$\def\R{\mathbb R}\def\lam{\lambda}\def\Om{\Omega}\def\pa{\partial}$ $\def\huga#1{\begin{gather} #1 \end{gather}}$ $\newcommand{\spr}[1]{\left\langle #1 \right\rangle}$ $\def\om{\omega}\newcommand{\reff}[1]{(\ref{#1})}$ $\def\er{{\rm e}}\def\ri{{\rm i}}\def\C{\mathbb C}$ $\newcommand{\bce}{\begin{center}}\newcommand{\ece}{\end{center}}$

pde2path - a Matlab package for continuation and bifurcation
in 2D elliptic systems


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Webpages of the old version pde2path 1.0, outdated! We strongly recommend to switch to pde2path 2.*


The system and the boundary conditions

pde2path treats PDE systems \begin{equation}\label{gform} G(u,\lam):=-\nabla\cdot(c\otimes\nabla u)+a u-b\otimes\nabla u-f=0, \end{equation} where $u=u(x)\in\R^N$, $x\in\Omega\subset\R^2$ some bounded domain, $\lam\in\R$ is a parameter, and $c(\cdot)\in\R^{N\times N\times 2\times 2}$, $b(\cdot)\in\R^{N\times N\times 2}$, $a(\cdot)\in\R^{N\times N}$ and $f(\cdot)\in\R^N$ can depend on $x,u,\nabla u$, and, of course, parameters.
The boundary conditions are \begin{align}\label{e:gnbc} {\bf n}\cdot (c \otimes\nabla u) + q u = g, \end{align} where ${\bf n}$ is the outer normal and again $q(\cdot)\in \R^{N\times N}$ and $g(\cdot)\in \R^N$ may depend on $x$, $u$, $\nabla u$ and parameters.
The meanings of $au$ and $f$ are standard, and the notations $\nabla\cdot(c\otimes\nabla u)$, $b\otimes\nabla u$ and ${\bf n}\cdot (c \otimes\nabla u)$ mean the following:
c-tensor. The $i^{{\rm th}}$ component of $\nabla\cdot(c\otimes\nabla u)$ is given by $$ [\nabla\cdot(c\otimes\nabla u)]_i:= \sum_{j=1}^N [\pa_xc_{ij11}\pa_x+\pa_x c_{ij12}\pa_y+\pa_yc_{ij21}\pa_x +\pa_yc_{ij22}\pa_y]u_j\,. $$ b-tensor. The $i^{{\rm th}}$ component of of $b\otimes\nabla u$ is given by $$ [b\otimes\nabla u]_i:=\sum_{j=1}^N [b_{ij1}\pa_x+b_{ij2}\pa_y]u_j. $$ Essentially, we added these advection terms to the pdetoolbox for the efficient evaluation of Jacobians.

Flux normal component. Finally, $i^{{\rm th}}$ component of ${\bf n}\cdot (c \otimes\nabla u)$ is given by $$ [{\bf n}\cdot (c \otimes\nabla u)]_i=\sum_{j=1}^N [n_1(c_{ij11}\pa_x+c_{ij12}\pa_y)+n_2(c_{ij21}\pa_x+c_{ij22}\pa_y)]u_j, $$ where ${\bf n}=(n_1,n_2)$.

See Demos for the implementation of \eqref{gform} and \eqref{e:gnbc} for pde2path for a number of examples.