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Eigensolutions of linear PDEs

Partial Differential Equation Toolbox™ solves eigenvalue equations of the form

$$\begin{array}{l}-\nabla \xb7\left(c\nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla \xb7\left(c\nabla u\right)+au={\lambda}^{2}mu\end{array}$$

and eigenvalue systems of the form

$$\begin{array}{l}-\nabla \xb7\left(c\otimes \nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla \xb7\left(c\otimes \nabla u\right)+au={\lambda}^{2}mu\end{array}$$

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.

PDE | Solve partial differential equations in 2-D regions |

**Vibration Of a Circular Membrane Using The MATLAB eigs Function**

Find vibration modes of a circular membrane.

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.

**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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