<p>Flow through the configuration of a three-layer channel containing a transition porous layer of variable permeability is revisited in this work to illustrate how Brinkman’s equation, which governs the flow in the transition layer, is reduced to the classic inhomogeneous differential equations of Airy and Weber, with constant forcing terms. Solutions to the homogeneous parts of these equations are given in terms of Airy’s and Weber’s special functions. Particular solutions to the resulting inhomogeneous equations give rise to three Nield-Kuznetsov functions of the first kind, which are analyzed in this work and extended to obtain general solutions to equations with variable forcing terms whose general solutions are expressed in terms of three Nield-Kuznetsov functions of the second kind. Methods of evaluation of all resulting special functions are presented.</p>