Abstract Transient analysis in the past of finite axially dispersed systems with or without a rate process has been subject to the use of the Danckwerts boundar conditions. We present here a formulation which suggests a general set of boundary conditions from which the Danckwerts conditions arise as a special case. The resulting boundary value problems neatly fit the mold of self-adjoint operator theory. An integral transform approach is outlined for those not familiar with operator theory. Discussion of actual solutions to different cases is deferred to two other papers which appear as Parts II and III.