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The method of successive approximations is used to show the existence of a unique solution to a model of a nonlinear Boltzmann equation under the homogeneous boundary and typical initial conditions. An explicit formula in terms of the prescribed functions for the calculation of an approximate solution and its error estimate are given. This formula reveals an interesting analogy between the initial-boundary value problem of the Boltzmann equation and the Cauchy problem for ordinary differentialdoi:10.1090/s0002-9947-1974-0347294-8 fatcat:6mngpjnj4jbmnic76ziomgzary