Solution Of Eigenvalue Problem Research Articles

Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian

In this paper, we investigate the existence of positive solutions for the eigenvalue problem of nonlinear fractional differential equation with generalized p-Laplacian operatorD0+β(ϕ(D0+αu(t)))=λf(u(t)),0<t<1,u(0)=u′(0)=u′(1)=0,ϕ(D0+αu(0))=(ϕ(D0+αu(1)))′=0,where 2<α⩽3,1<β⩽2 are real numbers, D0+α,D0+β are the standard Riemann–Liouville fractional derivatives, ϕ is a generalized p-Laplacian operator, λ>0 is a parameter, and f:(0,+∞)→(0,+∞) is continuous. By using the properties of Green function and Guo–Krasnosel’skii fixed-point theorem on cones, several new existence results of at least one or two positive solutions in terms of different eigenvalue interval are obtained. Moreover, the nonexistence of positive solution in term of the parameter λ is also considered.

Mixed Spectral-Element Method for 3-D Maxwell's Eigenvalue Problem

A high-order 3-D mixed spectral-element method (SEM) based on Gauss-Lobatto-Legendre polynomials in the mixed finite-element framework is proposed to remove the spurious eigenmodes in the numerical solution of the vector Maxwell eigenvalue problem with inhomogeneous, lossy isotropic, and anisotropic media. In order to suppress all zero and nonzero spurious modes that exist in the conventional finite-element and higher order SEMs, the proposed method not only employs the mixed-order curl-conforming vector basis functions for the electric field intensity, but also includes the divergence-free condition given by Gauss' law in a weak form. Several numerical examples are given to verify that the mixed SEM is free of any spurious eigenmodes and has spectral accuracy with analytic eigenvectors.

Analytical/numerical investigations of Kelvin-Helmholtz instability of superposed non-viscous fluids in the presence of magnetic field

The instability of two immiscible superposed, non-viscous, electrically conducting and counter-streaming fluids with upper fluid layer lighter than the lower fluid layer in the presence of uniform magnetic field has been investigated numerically. Using the linear theory and normal mode analysis, the exact solutions of eigen value problem have been obtained for stress free bounding surfaces and the dispersion relation so obtained has been analysed to examine the effects of magnetic field and velocity of streaming fluids on the growth rates of the unstable mode of perturbation on the physical system. It has been found that the magnetic field has a slight stabilizing effect on the system; whereas the velocity of the counter-streaming fluids has a large enough stabilizing effect towards neutral stability.

Subdomain Propagation Method for an Efficient Modal Analysis of Photonic Waveguides

The modal analysis is needed for the design of photonic waveguide devices to determine propagation constants of guided modes and their field pattern. Rigorous methods based on finite-difference or finite-element methods for the approximation of wave equations rely on the solution of eigenvalue problems. Imaginary-distance beam propagation methods require the solution of matrix equations of high order. The computation time needed for both approaches can be tedious. A modal analysis method based on a beam propagation scheme is proposed, which allows an efficient computation of field patterns and propagation constants, as the resulting algorithm is based on simple matrix vector multiplications. This allows the choice of a large number of discretization points and a dense discretization. The beam propagation operator is designed to include the eigenvalue spectrum of corresponding guided modes only. For this reason, the method can be interpreted as a subdomain propagation method. The concept will be validated by utilizing a single-mode and multimode rib waveguide.

Positive Solutions for Nonlinearq-Fractional Difference Eigenvalue Problem with Nonlocal Conditions

The problem of positive solutions for nonlinearq-fractional difference eigenvalue problem with nonlocal boundary conditions is investigated. Based on the fixed point index theory in cones, sufficient existence of positive solutions conditions is derived for the problem.

Backward Error of Polynomial Eigenvalue Problems Solved by Linearization of Lagrange Interpolants

This article considers the backward error of the solution of polynomial eigenvalue problems expressed as Lagrange interpolants. One of the most common strategies to solve polynomial eigenvalue problems is to linearize, which is to say that the polynomial eigenvalue problem is transformed into an equivalent larger linear eigenvalue problem, and solved using any appropriate eigensolver. Much of the existing literature on the backward error of polynomial eigenvalue problems focuses on polynomials expressed in the classical monomial basis. Hence, the objective of this article is to carry out the necessary extensions for polynomials expressed in the Lagrange basis. We construct one-sided factorizations that give simple expressions relating the eigenvectors of the linearization to the eigenvectors of the polynomial eigenvalue problem. Using these relations, we are able to bound the backward error of an approximate eigenpair of the polynomial eigenvalue problem relative to the backward error of an approximate eigenpair of the linearization. We develop bounds for the backward error involving both the norms of the polynomial coefficients and the properties of the Lagrange basis generated by the interpolation nodes. We also present several numerical examples to illustrate the numerical properties of the linearization and develop a balancing strategy to improve the accuracy of the computed solutions.

