The calculation of the transient voltage and current along a nonuniform LCRG transmission line of finite length is considered. The model equations form a system of first order coupled differential equations for the voltage and current, and it is solved by a combination of a wave splitting technique and an invariant imbedding approach as well as a Green's functions approach. Partial differential equations for the reflection kernel, transmission kernel and the Green's functions are derived. Results from two different types of splittings are compared. Numerical results for the direct and also for particular inverse problems are presented. Finally it is noted that the system of equations obtained in a special case corresponds to Maxwell's equations for plane wave propagation in a slab where the permittivity, conductivity, and permeability vary with the depth.