Eigenvalue Problems For Equations Research Articles

Thick-restarted joint Lanczos bidiagonalization for the GSVD

The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov subspaces. We consider the joint Lanczos bidiagonalization method, and analyze the feasibility of adapting the thick restart technique that is being used successfully in the context of other linear algebra problems. Numerical experiments illustrate the effectiveness of the proposed method. We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance. The analysis is done using a parallel implementation in the SLEPc library.

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: regularity and error analysis

Abstract Stochastic partial differential equation (PDE) eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic crystal structures. In this paper we present a multilevel quasi-Monte Carlo (MLQMC) method for approximating the expectation of the minimal eigenvalue of an elliptic eigenvalue problem with coefficients that are given as a series expansion of countably-many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discretizations of the spatial domain and truncations of the dimension of the stochastic parameter domain. To approximate the expectations, randomly shifted lattice rules are employed. This paper is primarily dedicated to giving a rigorous analysis of the error of this algorithm. A key step in the error analysis requires bounds on the mixed derivatives of the eigenfunction with respect to both the stochastic and spatial variables simultaneously. Under stronger smoothness assumptions on the parametric dependence, our analysis also extends to multilevel higher-order quasi-Monte Carlo rules. An accompanying paper (Gilbert, A. D. & Scheichl, R. (2023) Multilevel quasi-Monte Carlo methods for random elliptic eigenvalue problems II: efficient algorithms and numerical results. IMA J. Numer. Anal.) focusses on practical extensions of the MLQMC algorithm to improve efficiency and presents numerical results.

Positive and nodal solutions for parametric superlinear weighted [formula omitted]-equations

We consider a nonlinear eigenvalue problem for equations driven by a weighted (p,q)-Laplacian and a superlinear reaction. We prove a global (with respect to the parameter λ>0) existence and multiplicity result. We also generate nodal (sign-changing) solutions. Finally, we determine the topological properties of the solution set and prove the continuity properties of the solution multifunction.

Investigation of the Free Vibrations of Radial Functionally Graded Circular Cylindrical Beams Based on Differential Quadrature Method

In the current research, an effective differential quadrature method (DQM) has been developed to solve natural frequency and vibration modal functions of circular section beams along radial functional gradient. Based on the high-order theory of transverse vibration of circular cross-section beams, lateral displacement equation was reconstructed neglecting circumferential shear stress. Two equations coupled with deflection and rotation angles were derived based on elastic mechanics theory and further simplified into a constant coefficient differential equation with natural frequency as eigenvalue. Then, differential quadrature method was applied to transform the eigenvalue problem of the derived differential equation into a set of algebraic equation eigenvalue problems. Natural frequencies of the free vibrations of cylindrical beams with circular cross-sections were calculated at one time, and corresponding modal functions were solved together. The obtained numerical results indicated that the natural frequencies of functionally graded (FG) circular cylindrical beams obtained using differential quadrature method agreed with the results reported in related literatures. In addition, influences of varying gradient parameters on the modal shapes of circular cylindrical beams were found to be strongly consistent with previous studies. Numerical results further validated the feasibility and accuracy of the developed differential quadrature method in solving the transverse vibration of FG circular cross-section beams.

ElATools: A tool for analyzing anisotropic elastic properties of the 2D and 3D materials

