# NDEigensystem

NDEigensystem[[u[x,y,]],u,{x,y,}Ω,n]

gives the n smallest magnitude eigenvalues and eigenfunctions for the linear differential operator over the region Ω.

NDEigensystem[{1[u[x,y,],v[x,y,],],2[u[x,y,],v[x,y,],],},{u,v,},{x,y,}Ω,n]

gives eigenvalues and eigenfunctions for the coupled differential operators {op1,op2,} over the region Ω.

NDEigensystem[eqns,{u,},t,{x,y,}Ω,n]

gives the eigenvalues and eigenfunctions in the spatial variables {x,y,} for solutions u, of the coupled time-dependent differential equations eqns.

# Details and Options

• NDEigensystem gives lists {{λ1,,λn},{u1,,un}} of eigenvalues λi and eigenfunctions ui or {{λ1,,λn},{{u1,v1,},,{un,vn,}} in case of coupled systems.
• The equations eqns are specified as in NDSolve.
• An eigenvalue and eigenfunction pair {λi,ui} for the differential operator satisfy [u[x,y,]]==λi ui[x,y,].
• An eigenvalue and eigenfunctions pair {λi,{ui,vi,}} for coupled differential operators satisfy:
•  ℒ1[ui[x,y,…],vi[x,y,…],…]λi ui[x,y,…] ℒ2[ui[x,y,…],vi[x,y,…],…]==λi vi[x,y,…] ⋮
• Eigenvalues are sorted in order of increasing absolute value.
• With the default normalization, the eigenfunctions ui computed by NDEigensystem[[u[x,y,]],u,{x,y,}Ω,n] approximately satisfy .  »
• With the default normalization, the eigenfunctions {ui,vi,} for coupled differential operators approximately satisfy .
• Homogeneous DirichletCondition or NeumannValue boundary conditions may be included.  »
• When no boundary condition is specified on part of the boundary Ω, then this is equivalent to specifying a Neumann 0 condition.
• For a system of first-order time-dependent equations, the time derivatives D[u[t,x,y,],t], D[v[t,x,y,],t], are effectively replaced with λ u[x,y,], λ v[x,y,],.
• Systems of time-dependent equations that are higher than first order are reduced to a coupled first-order system with intermediate variables ut=u*,=, vt=v*,=, Only the functions u, v, are returned. »
• NDEigensystem accepts a Method option that may be used to control different stages of the solution. With Method->{s1->m1,s2->m2,}, stage si is handled by method mi. When stages are not given explicitly, NDEigensystem tries to automatically determine what stage to apply a given method to.
• Possible solution stages are:
•  "SpatialDiscretization" discretization of spatial operators "Eigensystem" computation of the eigensystem from the discretized system "Interpolation" creation of interpolating functions "VectorNormalization" normalization of the eigenvectors that are used to construct the eigenfunctions

# Examples

open allclose all

## Basic Examples(2)

Find the 4 smallest eigenvalues and eigenfunctions of the Laplacian operator on [0,π]:

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Visualize the eigenfunctions:

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Compute the first 6 eigenfunctions for a circular membrane with the edges clamped:

 In[1]:=
 In[2]:=
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Visualize the eigenfunctions:

 In[3]:=
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