Analytical insights into a fractional thin-film equation: Exact solutions and dynamics

Abstract

In this work, we examine a quadratic thin film equation with a constant negative absorption term. This equation extends a broad variety of the famous scalar reaction-diffusion equations appearing in nonlinear sciences and is derived from the estimations of lubrication theory to represent thin films of a Newtonian liquid dominated by surface tension effects. It is typically used to describe the behavior of light when it interacts with thin films, such as coatings on lenses or mirrors. The connection between thin film equations and optical quantum mechanics lies in the microscopic interactions between photons and the electrons in the thin film material. Employing the invariant subspace approach, we obtain explicit fractional exact solutions for the time-fractional case of the model containing the Riemann-Liouville derivative operator. Furthermore, we illustrate 3-D and 2-D plots of the obtained exact solutions for a better understanding of the physical phenomena.