Wolfram Language & System 10.3 (2015)|Legacy Documentation

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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 :

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Visualize the eigenfunctions:

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

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Visualize the eigenfunctions:

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Introduced in 2015
(10.2)