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You can enter coefficients for a system with *N* = `2`

equations in the PDE Modeler app. To do so, open the PDE Modeler app and select
`Generic System`

.

Then select **PDE > PDE Specification**.

Enter character expressions for coefficients using the form in Coefficients for Scalar PDEs in PDE Modeler 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.

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 Modeler app. Represent your problem coefficients at the command line.

You can enter scalars into the `c`

matrix,
corresponding to these equations:

$$\begin{array}{c}-\nabla \xb7\left({c}_{11}\nabla {u}_{1}\right)-\nabla \xb7\left({c}_{12}\nabla {u}_{2}\right)+{a}_{11}{u}_{1}+{a}_{12}{u}_{2}={f}_{1}\\ -\nabla \xb7\left({c}_{21}\nabla {u}_{1}\right)-\nabla \xb7\left({c}_{22}\nabla {u}_{2}\right)+{a}_{21}{u}_{1}+{a}_{22}{u}_{2}={f}_{2}\end{array}$$

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

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

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

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

These expressions correspond to the equations

$$\begin{array}{c}-\nabla \xb7\left(\left(\begin{array}{cc}4+\mathrm{cos}(xy)& 0\\ 0& 4+\mathrm{cos}(xy)\end{array}\right)\nabla {u}_{1}\right)-\nabla \xb7\left(\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\nabla {u}_{2}\right)=1\\ -\nabla \xb7\left(\left(\begin{array}{cc}.1& .2\\ .2& .3\end{array}\right)\nabla {u}_{1}\right)-\nabla \xb7\left(\left(\begin{array}{cc}7& .6\\ .5& \mathrm{exp}(x-y)\end{array}\right)\nabla {u}_{2}\right)=2\end{array}$$

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