Analytical solution of the vector model kinetic equations with constant kernel and their applications

Exact solutions are obtained for the first time for the half-space boundary-value problem for the vector model kinetic equations
$$\begin {gathered} \mu \frac {\partial }{\partial x}\Psi (x,\mu )+\Sigma \Psi (x,\mu )=C\int _{-\infty }^{\infty }\exp \left (-{\mu '}^2\right )\Psi (x,\mu ')\,d\mu ',\\ \lim _{x\to 0+}\Psi (x,\mu )=\Psi _0(\mu ),\qquad \mu >0,\\ \lim _{x\to +\infty }\Psi (x,\mu )=A,\qquad \mu <0, \end {gathered} $$
is obtained. Here $x>0$, $\mu \in (-\infty ,0)\cup (0,+\infty )$, $\Sigma =\operatorname {diag}\{\sigma _1,\sigma _2\}$, $C=\left [c_{ij}\right ]$ – $2\times 2$-matrix, $\Psi (x,\mu )$ is vector-column with elements $\psi _1$ and $\psi _2$. As an application, an exact solution is obtained for the first time to the problem of the diffusion slip of a binary gas for a model Boltzmann equation with collision operator in the form proposed by MacCormack.