Eigenvalue Problem Research Articles

Front propagation near the onset of instability

We describe the resulting spatiotemporal dynamics when a homogeneous equilibrium loses stability in a spatially extended system. More precisely, we consider reaction-diffusion systems, assuming only that the reaction kinetics undergo a transcritical, saddle-node, or supercritical pitchfork bifurcation as a parameter passes through zero. We construct traveling front solutions which describe the invasion of the now-unstable state by a nearby stable state. We show that these fronts are marginally spectrally stable near the bifurcation point, which, together with recent advances in the theory of front propagation into unstable states, establishes that these fronts govern the dynamics of localized perturbation to the unstable state. Our proofs are based on functional analytic tools to study the existence and eigenvalue problems for fronts, which become singularly perturbed after a natural rescaling.

Zero exclusion sectors for polynomials whose coefficients exhibit a +/-/+ sign pattern

The zeros of polynomials whose coefficients exhibit a +/-/+ sign pattern are sometimes excluded from sectors in the complex plane. It is shown how to obtain such sectors, as well as the number of zeros that lie between the sectors, along with bounds on their magnitudes. The results are illustrated with an application to polynomial eigenvalue problems.

New Results on a Nonlocal Sturm–Liouville Eigenvalue Problem with Fractional Integrals and Fractional Derivatives

In this paper, we investigate the eigenvalue properties of a nonlocal Sturm–Liouville equation involving fractional integrals and fractional derivatives under different boundary conditions. Based on these properties, we obtained the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville problem with a non-Dirichlet boundary condition. Furthermore, we discussed the continuous dependence of the eigenvalues on the potential function for a nonlocal Sturm–Liouville equation under a Dirichlet boundary condition.

A note on the failure of the Faber-Krahn inequality for the vector Laplacian

We consider a natural eigenvalue problem for the vector Laplacian related to stationary Maxwell's equations in a cavity and we prove that an analog of the celebrated Faber-Krahn inequality doesn't hold, regardless of whether a volume or perimeter constraint is applied.

Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation

Abstract The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper, we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus.

Riemannian acceleration with preconditioning for symmetric eigenvalue problems

Riemannian acceleration with preconditioning for symmetric eigenvalue problems

Experimental investigation of the second-mode internal solitary wave in continuous pycnocline and the applicability of weakly nonlinear theoretical models

The research on the propagation and evolution of the second-mode internal solitary waves(ISWs) is receiving more and more attention. In this study, second-mode internal solitary waves in continuous stratification are physically simulated in a laboratory-stratified fluid flume. Meanwhile, the second-mode ISWs and their induced flow field in the same stratification environment are solved based on the eigenvalue problem of the TG equation (Taylor-Goldstein), combined with the weakly non-linear ISW theoretical models. The experimental and theoretical results show that the symmetry of the second-mode ISW wave-flow field can be improved as the thickness ratio of the upper fluid layer and lower one approaches 1. The ISW speed and horizontal and vertical velocity range values in the continuous pycnocline are positively correlated with the changing ISW amplitude, while only the wavelength is negatively correlated with the iSW amplitude. The waveflow fields of the second-mode ISWs calculated by Korteweg-de Vries (KdV) and extended KdV (eKdV) models in the large amplitude cases are more consistent with the experimental results than those in the small amplitude cases. The two theoretical models used to describe second-mode ISWs can be significantly improved when the thickness ratio of the upper and lower fluid layers approaches 1. In this case, the eKdV model is more applicable than the KdV model.

Cluster perturbation theory. XI. Excitation-energy series using a variational excitation-energy function.

