Eigenfunction expansions for sixth-order boundary value problems arising in elastic-plated thin-film dynamics

Thin-film flows with a surface that has elastic bending resistance are governed by a sixth-order parabolic long-wave equation for the film height. Linearizing for small deflections about the equilibrium film height leads to a sixth-order boundary value problem (BVP). We discuss the boundary conditions (BCs) under which such sixth-order BVPs relevant to thin-film dynamics are self-adjoint. For a particular set of BCs, corresponding to an elastic-plated thin film in a closed trough, we explicitly derive a complete set of odd and even orthonormal eigenfunctions, which resemble trigonometric sines and cosines, as well as the so-called ``beam'' functions. Further, we derive explicitly the formulae for expressing derivatives of these eigenfunctions back into the same basis. Based on these novel, explicitly-constructed eigenfunctions and their derivative expansions, we propose a Galerkin spectral method for sixth-order BVPs relevant to thin-film dynamics. Importantly, due to the higher-order nature of the BVP, the coefficients of the spectral series decay rapidly in an algebraic manner, making the proposed expansions a highly-efficient computational tool. The proposed Galerkin spectral method and its convergence are demonstrated by solving model sixth-order problems.