Performance Analysis and Optimisation of Two-sided Factorization Algorithms for Heterogeneous Platform

Many applications, ranging from big data analytics to nanostructure designs, require the solution of large dense singular value decomposition (SVD) or eigenvalue problems. A first step in the solution methodology for these problems is the reduction of the matrix at hand to condensed form by two-sided orthogonal transformations. This step is standardly used to significantly accelerate the solution process. We present a performance analysis of the main two-sided factorizations used in these reductions: the bidiagonalization, tridiagonalization, and the upper Hessenberg factorizations on heterogeneous systems of multicore CPUs and Xeon Phi coprocessors. We derive a performance model and use it to guide the analysis and to evaluate performance. We develop optimized implementations for these methods that get up to 80% of the optimal performance bounds. Finally, we describe the heterogeneous multicore and coprocessor development considerations and the techniques that enable us to achieve these high-performance results. The work here presents the first highly optimized implementation of these main factorizations for Xeon Phi coprocessors. Compared to the LAPACK versions optmized by Intel for Xeon Phi (in MKL), we achieve up to 50% speedup.

A Fourier Continuation Method for the Solution of Elliptic Eigenvalue Problems in General Domains

We present a new computational method for the solution of elliptic eigenvalue problems with variable coefficients in general two-dimensional domains. The proposed approach is based on use of the novel Fourier continuation method (which enables fast and highly accurate Fourier approximation of nonperiodic functions in equispaced grids without the limitations arising from the Gibbs phenomenon) in conjunction with an overlapping patch domain decomposition strategy and Arnoldi iteration. A variety of examples demonstrate the versatility, accuracy, and generality of the proposed methodology.

A Study on Anisotropic Mesh Adaptation for Finite Element Approximation of Eigenvalue Problems with Anisotropic Diffusion Operators

Anisotropic mesh adaptation is studied for the linear finite element solution of eigenvalue problems with anisotropic diffusion operators. The $\mathbb{M}$-uniform mesh approach is employed with which any nonuniform mesh is characterized mathematically as a uniform one in the metric specified by a metric tensor. Bounds on the error in the computed eigenvalues are established for quasi-$\mathbb{M}$-uniform meshes. Numerical examples arising from the Laplace--Beltrami operator on parameterized surfaces and nonlinear diffusion problems in image processing and radiation hydrodynamics are presented. Numerical results show that anisotropic adaptive meshes can lead to more accurate computed eigenvalues than uniform or isotropic adaptive meshes. They also confirm the second order convergence of the error that is predicted by the theoretical analysis. The effects of approximation of curved boundaries on the computation of eigenvalue problems is also studied in two dimensions. It is shown that the initial mesh used to define the geometry of the physical domain should contain at least $\sqrt{N}$ boundary points to keep the effects of boundary approximation at the level of the error of the finite element approximation, where $N$ is the number of the elements in the final adaptive mesh. Only about $\sqrt[3]{N}$ boundary points in the initial mesh are needed for boundary value problems. This implies that the computation of eigenvalue problems is more sensitive to the boundary approximation than that of boundary value problems.

Solution of large scale parametric eigenvalue problems arising from brake squeal modeling

AbstractWe present a proper orthogonal decomposition (POD) based approach for the accurate solution of parametric eigenvalue problems arising from brake squeal modeling. We compare this approach to a traditional method based on modal decomposition which is very popular in industry. The proposed approach is more accurate than the traditional method especially for parametric studies and for studying non‐proportionally damped systems. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Improving robustness of the FEAST algorithm and solving eigenvalue problems from graphene nanoribbons

AbstractWe consider the FEAST eigensolver, introduced by Polizzi in 2009 [5]. We describe an improvement concerning the reliability of the algorithm and discuss an application in the solution of eigenvalue problems from graphene modeling. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On the use of matrix-free shift-invert strategies for global flow instability analysis

