## Equations You Can Solve Using Partial Differential Equation Toolbox

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.

### Coefficients

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.

### Boundary Conditions

For a system of PDEs, the generalized version of the Dirichlet boundary condition is
**hu** = **r**. The
condition represents the matrix **h** multiplying the
solution vector **u**, and equaling the vector **r**.

For a system of PDEs, the generalized version of the Neumann boundary condition is $$n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)+qu=g$$. For example, in case of circumferential and spherical boundaries, the generalized versions of the Neumann boundary condition are as follows:

If the boundary is a circumference (2-D case), the outward normal vector of the boundary of the boundary is given by $$n=\left(\mathrm{cos}(\phi ),\mathrm{sin}(\phi )\right)$$, the notation $$n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$$ means the

*N*-by-1 vector, for which the (*i*,1)-component is as follows:$$\sum _{j=1}^{N}\left(\mathrm{cos}(\phi ){c}_{i,j,1,1}\frac{\partial}{\partial x}+\mathrm{cos}(\phi ){c}_{i,j,1,2}\frac{\partial}{\partial y}+\mathrm{sin}(\phi ){c}_{i,j,2,1}\frac{\partial}{\partial x}+\mathrm{sin}(\phi ){c}_{i,j,2,2}\frac{\partial}{\partial y}\right)\text{\hspace{0.17em}}}{u}_{j$$

If the boundary is a spherical surface (3-D case), than the outward normal vector of the boundary is given by $$n=\left(\mathrm{sin}(\theta )\mathrm{cos}(\phi ),\mathrm{sin}(\theta )\mathrm{sin}(\phi ),\mathrm{cos}(\theta )\right)$$, and the notation $$n\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$$ means the

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

For each edge or face segment, there are a total of *N* boundary
conditions.