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

   https://doi.org/10.1088/1742-6596/3145/1/012005  

   Papanicolaou, N C; Christov, I C (November 2025, Journal of Physics: Conference Series)

   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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2. Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green’s function, and intermediate asymptotics for a finite thin film with elastic resistance

   https://doi.org/10.1007/s10665-024-10409-4  

   Papanicolaou, Nectarios C; Christov, Ivan C (February 2025, Journal of Engineering Mathematics)

   A linear sixth-order partial differential equation (PDE) of “parabolic” type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order eigenvalue problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed trough, and the eigenfunctions form a complete orthonormal set. Using these eigenfunctions, we derive the Green’s function for the governing sixth-order PDE on a finite interval and compare it to the known infinite-line solution. Further, we propose a Galerkin spectral method based on the constructed sixth-order eigenfunctions and their derivative expansions. The system of ordinary differential equations for the time-dependent expansion coefficients is solved by standard numerical methods. The numerical approach is applied to versions of the governing PDE with a second-order spatial derivative (in addition to the sixth-order one), which arises from gravity acting on the film. In the absence of gravity, we demonstrate the self-similar intermediate asymptotics of initially localized disturbances on the film surface, at least until the disturbances “feel” the finite boundaries, and show that the derived Green’s function is an attractor for such solutions. In the presence of gravity, we use the proposed Galerkin numerical method to demonstrate that self-similar behavior persists, albeit for shortened intervals of time, even for large values of the gravity-to-bending ratio. 

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3. Spreading and leveling of finite thin films with elastic interfaces: An eigenfunction expansion approach

   https://doi.org/10.1088/1742-6596/2910/1/012032  

   Papanicolaou, N C; Christov, I C (December 2024, Journal of Physics: Conference Series)

   The flow of a thin viscous liquid layer under an elastic film arises in natural processes, such as magmatic intrusions between rock strata, and industrial applications, such as coating surfaces with cured polymeric films. We study the linear dynamics of small perturbations to the equilibrium state of the film in a closed trough (finite film). Specifically, we are interested in the spreading (early-time) and leveling (late-time) dynamics as the film adjusts to equilibrium, starting from different initial perturbations. We consider both smooth and non-smooth spatially symmetric and localized initial conditions (perturbations). We find the exact series solutions for the film height, using the sixth-order complete orthonormal eigenfunctions associated with the posed initial-boundary-value problem derived in our previous work [Papanicolaou N C and Christov I C 2023J. Phys.: Conf. Ser.2675012016]. We show that the evolution of the perturbations begins with the spreading of the localized perturbations, followed by their mutual interaction as they spread, and finally, interactions with the confining lateral boundaries of the domain as the perturbations level. In particular, we highlight how the leading eigenvalues of the problem determine the scalings of certain figures of merit with time. 

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4. Series Solutions for Clamped Peridynamic Beams Using Fourth-Order Eigenfunctions

   https://doi.org/10.1007/978-3-031-75626-9_34  

   Wang, Ziyu; Christov, Ivan C (June 2025, Springer Nature Switzerland)

   Altenbach, Holm; Eremeyev, Victor A (Ed.)

   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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5. A non-trivial PT-symmetric continuum Hamiltonian and its eigenstates and eigenvalues

   https://doi.org/10.1063/5.0096250  

   Mead, Lawrence R.; Lee, Sungwook; Garfinkle, David (July 2022, Journal of Mathematical Physics)

   In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, C, for this system. All calculations are performed analytically and without approximation. 

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