Eigenvalue Problem Research Articles

Frequency Regulation in Nanocomposite Sandwich Joined Conical-Conical Shells with Open Cell Foam Partially Supported by Pasternak Elastic Foundation Using GDQ

This study investigated the natural vibration of a sandwich conical-conical shell that is partially supported by Winkler–Pasternak foundations. The sandwich system comprises an open-cell foam (OCF) core and laminated composite face sheets reinforced with graphene platelets using functionally graded models (FG-GPLRC). To account for through-the-thickness shear deformations and rotary inertias, the first-order theory of shells is combined with Donnell-type kinematic assumptions. Hamilton’s principle is utilized to establish the general motion equations and related boundary and continuity conditions. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature (GDQ) approach. An eigenvalue problem is formulated to analyze the corresponding mode shapes and vibration frequencies for the shell ends and continuity criteria under various boundary conditions. Parametric experiments are conducted for sandwich conical-conical shells partially supported by Winkler–Pasternak foundations. This study examined the impact of multiple factors on the frequency of a joined shell structure, including the total thickness, FG-GPL face sheet thickness, various boundary conditions, the length of each cone segment, and the Winkler and shear elastic foundation. Notably, this research also analyzed the effects of a partial elastic foundation on the structure’s frequency for the first time.

Fast Eigenvalue Decomposition of Arrowhead and Diagonal-Plus-Rank-k Matrices of Quaternions

Quaternions are a non-commutative associative number system that extends complex numbers, first described by Hamilton in 1843. We present algorithms for solving the eigenvalue problem for arrowhead and DPRk (diagonal-plus-rank-k) matrices of quaternions. The algorithms use the Rayleigh Quotient Iteration with double shifts (RQIds), Wielandt’s deflation technique and the fact that each eigenvector can be computed in O(n) operations. The algorithms require O(n2) floating-point operations, n being the order of the matrix. The algorithms are backward stable in the standard sense and compare well to the standard QR method in terms of speed and accuracy. The algorithms are elegantly implemented in Julia, using its polymorphism feature.

Graphs with large multiplicity of −2 in the spectrum of the eccentricity matrix

The eccentricity matrix of a simple connected graph is obtained from the distance matrix by keeping only the entries that are largest in at least one of their row or column. This matrix can be seen as a counterpart to the standard adjacency matrix, since the latter one is also obtained from the distance matrix but this time by keeping only the entries equal to 1. It is known that, for λ∉{−1,0} and a fixed i∈N, when n passes N there is only a finite number of graphs with n vertices having λ as an eigenvalue of multiplicity n−i in the spectrum of the adjacency matrix. This phenomenon motivates us to consider graphs with large multiplicity of an eigenvalue of the eccentricity matrix. In this context, we determine all connected graphs with n vertices for which −2 has the multiplicity n−i, where i≤5. Infinite families of graphs with this spectral property are encountered. Results of this paper can be compared to results concerning graphs with large multiplicity of zero in the spectrum of the adjacency matrix; at present the graphs for which this multiplicity is n−i, where i≤4, are known. Our results also become meaningful in the framework of the median eigenvalue problem since, for sufficiently large n, the median eigenvalue of the obtained graphs is always −2.

‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme

A new interpretation and applications of the ‘Diophantine’ and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of 0 , 1 , … , N , constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the ‘Diophantine’ and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm ‘eigenvectors’ of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

Dynamical analysis of a diffusive population-toxicant model with toxicant-taxis in polluted aquatic environments

This paper deals with a diffusive population-toxicant model in polluted aquatic environments, with a toxicant-taxis term describing a toxicant-induced behavior change, that is, the population tends to move away from locations with high-level toxicants. The global existence of solutions is established by the techniques of the semigroup estimation and Moser iteration. Based on a detailed study on the properties of the principal eigenvalue for non-self-adjoint eigenvalue problems, we investigated the local and global stability of the toxin-only steady-state solution and the existence of positive steady state, which yields sufficient conditions that lead to population persistence or extinction. Finally, by numerical simulations, we studied the effects of some key parameters, such as toxicant-taxis coefficient, advection rate, and effect coefficient of the toxicant on population growth, on population persistence. Both numerical and analytical results show that a weak chemotaxis effect, a small advection rate of the population, and a weak effect of the toxicant on population growth are favorable for population persistence.