A novel time-stepping shift-invert algorithm for linear stability analysis of laminar flows in complex geometries is presented. This method, based on a Krylov subspace iteration, enables the solution of complex non-symmetric eigenvalue problems in a matrix-free framework. Validations and comparisons to the classical exponential method have been performed in three different cases: (i) stenotic flow, (ii) backward-facing step and (iii) lid-driven swirling flow. Results show that this new approach speeds up the required Krylov subspace iterations and has the capability of converging to specific parts of the global spectrum. It is shown that, although the exponential method remains the method of choice if leading eigenvalues are sought, the performance of the present method could be dramatically improved with the use of a preconditioner. In addition, as opposed to other methods, this strategy can be directly applied to any time-stepper, regardless of the temporal or spatial discretization of the latter.

Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X-1A=L

Abstract In this paper, we present two new algebraic algorithms for the solution of the discrete algebraic Riccati equation. The first algorithm requires the nonsingularity of the transition matrix and is based on the solution of a standard eigenvalue problem for a new symplectic matrix; the proposed algorithm computes the extreme solutions of the discrete algebraic Riccati equation. The second algorithm solves the Riccati equation without the assumption of the nonsingularity of the transition matrix; the proposed algorithm is based on the solution of the matrix equation X + A*X-1A=L, where A is a singular matrix and L a positive definite matrix.

Parallel computing on graphics processing units and heterogeneous platforms

Parallel computing on graphics processing units and heterogeneous platforms

Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes

We construct quasimodes for the Klein-Gordon equation on the black hole exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasi-modes are associated with time-periodic approximate solutions of the Klein Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semi-classical non-linear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semi-classical parameter. Our construction results in exponentially small errors in the semi-classical parameter. This implies that general solutions to the Klein Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.

Fourier Expansion of the Beam Propagation Operator in the Eigenvalue Domain

Propagation methods in the frequency domain are powerful tools for the analysis of electromagnetic field propagation within photonic waveguide devices. They can be classified as wide angle beam propagation or eigenvector expansion methods. In comparison to the wide angle beam propagation methods, eigenvector expansion methods inherently include an accurate computation of radiation waves, guided modes, and evanescent waves. Common numerical methods rely on the solution of eigenvalue problems or the solution of matrix equations of high order. Consequently, the computation time needed can be tedious. A method is proposed, which exploits the inherent advantage of eigenvector expansion methods, but reduces the computation time considerably as the resulting propagation algorithm is based on simple matrix vector multiplications, allowing a large number of discretization points. The method takes advantage of the compact eigenvalue spectrum of the system matrix. The operator can be periodically expanded so that a Fourier decomposition can be applied. The expansion is carried out under consideration of the physical interpretation of eigenvalues for radiation waves, guided modes, and evanescent waves.

Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a relatively stiff structure on a soft elastic soil is derived through the application of a perturbation analysis. The full solution is obtained as the sum of the solution for an unconstrained elastic structure and small perturbing terms related to the ratio of the stiffness of the soil to that of the superstructure. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented earlier for the case of stiff soils. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.

Effect of twist and rotation on vibration of functionally graded conical shells

This paper presents the effect of twist and rotational speeds on free vibration characteristics of functionally graded conical shells employing finite element method. The objective is to study the natural frequencies, the influence of constituent volume fractions and the effects of configuration of constituent materials on the frequencies. The equation of dynamic equilibrium is derived from Lagrange’s equation neglecting the Coriolis effect for moderate rotational speeds. The properties of the conical shell materials are presumed to vary continuously through their thickness with power-law distribution of the volume fractions of their constituents. The QR iteration algorithm is used for solution of standard eigenvalue problem. Computer codes developed are employed to obtain the numerical results concerning the combined effects of twist angle and rotational speed on the natural frequencies of functionally graded conical shells. The mode shapes for typical shells are also depicted. Numerical results obtained are the first known non-dimensional frequencies for the type of analyses carried out here.

Eigenvalue problems for nonlinear third‐order m‐point p‐Laplacian dynamic equations on time scales

This work deals with the existence and uniqueness of a nontrivial solution for the third‐order p‐Laplacian m‐point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when λ is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley &amp; Sons, Ltd.

A multigrid method for eigenvalue problem

A multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem. In this scheme, solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space. Besides the multigrid scheme, all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems. The total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem. Therefore, this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving. Some numerical experiments are presented to validate the efficiency of the method.