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  • $\begingroup$ I am not exactly sure I understand what you are trying to do. Maybe you are looking for an eigenfunction expansion? There is an example of this in the documentation of NDEigensystem in the section Applications and then Eigenfunction Expansion. Hope this helps. $\endgroup$ Commented May 4, 2017 at 3:13
  • $\begingroup$ I think the solution to the above equation is trivial, which is $u=0$. However, if I am not wrong we can also get solution by adding the eigenfunctions? So I plan to do error analysis, for example I want to compare the exact analytical solution, which is $u=0$ with the sum of eigenfunctions. I assume the more eigenfunctions we add, the smaller the error is? I am just learning PDE, so that is what my understanding is. I am not sure if it is correct. So for example, if mathematica gives me these six eigenfunctions, how to add those functions? $\endgroup$ Commented May 4, 2017 at 3:24
  • $\begingroup$ I think there is some confusion here. $u=0$ is $u_{nm}$ with $n=m=0$. No sum just one term, no error too. However, what is the significance of this solution? The Helmholtz equation describes some physical process and for this matter $u=0$ is to be neglected. $\endgroup$ Commented May 4, 2017 at 5:03
  • $\begingroup$ The $u$ that I mean is the final analytical solution, i,e,, when the $m,n$ is infinite. Is it possible? Or my understanding is wrong? $\endgroup$ Commented May 4, 2017 at 5:06
  • $\begingroup$ Note 1 that the solution also depends on initial conditions. You have to supply two. 2 you have chosen a square which due to symmetry will sometimes have identical eigenvalues. See your second and third eigenvectors. These need special treatment when combining them. A rectangle is simpler. $\endgroup$ Commented May 4, 2017 at 5:55