Semi-analytical solutions for forced and free vibration of damped fluid-conveying pipe systems based on complex modal superposition method

In this paper, the response solutions of the damped fluid-conveying pipe system with elastic torsion constraints at both ends are analyzed. The pipe system considering gyroscopic effect induced by internal flow and damping effect is a typical damped gyroscopic system. This system cannot be decoupled in the modal space by the traditional modal analysis, and then the semi-analytical response solutions cannot be obtained by the classical modal superposition method. In order to remedy this problem, this paper proposes a new method based on complex modal superposition method and state-space method to give the semi-analytical response solutions of the damped fluid-conveying pipe system. The adjoint system is introduced through the concept of adjoint operator, and the orthogonality conditions of the damped fluid-conveying pipe system are derived in the state-space by using the eigenvalue relationship between the original system and the adjoint system. The Laplace transformation method is used to solve the eigenvalue problems of the original system and the adjoint system, and the implicit function equations about the eigenvalues and analytical eigenfunctions are obtained. According to the obtained orthogonality conditions, the semi-analytical response solutions of the system under arbitrary initial conditions and excitations are given, and the obtained solutions include transient and steady-state response solutions. In the numerical discussion part, the reliability and accuracy of the present method are verified. This research has positive significance for the dynamic analysis of fluid-conveying pipe systems, and the obtained orthogonality conditions have potential value for the nonlinear dynamics research.

On the stability analysis of a restrained functionally graded nanobeam in an elastic matrix with neutral axis effects

Abstract In this work, a general eigenvalue solution of an arbitrarily constrained nonlocal strain gradient nanobeam made of functionally graded material is presented for the first time for the stability response by the effect of the Winkler foundation. Elastic springs at the ends of the nanobeam are considered in the formulation, which have not been considered in most studies. In order to analyze deformable boundary conditions, linear equation systems are derived in terms of infinite power series by using the Fourier sine series together with the Stokes’ transform. The higher-order force boundary conditions are used to obtain a coefficient matrix including different end conditions, power-law index, elastic medium, and small-scale parameters. A general eigenvalue problem of technical interest, associated with nonlocal strain gradient theory, is mathematically evaluated and presented in detail. Parametric results are obtained to investigate the effects of material length scale parameter, Winkler stiffness, power-law index, nonlocal parameter, and elastic springs at the ends. In addition, the effects of the other higher-order elasticity theories simplified from nonlocal strain gradient theory are also investigated and some benchmark results are presented.

Stability analysis and numerical simulations of the infection spread of epidemics as a reaction–diffusion model

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results.

Finite element analysis for free vibration of pipes conveying fluids–physical significance of complex mode shapes

A finite element formulation is presented for the natural vibration analysis of pipes conveying fluids. The solution of the resulting quadratic eigenvalue problem generally yields complex eigenvalues and eigenvectors. The present study then develops a robust mathematical procedure that combines the real and imaginary components of the eigenvectors to form physically attainable (i.e., real) mode shapes. The procedure yields a family of solutions that is more general than previously known solutions. The well-known classical mode shape is shown to be recoverable as a special case from the present solution. The study provides new insights on the effects of viscous damping, axial compressive force, and the flexibility of intermediate pipe supports on the response. Additionally, the study develops a novel algorithm based on Hermitian angles between eigenvectors to automate the tracing of mode evolution in the frequency-velocity plots.

