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


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


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

Details and Options

  • NDEigenvalues gives a list {λ1,,λn} of the n smallest magnitude eigenvalues λi.
  • The equations eqns are specified as in NDSolve.
  • Eigenvalues are sorted in order of increasing absolute value.
  • 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.  »
  • NDEigenvalues 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, NDEigenvalues 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.
    "VectorNormalization"normalization of the eigenvectors that are used to construct the eigenfunctions.


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Basic Examples  (2)

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

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

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Scope  (12)

Options  (4)

Properties & Relations  (2)

Possible Issues  (7)

Introduced in 2015