We introduce a computational method and a user-friendly code with a terminal-based graphical user interface (GUI), named ElATools, developed to analyze mechanical and anisotropic elastic properties. ElATools enables facile analysis of the second-order elastic stiffness tensor of two-dimensional (2D) and three-dimensional (3D) crystal systems. It computes and displays the main mechanical properties including the bulk modulus, Young's modulus, shear modulus, hardness, p-wave modulus, universal anisotropy index, Chung-Buessem anisotropy index, log-Euclidean anisotropy parameter, Cauchy pressures, Poisson's ratio, and Pugh's ratio, using three averaging schemes of Voigt, Reuss, and Hill. It includes an online and offline database from the Materials Project with more than 13,000 elastic stiffness constants for 3D materials. The program supports output files of the well-known computational codes IRelast, IRelast2D, ElaStic, and AELAS. Four types of plotting and visualization tools are integrated to conveniently interface with GNUPLOT, XMGRACE, view3dscene and plotly libraries, offering immediate post-processing of the results. ElATools provides reliable means to investigate the mechanical stability based on the calculation of six (three) eigenvalues of the elastic tensor in 3D (2D) materials. It can efficiently identify anomalous mechanical properties, such as negative linear compressibility, negative Poisson's ratio, and highly-anisotropic elastic modulus in 2D and 3D materials, which are central properties to design and develop high-performance nanoscale electromechanical devices. Moreover, ElATools can predict the behavior of the sound velocities and their anisotropic properties, such as acoustic phase/group velocities and power flow angles in materials, by solving the Christoffel equation. Six case studies on selected material systems, namely, ZnAu2(CN)4, CrB2, δ-phosphorene, Pd2O6Se2 monolayer, and GaAs, and a hypothetical set of systems with cubic symmetry are presented to demonstrate the descriptive and predictive capabilities of ElATools. Program summaryProgram Title:ElAToolsCPC Library link to program files:https://doi.org/10.17632/njh92x66c6.1Developer's repository link:https://github.com/shahramyalameha/ElAToolsLicensing provisions: GNU General Public Licence 3.0Programming language: Fortran 90External routines/libraries:LaPack, Blas, and Plotly Javascript libraries, GNUPLOT, XMGRACE, view3dscene.Nature of problem: Identifying anisotropic elastic properties of 2D and 3D materials, and calculating acoustic phase and group velocities in homogeneous solids.Solution method: Second-order elastic stiffness tensor analysis using transformation law and calculations of the elastic surfaces properties. Solving the Christoffel equation eigenvalue problem using diagonalization and calculations of the sound velocities.Additional comments: Documentation available at https://yalameha.gitlab.io/elastictools/index.html

On some features of the eigenvalue problem for the [formula omitted] approximation of the neutron transport equation

In this work some characteristics of the eigenvalue problem for the neutron transport equations are considered. Various formulations are examined, discussing some theoretical and practical aspects. The standard multiplication eigenvalue that is particularly relevant for nuclear reactor physics applications is analysed, together with the time eigenvalue, including also the contribution of delayed neutrons. In addition, the less common collision and density eigenvalues are also discussed, highlighting interesting physical features. A semianalytical approach is developed allowing to evidence some interesting structures of the eigenvalue spectra.The study is carried out within the spherical harmonics approach. For the plane one dimensional geometry, the mathematical relationship between even and odd-order approximations for the homogeneous form of the equations for the eigenvalue formulation is investigated. It is shown that the even-order system of equations can be re-cast in the form of the contiguous lower odd-order one. Numerical results are obtained in the two-group energy model for various configurations for which a reference is available, providing also results for high-order approximations. The study includes a presentation and discussion of the spectra patterns for the various eigenvalue formulations.

A fractional q-difference equation eigenvalue problem with p(t)-Laplacian operator

This article is devoted to studying a nonhomogeneous boundary value problem involving Stieltjes integral for a more general form of the fractional q-difference equation with p(t)-Laplacian operator. Here p(t)-Laplacian operator is nonstandard growth, which has been used more widely than the constant growth operator. By using fixed point theorems of φ – (h, e)-concave operators some conditions, which guarantee the existence of a unique positive solution, are derived. Moreover, we can construct an iterative scheme to approximate the unique solution. At last, two examples are given to illustrate the validity of our theoretical results.