The Bézier curve and neural network model of the time-domain transient signals

The discussion is for the memory of Oleg Alexandrovich Tretyakov. In the discussion, one characteristic of the signal transfer of Professor Tretyakov's Evolutionary Approach to the Electromagnetics method is presented. Theoretically, the problem of the signal generated by a time-domain signal in a waveguide is addressed. The theoretic propagation is realized-exemplified through the actual TEC (TECU) map estimating by the Bézier curve and neural network. The striking aspect of the segmented prototype established with the Bézier approach is its adaptability. This mechanical curve, which does not need any preliminary preparation, is framed on differential geometric invariants. In the essay, a time-dependent complete set of magnetic waveguide modes is remembered. While the Dirichlet and Neumann eigenvalue problems determine the vector functions of the modes, the behavior of the time-evolving amplitudes is given by the Klein-Gordon equation. Examples of current data are discussed with the TEC map for the 2017 year. While the actual fluctuation of the interpolated CODE TEC atlas is illustrated, the mechanical Bézier curve family (untouched before) and the neural network introduce time-domain estimations to the reader. The parametric curve approach governs the Bézier model. The curve, which is C0 class segmented continuously, models on its way with new hourly components every twelve hours. The network model employs the solar wind parameters for the TEC atlas estimation. The reliability and consistency of the models are exhibited by the R correlation ratio, absolute error, and mean squared error. As a result, the R coefficient of the curve and network models vary around 91.2% and 98.8%, respectively. One can note the error of the network model falls to 1.1308 TECU. The outcomes are compatible with the former discussions.

Global existence and aggregation of chemotaxis–fluid systems in dimension two

This paper investigates the coupled Patlak–Keller–Segel–Navier–Stokes (PKS–NS) system in two-dimensional bounded domains with a focus on global well-posedness and the formation of striking patterns, in which the boundary conditions are Neumann conditions for the cell density and the chemical concentration, and the Navier slip boundary condition with zero friction for the fluid velocity. We prove that the solution of the system exists globally in time in the case of subcritical mass. Concerning the critical mass case, we construct the boundary spot steady states rigorously via the inner-outer gluing method.While studying the global existence and the concentration phenomenon of the chemotaxis–fluid model, we develop the global W2,p theory for the 2D stationary Stokes system subject to Navier boundary conditions with zero friction and further establish semigroup estimates of the nonstationary counterpart by analyzing the Stokes eigenvalue problem.

Nonlinearity-induced topological phase transition characterized by the nonlinear Chern number

As first demonstrated by the characterization of the quantum Hall effect by the Chern number, topology provides a guiding principle to realize the robust properties of condensed-matter systems immune to the existence of disorder. The bulk–boundary correspondence guarantees the emergence of gapless boundary modes in a topological system whose bulk exhibits non-zero topological invariants. Although some recent studies have suggested a possible extension of the notion of topology to nonlinear systems, the nonlinear counterpart of a topological invariant has not yet been understood. Here we propose a nonlinear extension of the Chern number based on the nonlinear eigenvalue problems in two-dimensional systems and show the existence of bulk–boundary correspondence beyond the weakly nonlinear regime. Specifically, we find nonlinearity-induced topological phase transitions, in which the existence of topological edge modes depends on the amplitude of oscillatory modes. We propose and analyse a minimal model of a nonlinear Chern insulator whose exact bulk solutions are analytically obtained. The model exhibits the amplitude dependence of the nonlinear Chern number, for which we confirm the nonlinear extension of the bulk–boundary correspondence. Thus, our result reveals the existence of genuinely nonlinear topological phases that are adiabatically disconnected from the linear regime.

Elliptic p-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by−Δp1u1=−|u1|p1−2u1in Ω,−Δp2u2=−|u2|p2−2u2in Ω,|∇u1|p1−2∇u1⋅ν=g1(x,u1,u2)on ∂Ω,|∇u2|p2−2∇u2⋅ν=g2(x,u1,u2)on ∂Ω, where ν(x) is the outer unit normal of Ω at x∈∂Ω, Δpi denotes the pi-Laplacian and gi:∂Ω×R×R→R are Carathéodory functions that satisfy general growth and structure conditions for i=1,2. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of gi near zero related to the first eigenvalue of the pi-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, (g1,g2)=∇g with a smooth function (s1,s2)↦g(x,s1,s2). By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the pi-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.

