# Documentation

## Systems in the PDE App

You can enter coefficients for a system with N = `2` equations in the PDE app, see Systems of PDEs. To do so, open the PDE app and select `Generic System`.

Then select PDE > PDE Specification.

Enter string expressions for coefficients using the form in Coefficients for Scalar PDEs in PDE App, with additional options for nonlinear equations. The additional options are:

• Represent the `i`th component of the solution `u` using `'u(i)'` for `i` = 1 or 2.

• Similarly, represent the `i`th components of the gradients of the solution `u` using `'ux(i)'` and `'uy(i)'` for `i` = 1 or 2.

 Note:   For elliptic problems, when you include coefficients `u(i)`, `ux(i)`, or `uy(i)`, you must use the nonlinear solver. Select Solve > Parameters > Use nonlinear solver.

Do not use quotes or unnecessary spaces in your entries.

For higher-dimensional systems, do not use the PDE app. Represent your problem coefficients at the command line.

You can enter scalars into the `c` matrix, corresponding to these equations:

$\begin{array}{c}-\nabla ·\left({c}_{11}\nabla {u}_{1}\right)-\nabla ·\left({c}_{12}\nabla {u}_{2}\right)+{a}_{11}{u}_{1}+{a}_{12}{u}_{2}={f}_{1}\\ -\nabla ·\left({c}_{21}\nabla {u}_{1}\right)-\nabla ·\left({c}_{22}\nabla {u}_{2}\right)+{a}_{21}{u}_{1}+{a}_{22}{u}_{2}={f}_{2}.\end{array}$

If you need matrix versions of any of the `cij` coefficients, enter expressions separated by spaces. You can give 1-, 2-, 3-, or 4-element matrix expressions. These mean:

• 1-element expression: $\left(\begin{array}{cc}c& 0\\ 0& c\end{array}\right)$

• 2-element expression: $\left(\begin{array}{cc}c\left(1\right)& 0\\ 0& c\left(2\right)\end{array}\right)$

• 3-element expression: $\left(\begin{array}{cc}c\left(1\right)& c\left(2\right)\\ c\left(2\right)& c\left(3\right)\end{array}\right)$

• 4-element expression: $\left(\begin{array}{cc}c\left(1\right)& c\left(3\right)\\ c\left(2\right)& c\left(4\right)\end{array}\right)$

For details, see c Coefficient for Systems.

For example, these expressions show one of each type (1-, 2-, 3-, and 4-element expressions)

These expressions correspond to the equations

$\begin{array}{c}-\nabla ·\left(\left(\begin{array}{cc}4+\mathrm{cos}\left(xy\right)& 0\\ 0& 4+\mathrm{cos}\left(xy\right)\end{array}\right)\nabla {u}_{1}\right)-\nabla ·\left(\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\nabla {u}_{2}\right)=1\\ -\nabla ·\left(\left(\begin{array}{cc}.1& .2\\ .2& .3\end{array}\right)\nabla {u}_{1}\right)-\nabla ·\left(\left(\begin{array}{cc}7& .6\\ .5& \mathrm{exp}\left(x-y\right)\end{array}\right)\nabla {u}_{2}\right)=2.\end{array}$