A Meshfree Maximum Entropy Method for the Solution of the Population Balance Equation

In this work, the number density function in the population balance equation (PBE) is approximated in terms of field nodes through a complete set of orthogonal basis functions in a semi-logarithmic space. We proposed the functional values at these field nodes to satisfy the maximum entropy solution. This hybridization of function approximation and information theories based on Shannon Maximum Entropy principle, allowed us to construct a sequence of positive continuous approximations of the PBE. The Lagrange multipliers, which result from the maximization of the Shannon entropy subject to the available average information, was estimated by solving a well-conditioned linear system of algebraic equations. As an application, this meshfree solution of the PBE is validated using an analytical solution of the microbial cell dynamics in a constant abiotic environment with simultaneous cell growth and division for which the analytical solution was derived by using the Adomian method.