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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.

## Apps

 PDE Modeler Solve partial differential equations in 2-D regions

## Topics

### Programmatic Workflow

Eigenvalues and Eigenmodes of L-Shaped Membrane

Use command-line functions to find the eigenvalues and the corresponding eigenmodes of an L-shaped membrane.

Eigenvalues and Eigenmodes of Square

Use command-line functions to find the eigenvalues and eigenmodes of a square domain.

Vibration of Circular Membrane

Find vibration modes of a circular membrane.

Vibration of Square Plate

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

### PDE Modeler App Workflow

Eigenvalues and Eigenmodes of L-Shaped Membrane: PDE Modeler App

Use the PDE Modeler app to find the eigenvalues and the corresponding eigenmodes of an L-shaped membrane.

Eigenvalues and Eigenmodes of Square: PDE Modeler App

Use the PDE Modeler app to find the eigenvalues and eigenmodes of a square domain.

L-Shaped Membrane with Rounded Corner

Find the eigenvalues for an L-shaped membrane with the rounded inner corner.

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