Group Analysis of the One-Dimensional Boltzmann Equation: IV. Complete Group Classification in the General Case

Abstract

We consider the one-dimensional Boltzmann equation \(f_t+cf_x+(\mathcal{F}f)_c=0\) with a function \(\mathcal{F}\) depending on (t,x,c,f) and obtain the complete group classification of such equations in the class of point changes of whole set of variables (t,x,c,f). for this, we impose additional conditions on the transformations for the invariance of (a) the relations dx = c dt and \(dc=\mathcal{F}dt\), (b) the lines dt = dx = 0, and (c) the form f dx dc, which fix the physical meaning of the used variables and the relations between them.