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Partial Differential Equation Toolbox™ solves scalar equations of the form

$$m\frac{{\partial}^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla \xb7\left(c\nabla u\right)+au=f$$

and eigenvalue equations of the form

$$\begin{array}{l}-\nabla \xb7\left(c\nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla \xb7\left(c\nabla u\right)+au={\lambda}^{2}mu\end{array}$$

For scalar PDEs, there are two choices of boundary conditions for each edge or face:

Dirichlet — On the edge or face, the solution

*u*satisfies the equation*hu*=*r*,where

*h*and*r*can be functions of space (*x*,*y*, and, in 3-D case,*z*), the solution*u*, and time. Often, you take*h*= 1, and set*r*to the appropriate value.Generalized Neumann boundary conditions — On the edge or face the solution

*u*satisfies the equation$$\overrightarrow{n}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(c\nabla u\right)+qu=g$$

$$\overrightarrow{n}$$ is the outward unit normal.

*q*and*g*are functions defined on ∂Ω, and can be functions of*x*,*y*, and, in 3-D case,*z*, the solution*u*, and, for time-dependent equations, time.

The toolbox also solves systems of equations of the form

$$m\frac{{\partial}^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla \xb7\left(c\otimes \nabla u\right)+au=f$$

and eigenvalue systems of the form

$$\begin{array}{l}-\nabla \xb7\left(c\otimes \nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla \xb7\left(c\otimes \nabla u\right)+au={\lambda}^{2}mu\end{array}$$

A system of PDEs with *N* components is *N* coupled
PDEs with coupled boundary conditions. Scalar PDEs are those with *N* = 1, meaning just
one PDE. Systems of PDEs generally means *N* > 1. The documentation
sometimes refers to systems as multidimensional PDEs or as PDEs with
a vector solution *u*. In all cases, PDE systems
have a single geometry and mesh. It is only *N*,
the number of equations, that can vary.

The coefficients *m*, *d*, *c*, *a*,
and *f* can be functions of location (*x*, *y*,
and, in 3-D, *z*), and, except for eigenvalue problems,
they also can be functions of the solution *u* or
its gradient. For eigenvalue problems, the coefficients cannot depend
on the solution `u`

or its gradient.

For scalar equations, all the coefficients except *c* are
scalar. The coefficient *c* represents a 2-by-2 matrix
in 2-D geometry, or a 3-by-3 matrix in 3-D geometry. For systems of *N* equations,
the coefficients **m**, **d**, and **a** are *N*-by-*N* matrices, **f** is an *N*-by-1 vector, and **c** is a 2*N*-by-2*N* tensor
(2-D geometry) or a 3*N*-by-3*N* tensor
(3-D geometry). For the meaning of $$c\otimes u$$,
see c Coefficient for specifyCoefficients.

When both *m* and *d* are `0`

,
the PDE is stationary. When either *m* or *d* are
nonzero, the problem is time-dependent. When any coefficient depends
on the solution *u* or its gradient, the problem
is called nonlinear.

For systems of PDEs, there are generalized versions of the Dirichlet and Neumann boundary conditions:

**hu**=**r**represents a matrix**h**multiplying the solution vector**u**, and equaling the vector**r**.$$n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)+qu=g$$. For 2-D systems, the notation $$n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$$ means the

*N*-by-1 matrix with (*i*,1)-component$$\sum _{j=1}^{N}\left(\mathrm{cos}(\alpha ){c}_{i,j,1,1}\frac{\partial}{\partial x}+\mathrm{cos}(\alpha ){c}_{i,j,1,2}\frac{\partial}{\partial y}+\mathrm{sin}(\alpha ){c}_{i,j,2,1}\frac{\partial}{\partial x}+\mathrm{sin}(\alpha ){c}_{i,j,2,2}\frac{\partial}{\partial y}\right)\text{\hspace{0.17em}}}{u}_{j$$

where the outward normal vector of the boundary $$n=\left(\mathrm{cos}(\alpha ),\mathrm{sin}(\alpha )\right)$$.

For 3-D systems, the notation $$n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$$ means the

*N*-by-1 vector with (*i*,1)-component$$\begin{array}{l}{\displaystyle \sum _{j=1}^{N}\left(\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,1}\frac{\partial}{\partial x}+\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,2}\frac{\partial}{\partial y}+\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,3}\frac{\partial}{\partial z}\right){u}_{j}}\\ +{\displaystyle \sum _{j=1}^{N}\left(\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,1}\frac{\partial}{\partial x}+\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,2}\frac{\partial}{\partial y}+\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,3}\frac{\partial}{\partial z}\right){u}_{j}}\\ +{\displaystyle \sum _{j=1}^{N}\left(\mathrm{cos}\left(\theta \right){c}_{i,j,3,1}\frac{\partial}{\partial x}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,2}\frac{\partial}{\partial y}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,3}\frac{\partial}{\partial z}\right){u}_{j}}\end{array}$$

where the outward normal vector of the boundary $$n=\left(\mathrm{sin}(\phi )\mathrm{cos}(\theta ),\mathrm{sin}(\phi )\mathrm{sin}(\theta ),\mathrm{cos}(\phi )\right)$$.

For each edge or face segment, there are a total of

*N*boundary conditions.