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 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.

Examples

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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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