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


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


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:
  • 1[ui[x,y,],vi[x,y,],]λi ui[x,y,]
    2[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:
  • "PDEDiscretization"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


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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:

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Visualize the eigenfunctions:

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Scope  (14)

Options  (7)

Applications  (8)

Properties & Relations  (3)

Possible Issues  (9)

Neat Examples  (1)

See Also

NDEigenvalues  NDSolve  DEigensystem  Eigensystem  DirichletCondition  NeumannValue  PeriodicBoundaryCondition  ImplicitRegion


Introduced in 2015