An exact solution of delay-differential equations in association models

Abstract

In modeling automatic engines, like in physiological or biological systems or ecology dynamics, oftently is necessary to include delay effects in the equations. This effect is related to reaction or transfer times and can be extended to the spatial case, for example, in cases such as the influence in local green biomass density due to dispersal of seeds. Spatial delay effects are present in liquid mixtures models such as the Cummings-Stell model (CSM) for associating molecules: in a series of publications they solve, for particular cases, an equation with spatial delay that must be satisfied by an auxiliary (Baxter's) function. In this paper, we present an analytical and general solution of the first order Delay-Differential Equation (or Differential-Difference Equation) DDE, for the auxiliary Baxter's function that appears in the CSM. A $n$-partition of the domain leaves a set of DDE's defined in the subintervals. We use recursive properties of these auxiliary functions and a matrix composed by differential and shift operators (MDSO) in order to obtain the solution of the original problem with an arbitrary value of $n$. The problem of solving spatial DDE's is common to other models of associative fluids, such as homogeneous and inhomogeneous mixtures of sticky shielded hard spheres, or models of chemical ion association and dipolar dumbbells and polymers. In all the cases the location of the potential, $L = m \sigma / n$, has different physical effects.