi}, 6]

{9.12781*10^-18 - 1.27262*10^-7 I, 
 9.12781*10^-18 + 1.27262*10^-7 I, -1.87511*10^-20 - 
  1. I, -1.87511*10^-20 + 1. I, 2.74571*10^-17 - 2.00001 I, 
 2.74571*10^-17 + 2.00001 I}

is the same as this:

NDEigenvalues[{v[x], Laplacian[u[x], {x}]}, {u[x], v[x]}, {x, 0, Pi},
  6]

{9.12781*10^-18 - 1.27262*10^-7 I, 
 9.12781*10^-18 + 1.27262*10^-7 I, -1.87511*10^-20 - 
  1. I, -1.87511*10^-20 + 1. I, 2.74571*10^-17 - 2.00001 I, 
 2.74571*10^-17 + 2.00001 I}

In the Applications -> Structural Mechanics section is another example.

I hope that helps.

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