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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.
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This content will become publicly available on June 18, 2026
Series Solutions for Clamped Peridynamic Beams Using Fourth-Order Eigenfunctions
We propose an analytical approach to solving nonlocal generalizations of the Euler–Bernoulli beam. Specifically, we consider a version of the governing equation recently derived under the theory of peridynamics. We focus on the clamped–clamped case, employing the natural eigenfunctions of the fourth derivative subject to these boundary conditions. Static solutions under different loading conditions are obtained as series in these eigenfunctions. To demonstrate the utility of our proposed approach, we contrast the series solution in terms of fourth-order eigenfunctions to the previously obtained Fourier sine series solution. Our findings reveal that the series in fourth-order eigenfunctions achieve a given error tolerance (with respect to a reference solution) with ten times fewer terms than the sine series. The high level of accuracy of the fourth-order eigenfunction expansion is due to the fact that its expansion coefficients decay rapidly with the number of terms in the series, one order faster than the Fourier series in our examples.
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- Award ID(s):
- 2245343
- PAR ID:
- 10612274
- Editor(s):
- Altenbach, Holm; Eremeyev, Victor A
- Publisher / Repository:
- Springer Nature Switzerland
- Date Published:
- ISBN:
- 978-3-031-75626-9
- Page Range / eLocation ID:
- 661-678
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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