Investigation of generalised Fredholm equations and their direct solution subject to constraints

Certain generalisations of the Bolzmann equation have been developed to study the mesoscopic dynamics of systems that are much more complex than simple gases [1, 2, 3]. In a recent study of one such system [4] linked to polymer dynamics in micromoulding, an application of the Chapman-Enskog expansion was implemented. This method produces a linear integral equation, itself a generalisation of the Fredholm equation of the second kind, that is usually solved by ad hoc methods, in part by appealing to constraints that appear to restrict the equation in a countably infinite number of degrees of freedom. A conclusion that may be reached is that the unconstrained Fredholm-like equation has an infinite dimensional solution space. Whether or not this is the case, no general solution method for Fredholm-like equations with infinite dimensional solution spaces was available and the ad hoc method usually used was not applicable in our case. Here we develop a method for determining a solution