A class of partial differential equations is considered that is directly connected with the transport equation. It is shown that if the initial-boundary conditions are specified on a given net as univariate quadratic splines, then there exists a bivariate quadratic spline unique on the net that satisfies exactly the initial boundary conditions and satisfies the differential equation at the nodes of the net. The spline is then constructed by an exact finite-difference scheme. As a first application we provide a new algorithm for a spherically symmetric problem in neutron transport theory. This is further illustrated by numerical examples.