Wolfram Language & System 11.0 (2016)|Legacy Documentation

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NDEigenvalues

NDEigenvalues[[u[x,y,]],u,{x,y,}Ω,n]
gives the n smallest magnitude eigenvalues for the linear differential operator over the region Ω.

NDEigenvalues[{1[u[x,y,],v[x,y,],],2[u[x,y,],v[x,y,],],},{u,v,},{x,y,}Ω,n]
gives eigenvalues for the coupled differential operators {op1,op2,} over the region Ω.

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

Details and OptionsDetails 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:
  • "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 of the Laplacian operator on [0,π]:

In[1]:=
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Compute the first 6 eigenvalues for a circular membrane with the edges clamped:

In[1]:=
Click for copyable input
Out[1]=
Introduced in 2015
(10.2)