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# Documentation

## Coefficients for Systems of PDEs

As Systems of PDEs describes, toolbox functions can address the case of systems of N PDEs. How do you represent the coefficients of your PDE in the correct form? In general, an elliptic system is

$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f,$

The notation $\nabla \cdot \left(c\otimes \nabla u\right)$ means the N-by-1 matrix with (i,1)-component

$\sum _{j=1}^{N}\left(\frac{\partial }{\partial x}{c}_{i,j,1,1}\frac{\partial }{\partial x}+\frac{\partial }{\partial x}{c}_{i,j,1,2}\frac{\partial }{\partial y}+\frac{\partial }{\partial y}{c}_{i,j,2,1}\frac{\partial }{\partial x}+\frac{\partial }{\partial y}{c}_{i,j,2,2}\frac{\partial }{\partial y}\right){u}_{j}$

Other problems with N > 1 are the parabolic system

$d\frac{\partial u}{\partial t}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f,$

the hyperbolic system

$d\frac{{\partial }^{2}u}{\partial {t}^{2}}-\nabla \cdot \left(c\otimes \nabla u\right)+au=f,$

and the eigenvalue system

$-\nabla \cdot \left(c\otimes \nabla u\right)+au=\lambda du.$

To solve a PDE using this toolbox, you convert your problem into one of the forms the toolbox accepts. Then express your problem coefficients in a form the toolbox accepts.

The question is how to express each coefficient: d, c, a, and f. For answers, see f for Systems, c for Systems, and a or d for Systems.

 Note:   If any coefficient depends on time or on the solution u or its gradient, then all coefficients should be NaN when either time or the solution u is NaN. This is the way that solvers check to see if the equation depends on time or on the solution.