Non-self-adjoint sixth-order eigenvalue problems arising from clamped elastic thin films on closed domains

TL;DR

This paper develops a spectral method using biorthogonal eigenfunctions to solve non-self-adjoint sixth-order boundary value problems from elastic thin film models, achieving high accuracy and rapid convergence.

Contribution

It introduces a novel Petrov--Galerkin spectral method based on biorthogonal eigenfunctions for sixth-order problems with non-self-adjoint operators, including explicit formulas and validation.

Findings

Spectral method achieves rapid convergence exceeding sixth-order rate.

Eigenfunctions form biorthogonal sets for series expansion.

Method accurately solves model problems with exact solutions.

Abstract

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. We consider the case in which the elastic interface is clamped at the lateral walls of a closed trough and thus encloses a finite amount of fluid. For a slender film undergoing infinitesimal deformations, the displacement of the elastic surface from its initial equilibrium position obeys a sixth-order (in space) initial boundary value problem (IBVP). To solve this IBVP, we construct a set of odd and even eigenfunctions that intrinsically satisfy the boundary conditions (BCs) of the original IBVP. These eigenfunctions are the solutions of a non-self-adjoint sixth-order eigenvalue problem (EVP). To use the eigenfunctions for series expansions, we also construct and solve the adjoint EVP, leading to another set of even and odd eigenfunctions, which are orthogonal to the original set (biorthogonal). The eigenvalues of the adjoint EVP are the same as those of the original EVP, and we find accurate asymptotic formulas for them. Next, employing the biorthogonal sets of eigenfunctions, a Petrov--Galerkin spectral method for sixth-order problems is proposed, which can also handle lower-order terms in the IBVP. The proposed method is tested on two model sixth-order BVPs, which admit exact solutions. We explicitly derive all the necessary formulas for expanding the quantities that appear in the model problems into the set(s) of eigenfunctions. For both model problems, we find that the approximate Petrov--Galerkin spectral solution is in excellent agreement with the exact solution. The convergence rate of the spectral series is rapid, exceeding the expected sixth-order algebraic rate.