摘要
Biochemical oxygen demand (BOD) is one of the key variables used to evaluate the quality of river water and in recent years great efforts have been made to model BOD dynamics in a probabilistic way. Different approaches have been proposed and several sources of randomness have been considered. We focus on the case in which the uncertainty arises due to both the initial condition and to point inputs of BOD along the river. No constraints are assumed about the probability distribution functions of the initial condition and inputs; moreover, both non-linear and linear decay reactions of the BOD along the stream are considered. Under these conditions, a stochastic differential approach of the point-process type is proposed and a general semi-analytical solution for the evolution of the BOD probability distribution is obtained. In particular, when the decay reaction is linear a simple analytical solution is deduced, and analytical expressions for the evolution of the moments is given. Some examples are presented in the last part of the work, which would suggest the usefulness of the proposed solution whenever the uncertainty of the inputs and initial conditions are predominant with respect to the randomness in the decay reaction.