Partial Differential Equation Toolbox 

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.
The basic scalar equation of the toolbox is the elliptic PDE
where is the vector , and c is a 2by2 matrix function on , the bounded planar domain of interest. c, a, and f can be complex valued functions of x and y.
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.
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.