# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# 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 over the region Ω.

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

## Details and OptionsDetails and Options

• NDEigensystem gives lists of eigenvalues and eigenfunctions or in case of coupled systems.
• The equations eqns are specified as in NDSolve.
• An eigenvalue and eigenfunction pair for the differential operator satisfy .
• An eigenvalue and eigenfunctions pair for coupled differential operators satisfy:
• Eigenvalues are sorted in order of increasing absolute value.
• With the default normalization, the eigenfunctions computed by NDEigensystem[[u[x,y,]],u,{x,y,}Ω,n] approximately satisfy .  »
• With the default normalization, the eigenfunctions for coupled differential operators approximately satisfy .
• Homogeneous DirichletCondition or NeumannValue boundary conditions may be included. Inhomogeneous boundary conditions will be replaced with corresponding homogeneous boundary conditions. »
• 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 .
• Systems of time-dependent equations that are higher than first order are reduced to a coupled first-order system with intermediate variables , , 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 is handled by method . 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 "VectorNormalization" normalization of the eigenvectors that are used to construct the eigenfunctions

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Find the 4 smallest eigenvalues and eigenfunctions of the Laplacian operator on :

 Out[1]=

Visualize the eigenfunctions:

 Out[2]=

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

 Out[2]=

Visualize the eigenfunctions:

 Out[3]=