You can enter coefficients for a system with N = 2
equations
in the PDE app. To do so, open the PDE 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 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 |
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:
$$\begin{array}{c}-\nabla \xb7\left({c}_{11}\nabla {u}_{1}\right)-\nabla \xb7\left({c}_{12}\nabla {u}_{2}\right)+{a}_{11}{u}_{1}+{a}_{12}{u}_{2}={f}_{1}\\ -\nabla \xb7\left({c}_{21}\nabla {u}_{1}\right)-\nabla \xb7\left({c}_{22}\nabla {u}_{2}\right)+{a}_{21}{u}_{1}+{a}_{22}{u}_{2}={f}_{2}\end{array}$$
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
$$\begin{array}{c}-\nabla \xb7\left(\left(\begin{array}{cc}4+\mathrm{cos}(xy)& 0\\ 0& 4+\mathrm{cos}(xy)\end{array}\right)\nabla {u}_{1}\right)-\nabla \xb7\left(\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\nabla {u}_{2}\right)=1\\ -\nabla \xb7\left(\left(\begin{array}{cc}.1& .2\\ .2& .3\end{array}\right)\nabla {u}_{1}\right)-\nabla \xb7\left(\left(\begin{array}{cc}7& .6\\ .5& \mathrm{exp}(x-y)\end{array}\right)\nabla {u}_{2}\right)=2\end{array}$$