This work is an extension of the authors’ earlier investigations into a nonlinear theory of propagation of impurities in solids. Two mathematical models governed by partial differential equations are studied in this paper. Both models describe the dynamics of the impurities’ propagation in a semi-infinite prismatic solid with an impenetrable side surface. The difference between the models is associated with the impurity source that is located at the boundary of the solid. In the first case the source is assumed to be present during a finite time interval, whereas in the second case the impurity source is assumed to be stationary and provide a constant impurity flow. An approach is presented in this paper that allows to transform the nonlinear partial differential equations into nonlinear ordinary differential equations and derive solutions to the latter. A methodology is presented for the formulation of the initial conditions for the nonlinear ordinary differential equations and criteria are formulated for evaluation of the correctness of those conditions. The obtained solutions provide information regarding the time-dependent concentration of the impurities and their finite propagation speed. It is also shown in the paper that the proposed solution approach is also efficient in application to the linear theory of diffusion.