# DEigenvalues

DEigenvalues[[u[x,y,],u,{x,y,}Ω,n]

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

DEigenvalues[eqns,u,t,{x,y,}Ω,n]

gives the eigenvalues for solutions u of the time-dependent differential equations eqns.

# Details and Options

• DEigenvalues can compute eigenvalues for ordinary and partial differential operators with given boundary conditions.
• DEigenvalues gives a list {λ1,,λn} of the n smallest magnitude eigenvalues λi.
• An eigenvalue and eigenfunction pair {λi,ui} for the differential operator satisfy [ui[x,y,]]==λi ui[x,y,].
• 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 the boundary Ω, then this is equivalent to specifying a Neumann 0 condition.
• The equations eqns are specified as in DSolve.
• N[DEigenvalues[]] calls NDEigenvalues for eigenvalues that cannot be computed symbolically.
• The Assumptions option can be used to specify assumptions on parameters.

# Examples

open allclose all

## Basic Examples(2)

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

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

## Scope(17)

### 1D(8)

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 5 smallest eigenvalues in an interval:

Specify a Laplacian operator:

Specify homogeneous Neumann boundary conditions:

Find the 5 smallest eigenvalues in an interval:

This is equivalent:

Specify a Laplacian operator:

Specify a homogeneous Dirichlet boundary condition:

Specify a homogeneous Neumann boundary condition:

Find the 5 smallest eigenvalues in an interval:

Find the eigenvalue:

Find symbolic expressions for the eigenvalues of a Laplace operator:

Specify a Laplacian operator:

Specify a homogeneous Dirichlet boundary condition:

Specify a homogeneous nonzero Neumann boundary condition:

Find the 5 smallest eigenvalues in an interval:

Specify an Airy operator:

Specify a homogeneous Dirichlet boundary condition:

Find the 3 smallest eigenvalues in an interval:

The eigenvalues are roots of a transcendental equation:

Numerical approximations for the eigenvalues:

Specify an Airy operator:

Specify a homogeneous Neumann boundary condition:

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

The eigenvalues are roots of a transcendental equation:

Specify a heat equation with homogeneous Dirichlet boundary conditions:

Find the 4 smallest eigenvalues:

### 2D(5)

Specify a Laplacian operator with homogeneous Dirichlet boundary conditions:

Find the 9 smallest eigenvalues in a rectangle:

Specify a Laplacian operator with homogeneous Neumann boundary conditions:

Find the 4 smallest eigenvalues in a rectangle:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 4 smallest eigenvalues of the operator in a unit disk:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 6 smallest eigenvalues of the operator in a triangle:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 4 smallest eigenvalues in a sector of a disk:

### 3D(4)

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 7 smallest eigenvalues in a cuboid:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 5 smallest eigenvalues in a cylinder:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 7 smallest eigenvalues in a ball:

Specify a Laplacian operator:

Specify homogeneous Dirichlet boundary conditions:

Find the 7 smallest eigenvalues in a prism:

## Properties & Relations(3)

Use NDEigenvalues to find numerical eigenvalues and eigenvectors:

Exact eigenvalues:

Numerical eigenvalues:

Use DEigensystem to find the eigensystem for a differential operator:

Find the eigenvalues:

Find the eigensystem:

Apply N[DEigenvalues[] to invoke NDEigenvalues if symbolic evaluation fails:

## Possible Issues(2)

Inhomogeneous Dirichlet conditions are replaced with homogeneous ones:

The same result:

Inhomogeneous Neumann values are replaced with homogeneous ones:

The same result:

Wolfram Research (2015), DEigenvalues, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigenvalues.html.

#### Text

Wolfram Research (2015), DEigenvalues, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigenvalues.html.

#### CMS

Wolfram Language. 2015. "DEigenvalues." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DEigenvalues.html.

#### APA

Wolfram Language. (2015). DEigenvalues. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DEigenvalues.html

#### BibTeX

@misc{reference.wolfram_2023_deigenvalues, author="Wolfram Research", title="{DEigenvalues}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/DEigenvalues.html}", note=[Accessed: 02-October-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_deigenvalues, organization={Wolfram Research}, title={DEigenvalues}, year={2015}, url={https://reference.wolfram.com/language/ref/DEigenvalues.html}, note=[Accessed: 02-October-2023 ]}