Computational Physics Lectures: Partial differential equations

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Famous PDEs, general equation in two dimensions

A general partial differential equation with two given dimensions reads

$\begin{equation*} A(x,y)\frac{\partial^2 u}{\partial x^2}+B(x,y)\frac{\partial^2 u}{\partial x\partial y} +C(x,y)\frac{\partial^2 u}{\partial y^2}=F(x,y,u,\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}), \end{equation*}$ and if we set

$\begin{equation*} B=C=0, \end{equation*}$ we recover the

$1+1$ -dimensional diffusion equation which is an example of a so-called parabolic partial differential equation. With

$\begin{equation*} B=0, \hspace{1cm} AC < 0 \end{equation*}$ we get the

$2+1$ -dim wave equation which is an example of a so-called elliptic PDE, where more generally we have

$B^2 > AC$ . For

$B^2 < AC$ we obtain a so-called hyperbolic PDE, with the Laplace equation in Eq. (3) as one of the classical examples. These equations can all be easily extended to non-linear partial differential equations and

$3+1$ dimensional cases.