<p>In this paper, we propose a modified Bhatnagar-Gross-Krook (MBGK) model for the Boltzmann equation. Within the framework of the MBGK-type model, we address the classical problem of gas flow along a flat solid wall. This problem is reduced to a nonlinear system of integral equations. By employing the trivial solution of the original BGK model corresponding to the equilibrium Maxwellian distribution, we propose an approximate scheme for solving the resulting system. As a result, the problem is transformed into the sequential solution of a system of uncoupled, conservative, nonlinear integral equations for the gas density and average mass velocity. We prove a theorem on the existence of a solution within the space of positive and bounded functions. A bounded and physically meaningful positive solution is obtained as a consequence of the proposed MBGK-type model. We derive a uniform estimate between the exact and approximate solutions, where the error tends to zero with decreasing geometrical progression rate. Finally, based on a constructive existence theorem, an extensive set of numerical simulations is carried out. The obtained uniform estimate between the exact and the approximate solutions allows one to compute the solution numerically with arbitrary prescribed accuracy.</p>