I am trying to solve Boltzmann eqn:  $\frac{e}{m}\tau\vec{E}_x\cdot \nabla_\vec{v}{f(\vec{v})}=f(\vec{v})-f_0(\vec{v})$where  $f_0(v)=\exp{\frac{-m \vec{v}^2}{T}}$ and  $\vec{v}=\{\vec{v}_x,\vec{v}_y\}$ (i.e., $\vec{v}$ is in 2D.)  For simplicity, let's assume $\vec{v}_y=0$.  Assume $\tau$ and $T$ are constants. Then for any given $\vec{v}$ and $|E|$, $f_0(\vec{v})$ is completely known.I am trying to set up a self-consistent code in Matlab to solve this equation for any given set of $\vec{v}$. However, I am not sure how to get Matlab to solve this? the ODE command, ode45, is not exactly for this task. Any ideas?EDIT: I need to do a simulation, so I will choose the values of $\vec{v}$ and do the simulation accordingly. All other constants are assumed known ($e,m,\tau,T$ and $|E|$).
Self-Consistent solution of Boltzmann eqn. using Matlab
matlab