Orthonormal eigenfunction expansions for sixth-order boundary value problems

TL;DR

This paper introduces a spectral Galerkin method using orthonormal eigenfunction expansions tailored for sixth-order boundary value problems, particularly applicable to thin-film flow models with elastic surface resistance.

Contribution

It derives a complete set of orthonormal eigenfunctions satisfying relevant boundary conditions and applies them in a spectral method for sixth-order BVPs, demonstrating convergence.

Findings

Eigenfunctions intrinsically satisfy boundary conditions

Spectral method achieves convergence on model problems

Provides a systematic approach for sixth-order BVPs

Abstract

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. To solve such problems, we first derive a complete set of odd and even orthonormal eigenfunctions -- resembling trigonometric sines and cosines, as well as the so-called ``beam'' functions. These functions intrinsically satisfy boundary conditions (BCs) of relevance to thin-film flows, since they are the solutions of a self-adjoint sixth-order Sturm--Liouville BVP with the same BCs. Next, we propose a Galerkin spectral approach for sixth-order problems; namely the sought function as well as all its derivatives and terms appearing in the differential equation are expanded into an infinite series with respect to the derived complete orthonormal (CON) set of eigenfunctions. The unknown coefficients in the series expansion are determined by solving the algebraic system derived by taking successive inner products with each member of the CON set of eigenfunctions. The proposed method and its convergence are demonstrated by solving two model sixth-order BVPs.