Traditionally, excitation energies in coupled-cluster (CC) theory have been calculated by solving the CC Jacobian eigenvalue equation. However, based on our recent work [Jørgensen et al., Sci. Adv. 10, eadn3454 (2024)], we propose a reformulation of the calculation of excitation energies where excitation energies are determined as a conventional molecular property. To this end, we introduce an excitation-energy function that depends on the CC Jacobian and the right and left eigenvectors for the Jacobian eigenvalue problem. This excitation-energy function is variational with respect to the right and left eigenvectors but not with respect to the cluster amplitudes. Instead, the cluster amplitudes satisfy the cluster-amplitude equations, and we set up an excitation-energy Lagrangian by adding to the excitation-energy function the cluster-amplitude equations with an undetermined multiplier for each cluster-amplitude constraint. The excitation-energy Lagrangian is variational in all its parameters. Based on the variational property of the Lagrangian, we have determined two quadratically convergent excitation-energy series: the total-order cluster-perturbation (tCP) and variational cluster-perturbation (vCP) excitation-energy series. Calculations of the excitation energies of three small molecules have shown that the vCP series is to be preferred over the tCP series. The test calculations have been carried out for CPS(D) expansions [targeting the CC singles-and-doubles (CCSD) wave function from the CC singles wave function] and the CPSD(T) expansion [targeting the CC singles-doubles-triples (CCSDT) wave function from the CCSD wave function]. For the S(D) and SD(T) orbital excitation space calculations, we obtain in the second vCP iteration excitation energies with a mean deviation from CCSD excitation energies of about 0.04eV for the S(D) orbital spaces, and for the SD(T) orbital space calculation, we obtain a mean deviation from the CCSDT excitation energies of 0.001eV.

Single-Source Localization as an Eigenvalue Problem

Single-Source Localization as an Eigenvalue Problem

Heat transfer in the entrance region of annular laminar flow with constant and different heat flux on two walls

AbstractHeat transfer differs in the regions where the flow is developed and developing thermally. These regions can be differentiated by using the thermal entry length. Many researchers have presented correlations to determine the thermal entry length for natural and forced convection. In this study, heat transfer in the entrance region of a concentric annuli is investigated. It is accepted that beginning from the inlet of annuli the flow is developed hydrodynamically and it is developing thermally. Heat transfer is investigated where the internal or external surfaces of the annuli are at constant but different heat fluxes. The fluid velocity is assumed to be constant or radially variable. Due to thermal boundary conditions, one thermal boundary layer appears on the outer cylinder surface, another on the inner cylinder surface. The edge of two boundary layers will be adiabatic and naturally, the temperature of fluid between the two edges will be equal to free stream temperature. Transformation, Separation of Variables method, eigenvalue problem, Sturm‐Liouville system, Bessel differential equation and properties of orthogonal functions are used in solution of the problem. Exact and analytical solutions of the momentum and energy equations are presented. Velocity and temperature distributions, local Nusselt numbers and convection heat transfer coefficients are calculated for the internal and external surfaces of annuli.

The role of symmetry in solving large-scale eigenvalue problems: Applications to quantum mechanics and vibrational analysis

The role of symmetry in solving large-scale eigenvalue problems: Applications to quantum mechanics and vibrational analysis

New insights into the Riesz space fractional variational problems and Euler–Lagrange equations

In this paper, we investigate the solvability of a constrained variational problem with a Lagrangian dependent on the Riesz–Caputo derivative. Our approach leverages the direct method in the calculus of variations and the theory of fractional calculus. The main objective of this study is to establish a compactness property of the Riesz fractional integral operator, which enables us to discover extremum points of the constrained fractional variational problem without imposing the convexity condition on the fractional operator variable of the associated Lagrangians. Following this, we derive the Euler–Lagrange equations in their weak form, highlighting their significance in determining minimizers of the variational problem. Finally, we explore a compelling application of fractional variational calculus, specifically examining the intriguing relationship between the fractional Sturm–Liouville eigenvalue problem and constrained fractional variational problems. Our findings provide a new perspective on the solvability of constrained fractional variational problems and offer insights into the application of the direct method in such problems.

Modeling dispersion of circumferential waves in underwater targets with spectral methods.

