Graph representation for asymptotic expansion in homogenisation of nonlinear first-order equations

Abstract

Homogenisation of a linear transport equation leads to an integro-differential equation with the differential part of the same type as the starting equation. The (non-periodic) homogenisation of semilinear transport equations is open. In order to pinpoint technical difficulties, as a first step in that direction, following the approach of Tartar we consider an ordinary differential equation with an oscillating coefficient a $$\left\{\begin{array}{ll} u' +au^2 &=f \\ u(0) &=v \end{array}\right. $$ instead, and expand the solution in terms of a small parameter (the amplitude of oscillations in a). The crucial observation we made is a correspondence between multiple integrals representing the terms in asymptotic expansion of the solution and certain graphs, which allows easy manipulation of otherwise highly complicated expressions, and leads to efficient computation of the terms in expansion.