  In the present work the classical problem of the kinetic theory of gases (the
Smoluchowsky' problem about temperature jump in rarefied gas) is considered.
The rarefied gas fills half-space over a flat firm surface. logarithmic
gradient of temperature is set far from surface. The kinetic equation with
modelling integral of collisions in the form of BGK-model (Bhatnagar, Gross and
Krook) is used. The general mirror-diffuse boundary conditions of molecules
reflexions of gas from a wall on border of half-space (Maxwell conditions) are
considered. Expanding distribution function on two orthogonal directions in
space of velocities, the Smoluchowsky' problem to the solution of the
homogeneous vector one-dimensional and one-velocity kinetic equation with a
matrix kernel is reduced. Then generalization of source-method is used and
boundary conditions include in non-homogeneous vector kinetic equation. The
solution in the form of Fourier integral is searched. The problem is reduced to
the solution of vector Fredholm integral equation of the second sort with
matrix kernel. The solution of Fredholm equation in the form of Neumann's
polynoms with vector coefficients is searched. The system vector algebraic
interengaged equations turns out. The solution of this system is under
construction in the form of Neumann's polynoms. Comparison with well-known
Barichello - Siewert' high-exact results is made. Zero and the first approach
of jumps of temperature and numerical density are received. It is shown, that
transition from the zero to the first approach raises 10 times accuracy in
calculation coefficients of temperature and concentration jump.
