VisualPDE solves systems of PDEs that look like generalised reaction–diffusion equations. It can do this in 1D or 2D.

The simplest type of system is just a single PDE in a single unknown, $u$,

$\pd{u}{t} = \vnabla \cdot (D_u \vnabla u) + f_u,$

where $D_u$ and $f_u$ are functions of $u$, $t$, and space that you can choose. For example, if $f_u=0$ and $D_u$ is a constant, you have the heat equation.

The most complicated type is a coupled system of PDEs in four unknowns, $u$, $v$, $w$ and $q$:

\begin{aligned} t_u\pd{u}{t} &= \vnabla \cdot(D_{uu}\vnabla u+D_{uv}\vnabla v+D_{uw}\vnabla w+D_{uq}\vnabla q) + f_u,\\ \text{one of}\left\{\begin{matrix}\displaystyle t_v\pd{v}{t} \\ v\end{matrix}\right. & \begin{aligned} &= \vnabla \cdot(D_{vu}\vnabla u+D_{vv}\vnabla v+D_{vw}\vnabla w+D_{vq}\vnabla q) + f_v \vphantomt_v\pd{v}{t}}, \\ &= \vnabla \cdot(D_{vu}\vnabla u+D_{vw}\vnabla w+D_{vq}\vnabla q) + f_v, \end{aligned}\\ \text{one of}\left\{\begin{matrix}\displaystyle t_w\pd{w}{t} \\ w\end{matrix}\right. & \begin{aligned} &= \vnabla \cdot(D_{wu}\vnabla u+D_{wv}\vnabla v+D_{ww}\vnabla w+D_{wq}\vnabla q) + f_w \vphantomt_w\pd{w}{t}}, \\ &= \vnabla \cdot(D_{wu}\vnabla u+D_{wv}\vnabla v+D_{wq}\vnabla q) + f_w, \end{aligned}\\ \text{one of}\left\{\begin{matrix}\displaystyle t_q\pd{q}{t} \\ q\end{matrix}\right. & \begin{aligned} &= \vnabla \cdot(D_{qu}\vnabla u+D_{qv}\vnabla v+D_{qw}\vnabla w+D_{qq}\vnabla q) + f_q \vphantomt_q\pd{q}{t}}, \\ &= \vnabla \cdot(D_{qu}\vnabla u+D_{qv}\vnabla v+D_{qw}\vnabla w) + f_q, \end{aligned} \end{aligned

where the diffusion coefficients ($D_{uu}$ etc.), the timescales ($t_u$ etc.) and the interaction/kinetic terms ($f$, $g$, $h$, $j$) can depend on the unknowns, space, and time. In matrix form, we can summarise this by saying we solve systems of the form

$\m{M} \pd{\v{u}}{t} = \vnabla\cdot(\m{D}\vnabla\v{u}) + \v{f},$

where

• $\v{u}$ is a vector of one, two, three or four unknowns,
• $\m{M}$ is a diagonal matrix with potentially some zeros on the diagonal; you might know this as a ‘mass matrix’,
• $\m{D}$ is a possibly non-constant matrix that may contain zeros; you might know this as a ‘diffusion tensor’,
• $\v{f}$ is a vector of one, two, three or four components that contains our interaction or kinetic terms.

VisualPDE allows you to easily change the number of components and the boundary conditions. You can set initial conditions just by clicking the screen.