Nico Scheerlinck

Numerical solution of the Fokker-Planck equation for postharvest applications

In recent years many mathematical models have been developed to predict the change of fruit quality attributes such as firmness during maturation and cold storage. Instead of modelling the full chain of underlying biochemical processes, typically models of minimum complexity are developed to describe the basic dynamics of fruit quality attributes by means of a limited number of state variables. However, these models are deterministic and do not take into account the large natural variability of fruit quality attributes. In this research the evolution of fruit quality attributes was represented by means of a system of differential equations in which the initial conditions and the model parameters were specified as random variables together with their probability density functions. A fundamental approach from stochastic systems theory is used to compute the propagation of these probability density functions through a dynamic model for apple firmness during maturation. This approach considers the solution of the Fokker-Planck or forward diffusion equation. Because the Fokker-Planck equation is a first order partial differential equation of hyperbolic type, usually the solution can not be derived in closed form. Numerical tools are developed to discretise the computational domain and are based on special finite difference schemes. The main features of this stochastic approach were demonstrated for the time evolution of soluble solids content during apple maturation.