Solutions of Surface-Wave Dispersion and Attenuation in Stratified Viscoelastic Media Using a Spectral-Element Approach

ABSTRACT Efficient and accurate calculation for the dispersion and attenuation of the surface waves in viscoelastic media is numerically challenging because the eigen wavenumbers are located in the complex domain. In this study, we propose a semianalytical spectral-element method (SASEM), which can determine the complex eigen wavenumbers by solving linear eigenvalue problems. By simplifying the structure of the eigenvalue problem, we significantly improve the calculation efficiency. The implementation of the frequency-dependent automatic discretization, semi-infinite element, and mode filter guarantees the correctness and accuracy of the modal solutions. Because no root-searching schemes are required, the root-skipping problem is naturally avoided. The numerical tests show that the SASEM can provide sufficiently accurate solutions with much less computation cost than traditional Muller’s method. Meanwhile, SASEM exhibits high flexibility when applied to media the parameters for which vary continuously with depth. To demonstrate the effectiveness of SASEM for complicated dispersion features, the dispersion curves and eigen wavefields of the viscoelastic media with a low-velocity layer are also analyzed. The results of numerical tests indicate the versatility, efficiency, and accuracy of our method. With further study, the proposed SASEM has the potential to become a promising tool for the investigation and retrieval of viscoelastic subsurface structures.

Numerical optimization of Neumann eigenvalues of domains in the sphere

This paper deals with the numerical optimization of the first three eigenvalues of the Laplace-Beltrami operator of domains in the Euclidean sphere of R3 with Neumann boundary conditions. We address two approaches: the first one is a generalization of the initial problem leading to a density method and the other one is a shape optimization procedure using the level-set method. The original goal of those methods was to investigate the conjecture according to which the geodesic ball was optimal for the first non-trivial eigenvalue under certain conditions. These computations provide strong insight into the optimal shapes of those eigenvalue problems and show a rich variety of geometries regarding the proportion of the surface area of the sphere occupied by the domain. In the last part, the same algorithms are used to carry out the same investigations on a torus.

Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces

This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.

A numerical assembly technique for free vibration of plane frames using a shifted and deflated Newton method

A semi-analytic method, namely the Numerical Assembly Technique (NAT) is extended to compute natural frequencies of arbitrary planar frame structures. The frame structure is systematically defined by a set of nodes, beams, bearings, springs, and external loads and the corresponding boundary and interface conditions are formulated. We consider the Timoshenko-Ehrenfest beam theory and take axial, bending and shear deformations as well as linear and rotational inertia into account. Since we use the exact solutions of the governing differential equations in the NAT, the accuracy of the results is independent from the spatial discretization. However, the resulting eigenvalue problem is non-linear and requires a numerical algorithm to solve. We consider three different representations of exact homogeneous solutions and study the numerical stability of the resulting system matrix. To solve for the eigenvalues we propose a Newton method enhanced with a shifted deflation algorithm. The stability and the efficiency of this approach is illustrated in numerical examples.

Conforming element method and shifted-inverse two-grid dicretization for the fourth-order Steklov eigenvalue problem

Conforming element method and shifted-inverse two-grid dicretization for the fourth-order Steklov eigenvalue problem

Finite element analysis of the Oseen eigenvalue problem

We propose and analyze a finite element method for the Oseen eigenvalue problem. This problem is an extension of the Stokes eigenvalue problem, where the presence of the convective term leads to a non-symmetric problem and hence, to complex eigenvalues and eigenfunctions. With the aid of the compact operators theory, we prove that for inf-sup stable finite elements the convergence holds and hence, error estimates for the eigenvalues and eigenfunctions are derived. We also propose an a posteriori error estimator which results to be reliable and efficient. We report a series of numerical tests in two and three dimension in order to assess the performance of the method and the proposed estimator.