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Numerical Solution of the Fokker-Planck Equation for Postharvest Applications
Published by the American Society of Agricultural and Biological Engineers, St. Joseph, Michigan www.asabe.org
Citation: Paper number 033099, 2003 ASAE Annual Meeting . (doi: 10.13031/2013.14055) @2003Authors: Nico Scheerlinck, Ann Peirs, Michèle Desmet, Sofie Clauwers, Bart M. Nicolaï
Keywords: Postharvest technology, Fokker-Planck equation, stochastic models, numerical methods, Monte Carlo method
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.