The dispersion of circumferential waves propagating around cylindrical and spherical underwater targets with an arbitrary number of elastic and fluid layers is modeled using the spectral collocation method. The underlying differential equations are discretized by Chebyshev interpolation and the corresponding differentiation matrices, and the calculation of the dispersion curves is transformed into a generalized eigenvalue problem. Furthermore, for targets in infinite fluid, the perfect matched layer is used to emulate the Sommerfeld radiation condition. For solid targets, a transformation of potential functions, along with the corresponding boundary condition, is introduced to eliminate the singularity of the low-order modes at the origin. Numerical results are presented and compared with results obtained by the winding number integral method to verify the accuracy and efficiency of the approach.

A Posteriori Analysis of the Mixed Finite Element Method for General Second-Order Elliptic Eigenvalue Problems

This paper realizes a posteriori analysis for the lowest-order Raviart-Thomas mixed finite element approximation of a class of general second-order elliptic eigenvalue problems. Based on the superconvergence results between the discrete eigenfunctions and the interpolant of eigenfunction, we derive a residual-type error indicator. This error indicator is fully computable and has a clear expression on each element, we verify its reliability and efficiency. Furthermore, we implement adaptive computation, and numerical experiments demonstrate the robustness of the method.

An efficient algorithm for the eigenvalue problem of a Hermitian quaternion matrix in quantum chemistry

An efficient algorithm for the eigenvalue problem of a Hermitian quaternion matrix in quantum chemistry

Local Discontinuous Adaptive Finite Element Method for Steklov Eigenvalue Problems

Using the flexibility of the finite element method to solve the solution problems on different shaped and natured elements, the local discontinuous Galerkin method can handle very complex boundary problems. Using the local discontinuous Galerkin method to perform a priori error estimation for the Steklov eigenvalue problem, we obtain a reasonable error estimation subspace, which can effectively solve the validity and reliability of the eigenfunction indicator subspace and the reliability of the eigenvalue error estimation indicator. We use precise numerical data obtained from MATLAB experiments as the basis for judging whether the conclusion is reasonable. Finally, combining theoretical analysis, we show that the method achieves optimal convergence order.

Axial Vibration of a Viscoelastic FG Nanobeam with Arbitrary Boundary Conditions

ObjectiveThis study investigates the axial vibration of a viscoelastic functionally graded (FG) nanobeam under deformable boundary conditions for the first time. The primary focus is on exploring the effects of damping and scale parameters on the dynamic behavior of the nanobeam.MethodsThe governing equation of the viscoelastic FG nanobeam is formulated by incorporating nonlocal elasticity theory and the Kelvin-Voigt viscoelastic model. The Fourier sine series is chosen as the axial displacement function, with higher-order derivatives obtained using Stokes transforms. The Fourier coefficient is determined through the governing equation and incorporated into the deformable boundary conditions. The resulting eigenvalue problem provides solutions for both rigid and constrained general boundary conditions.ConclusionsThe study presents solutions for various boundary conditions, comparing the results with existing literature. The analysis reveals significant findings, including the observation that damping has a greater influence on fundamental frequencies in higher modes, and that the impact of damping decreases as the nonlocal scale parameter increases. These findings are presented through tables and graphs to highlight the effects of damping and scale parameters on the dynamic behavior of the nanobeam.

An analysis of the Sturm-Liouville eigenvalue problem of a cylinder between two spheres of equal radius

An analysis of the Sturm-Liouville eigenvalue problem of a cylinder between two spheres of equal radius

The Two-Grid Scheme of the Interior Penalty Discontinuous Galerkin Finite Element Method for Second-Order Elliptic Eigenvalue Problems

This paper investigates a class of second-order elliptic eigenvalue problems with variable coefficients and proposes a two-grid discretization scheme based on shifted inverse iteration. First, the corresponding interior penalty discontinuous Galerkin (IPDG) finite element formulation is presented, and the existence and uniqueness of the weak solution for this method are theoretically proven. Second, an a priori error estimate is derived, followed by an error analysis for the proposed scheme. Finally, numerical experiments are conducted to validate the effectiveness of the proposed algorithm.

Solving coupled differential eigenvalue problems using the differential transformation method numerical example: Dynamic analysis of multi-span beams

Solving coupled differential eigenvalue problems using the differential transformation method numerical example: Dynamic analysis of multi-span beams