On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

We investigate the eigenvalue problem for Kirchhoff type equations involving a superlinear perturbation, namely, −a∫RN|∇u|2dx+1Δu+μV(x)u=λf(x)u+g(x)|u|p−2u in RN, where V∈C(RN) is a potential well with the bottom Ω≔int{x∈RN|V(x)=0}. When N = 3 and 4 < p < 6, for each a > 0 and μ sufficiently large, we obtain at least one positive solution for 0 < λ ≤ λ1(fΩ), while at least two positive solutions exist for λ1(fΩ) < λ < λ1(fΩ) + δa without any assumption on the integral ∫Ωg(x)ϕ1pdx, where λ1(fΩ) > 0 is the principal eigenvalue of −Δ in H01(Ω) with the weight function fΩ ≔ f|Ω and ϕ1 > 0 is the corresponding principal eigenfunction. When N ≥ 3 and 2 < p < min{4, 2*}, for μ sufficiently large, we conclude that (i) at least two positive solutions exist for 0 < a < a0(p) and 0 < λ < λ1(fΩ); (ii) if ∫Ωg(x)ϕ1pdx<0, at least three positive solutions exist for 0 < a < a0(p) and λ1(fΩ)≤λ<λ1(fΩ)+δ̄a; and (iii) if ∫Ωg(x)ϕ1pdx>0, at least two positive solutions exist for a ≥ a0(p) and 0≤λa+<λ<λ1(fΩ).

Low Perturbations for a Class of Nonuniformly Elliptic Problems

We introduce and study a new functional which was motivated by our paper on the Caffarelli-Kohn-Nirenberg inequality with variable exponent (Bahrouni, R\u{a}dulescu and Repov\v{s}, Nonlinearity 31 (2018), 1518-1534). We also study the eigenvalue problem for equations involving this new functional.

Vertex combination approach for uncertainty propagation analysis in spacecraft structural system with complex eigenvalue

Uncertainty propagation analysis is crucial to the safety and optimal design of spacecraft structural system in which high accuracy modelling is required. In consideration of convex uncertainties, a vertex combination approach based on the convex set theory is proposed to evaluate the influence of uncertain parameters on the complex eigenvalue of spacecraft structural system with non-proportional damping. The uncertain parameters are dealt with as a closed and bounded convex set. In the proposed method, complex eigenvalue problem with convex uncertainties can be converted into a family of equivalent eigenvalue problems without uncertainties, and the bound evaluation of eigenvalue is conducted by determining the extreme values of the deterministic complex eigenvalue problems. The advantage of this work is easy to integrate with existing codes for large-scale structural system, which allows us to carry out the extension of such deterministic finite element codes to deal with uncertain structural system with complex eigenvalue. Three numerical examples are performed to demonstrate the effectiveness and the accuracy of the proposed method.

Nonlinear propagation of coupled surface TE and leaky TM electromagnetic waves

ABSTRACT Propagation of the coupled electromagnetic wave, which is a superposition of TE surface and TM leaky waves, in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear inhomogeneous medium is studied (if the permittivity is linear, the coupled wave does not exist). Nonlinear coupled TE–TM wave is characterised by two (independent) frequencies and two (coupled) propagation constants (propagation constants). The physical problem is reduced to a nonlinear two-parameter transmission eigenvalue problem for Maxwell's equations. Existence of coupled TE–TM waves is proved. Intervals of localisation of propagation constants are found.

Krylov methods for adjoint‐free singular vector based perturbations in dynamical systems

AbstractThe estimation of weather forecast uncertainty with ensemble systems requires a careful selection of perturbations to establish a reliable sampling of the error growth potential in the phase space of the model. Usually, the singular vectors of the tangent linear model propagator are used to identify the fastest growing modes (classical singular vector perturbation (SV) method). In this article we present an efficient matrix‐free block Krylov method for generating fast‐growing perturbations in high‐dimensional dynamical systems. A specific matrix containing the nonlinear evolution of perturbations is introduced, which we call Evolved Increment Matrix (EIM). Instead of solving an equivalent eigenvalue problem, we use the Arnoldi method for a direct approximation of the leading singular vectors of this matrix, which however is never computed explicitly. This avoids linear and adjoint models but requires forecasts with the full nonlinear system. The performance of the approximated perturbations is compared with singular vectors of a full EIM (not with the classical SV method). We show promising results for the Lorenz 96 differential equations and a shallow‐water model, where we obtain good approximations of the fastest‐growing perturbations by using only a small number of Arnoldi iterations.

Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid

The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids.

Modifid Interpolatory Projection Method for Weakly Singular Integral Equation Eigenvalue Problems

This paper deals with eigenvalue problems for linear Fredholm integral equations of the second kind with weakly singular kernels. A new discrete method is proposed for the approximation of eigenvalues. Compactness of the integral operator in L1[0, 1] space is obtained. This method is based on the approximation of the integral operator by modified interpolatory projection. Different from traditional methods, norm convergence of operator approximation is proved theoretically. Further, convergence of eigenvalue approximation is obtained by analytical tools. Numerical examples are presented to illustrate the theoretical results and the efficiency of the method.

The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations

In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert–Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.

Spectra of Nonselfadjoint Eigenvalue Problems for Elliptic Systems in Mathematical Models of the Wave Propagation in Open Waveguides

Statements and analysis are presented of nonselfadjoint eigenvalue problems for elliptic equations and systems, including singular Sturm–Liouville problems on the line, that arise in mathematical models of the wave propagation in open metal-dielectric waveguides. Existence of real and complex spectra are proved and their distribution is investigated for canonical structures possessing circular symmetry of boundary contours.

Excitonic States and Photoluminescence Spectra of a Quantum Dot Molecule Exposed to the External Static Electric Field

The influence of a static electric field on the energy levels, wave functions, and optical properties of excitons in a quantum dot molecule is analyzed. Taking into account the cylindrical symmetry of the molecule, the wave functions and excitonic energies are calculated when the external constant electric field is applied along the symmetry axis. Two dots separated by a narrow barrier are considered as one system of three‐dimensional potential wells for an electron and a hole. The six‐dimensional Schrödinger equation is transformed into the equivalent eigenvalue problem given by the system of the coupled two‐dimensional second order differential equations as the separation of the relative‐ and center‐of‐mass motion of an electron and a hole is not possible in this case. The differential equations are solved numerically using finite differences method. Having the eigenfunctions and eigenvalues, the excitonic photoluminescence intensity is calculated and analyzed. As an example, the configuration indicating on the possibility of non‐monotonic behavior of the photoluminescence intensity versus electric field is discussed. The comparison of numerical results with experimental data for the GaInAs quantum dot molecule is presented. Satisfactory agreement with the available experimental data is obtained.

On positive solution to multi-point fractional h-sum eigenvalue problems for caputo fractional h-difference equations

In this article, we study the existence of at least one positive solution to a multi-point fractional h-sum eigenvalue problem for Caputo fractional h-difference equation, by using the Guo-Krasnoselskii?s fixed point theorem. Moreover, we present some examples to display the importance of these results.

Semiclassical Asymptotics of Solutions to Hartree Type Equations Concentrated on Segments

We study the nonlinear eigenvalue problem for two-dimensional Hartree type equations with selfaction potentials possessing logarithmic singularity and depending on the distance between points. To find a series of asymptotic eigenvalues, we derive a counterpart of the Bohr–Sommerfeld quantization rule. The corresponding asymptotic eigenfunctions are localized near a plane segment.

Existence of two nontrivial solutions for sufficiently large values of the spectral parameter in eigenvalue problems for equations with discontinuous right-hand sides

The question on the existence of solutions to eigenvalue problems is treated for nonlinear equations with discontinuous operators in a real Hilbert space. Using a variational method, theorems on the existence of two nontrivial solutions for sufficiently large values of the spectral parameter are proved. As an application, eigenvalue problems for elliptic-type equations with nonlinear terms which are discontinuous in the phase variable are investigated. Bibliography: 22 titles.