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# Partial Differential Equation Toolbox

## Defining and Solving PDEs

With the Partial Differential Equation Toolbox, you can define and numerically solve different types of PDEs, including elliptic, parabolic, hyperbolic, eigenvalue, nonlinear elliptic, and systems of PDEs with multiple variables.

### Elliptic PDE

The basic scalar equation of the toolbox is the elliptic PDE

where is the vector , and c is a 2-by-2 matrix function on , the bounded planar domain of interest. c, a, and f can be complex valued functions of x and y.

### Parabolic, Hyperbolic, and Eigenvalue PDEs

The toolbox can also handle the parabolic PDE

the hyperbolic PDE

and the eigenvalue PDE

where d is a complex valued function on and is the eigenvalue. For parabolic and hyperbolic PDEs, c, a, f, and d can be complex valued functions of x, y, and t.

### Nonlinear Elliptic PDE

A nonlinear Newton solver is available for the nonlinear elliptic PDE

where the coefficients defining c, a, and f can be functions of x, y, and the unknown solution u. All solvers can handle the PDE system with multiple dependent variables

You can handle systems of dimension two from the PDE app. An arbitrary number of dimensions can be handled from the command line. The toolbox also provides an adaptive mesh refinement algorithm for elliptic and nonlinear elliptic PDE problems.