5.1.1 Example 1

The equations to solve are the following on the unit square in 2D.\begin{align*} \frac{\partial v\left ( x,y,t\right ) }{\partial t} & =D\nabla ^{2}v+\left ( a-v\right ) \left ( v-1\right ) v-w+I\\ \frac{\partial w\left ( x,y,t\right ) }{\partial t} & =\epsilon \left ( v-\gamma w\right ) \end{align*}

Using \(a=0.1,\gamma =2,\epsilon =0.005,D=5\times 10^{-5},I=0,\) hence the PDE’s are\begin{align*} \frac{\partial v\left ( x,y,t\right ) }{\partial t} & =\left ( 5\times 10^{-5}\right ) \nabla ^{2}v+\left ( 0.1-v\right ) \left ( v-1\right ) v-w\\ \frac{\partial w\left ( x,y,t\right ) }{\partial t} & =0.005\left ( v-2w\right ) \end{align*}

Initial conditions, \(t=0\)\begin{align*} v\left ( x,y,0\right ) & =\exp \left ( -100\left ( x^{2}+y^{2}\right ) \right ) \\ w\left ( x,y,0\right ) & =0 \end{align*}

Boundary conditions are homogeneous Neumann for \(v\). (I solved this numerically, fractional step method. ADI for the diffusion solve).