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# Eigenvalue Problems

Eigensolutions of linear PDEs

Partial Differential Equation Toolbox™ solves eigenvalue equations of the form

$\begin{array}{l}-\nabla ·\left(c\nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla ·\left(c\nabla u\right)+au={\lambda }^{2}mu\end{array}$

and eigenvalue systems of the form

$\begin{array}{l}-\nabla ·\left(c\otimes \nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla ·\left(c\otimes \nabla u\right)+au={\lambda }^{2}mu\end{array}$

Here, λ is an unknown complex number. The coefficients m, d, c, a, and f cannot depend on the solution `u` or its gradient.

In solid mechanics, an eigenvalue problem is associated with wave phenomena describing, for example, the natural modes of a vibrating membrane. In quantum mechanics, λ is the energy level of a bound state in the potential well a(x), where x represents a 2-D or 3-D point.

For details, see Eigenvalue Equations.

## Apps

 PDE Solve partial differential equations in 2-D regions

## Topics

### Programmatic Workflow

Vibration Of a Circular Membrane Using The MATLAB eigs Function

Find vibration modes of a circular membrane.

Vibration of a Square Plate

Calculate the vibration modes and frequencies of a 3-D simply supported, square, elastic plate.

Eigenvalues and Eigenfunctions for the L-Shaped Membrane

Find the eigenvalues and the corresponding eigenfunctions of an L-shaped membrane using the PDE app and the command line.

Eigenvalues and Eigenmodes of a Square

Find the eigenvalues and eigenmodes of a square domain using the PDE app and the command line.

### PDE App Workflow

Eigenvalues and Eigenfunctions for the L-Shaped Membrane

Find the eigenvalues and the corresponding eigenfunctions of an L-shaped membrane using the PDE app and the command line.

L-Shaped Membrane with a Rounded Corner

Find the eigenvalues for an L-shaped membrane where the inner corner at the "knee" is rounded.

Eigenvalues and Eigenmodes of a Square

Find the eigenvalues and eigenmodes of a square domain using the PDE app and the command line.

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