Lowrank methods for parameterdependent eigenvalue problems. In the moleculefield example introduced above, small values of n will. In this article, we are concerned with finding effective ways to solve time dependent problems using rbfs. We first analyze the properties on the analytic perturbation of the invariant pair of a nonli.
We associate to each of these problems a quadratic operator pencil with selfadjoint operators. In,anoperatortheoretic formulation of the problems with the spectral parameter contained in only one of the boundary conditions has been given. Parameterdependent matrix eigenvalue problems and their applications in structural dynamics n. This example makes it clear that the sturmliouville problems are of. Since the eigenvalues depend continuously on the parameter, a stable equilibrium may lose stability and bifurcate into another neighboring solution when qr. Computing the spectrum of nonselfadjoint sturmliouville. Transmission eigenvalues for maxwells equations in. Pdf parameterdependent matrix eigenvalue problems and their. The basic example of such problems are socalled resonance problems 18. Asymptotics of the eigenvalues of a selfadjoint fourth order boundary value problem with four eigenvalue parameter dependent boundary conditions manfredmollerandbertinzinsou e john knopfmacher centre for applicable analysis and number eor y, school of mathematics, university of the witwatersrand wits, private bag, johannesburg, south africa. Research article asymptotics of the eigenvalues of a self.
Parameterdependent parallel block sparse arnoldi and dohler. Since the eigenvalues depend continuously on the parameter, a stable equilibrium may lose stability and bifurcate into another neighboring solution when qr crosses a certain critical value 7cr. Most notably, it plays a crucial role in a posteriori estimation of reduced basis methods for parametrized partial differential equations. One of the separated boundary conditions depends linearly on the eigenvalue parameter these problems can be represented by an operator polynomial l. In this thesis, we study the problem of coalescing singular values of smooth realvalued parameter dependent matrices, or ofcoalescing eigenvalues of smoothrealvalued parameter dependent symmetric matrices. Eigenvalues of vectorial sturmliouville problems with parameter. In analyzing stability of parameterdependent systems, a.
We study the spectrum of a parameter dependent sturmliouville problem by using the continued fractions, through which necessary and su. Eigenvalue coalescence for parameter dependent matrices. Group velocity of elastic waves as eigensensitivity of a. Paper open access an eigenvalue approximation for parameter. Li, discrete sturmliouville problems with nonlinear parameter in the boundary conditions, j. This is a generalized eigenvalue problem and many globally convergent algorithms exist for its solution i, 21.
Transmission eigenvalues for maxwells equations in isotropic. Selfadjoint fourth order differential operators with. A new method is proposed to compute the eigenpairs of a parameter. Elliptic eigenvalue problems with eigenparameter dependent. Pdf numerical solution of linear and nonlinear eigenvalue. Weve reduced the problem of nding eigenvectors to a problem that we already know how to solve. Eigenvalues of vectorial sturmliouville problems with parameter dependent boundary conditions.
We generalize the regularized sampling method introduced in 2005 by the author to compute the eigenvalues of scalar sturmliouville problems slps to the case of vectorial slp with parameter dependent boundary conditions. Asymptotics of the eigenvalues of a selfadjoint fourth order boundary value problem with four eigenvalue parameter dependent boundary conditions. These yield generalized eigenvalue problems, of which the formulation and di erent solution. Mennicken and m oller 5 developed useful tools for the identi cation of the. One of the separated boundary conditions depends linearly on the eigenvalue parameter these problems can be represented by an operator polynomial \l\lambda\lambda 2mi\alpha\lambda ka\, where \\alpha0\, m and k are. Eigenvalues of vectorial sturmliouville problems with. A collection of nonlinear eigenvalue problems university of. Eigenvalue problems of the parameter dependent system of. A parameter dependent hamiltonian generally arises in the context of adia.
Spectral asymptotics of eigenvalue problems with non. Mar 14, 2017 fourth order eigenvalue problems with periodic and separated boundary conditions are considered. The fundamental analytical object of such a dependent bvps is the characteristic determinant, and it is proved that any characteristic determinant on a jordan curve that contains no eigenvalues of the problem can be characterized geometrically as the determinant of a transition func. In many problems in engineering and physical sciences, we are usually interested in knowing how the sensitivity of the physical quantity behaves for small changes in the parameters. Approximation of pde eigenvalue problems involving. We shall extend the method based on sampling theory 1 see also 24 to compute the eigenvalues of sturm liouville problems with parameter dependent potential and boundary conditions for motivations, see for example 5,6,10,11, for more on sampling theory see 12. The existence of eigenvalues, estimates of eigenvalues and eigenfunctions, oscillation properties of eigenfunctions, and expansion theorems were considered in 4, 7, 9, 12, 17, 18, 21, 24 for fourthorder ordinary di. For time dependent problems, in particular, differentiation matrices often have unstable eigenvalues requiring severe dissipation in time. In particular, we will prove that provided some assumptions, at least at low frequency, the transmission eigenvalues form a discrete set. But theoretical and numerical investigation in three parameter eigenvalue problems are still very few. On rayleighritz method in threeparameter eigenvalue problems. The solution of dudt d au is changing with time growing or decaying or oscillating.
Eigenvalue problems of the parameter dependent system of ordinary di. Spectral methods for nonstandard eigenvalue problems. So lets compute the eigenvector x 1 corresponding to eigenvalue. Boundary value problems with eigenvaluedependent boundary and.
Spectral asymptotics of selfadjoint fourth order boundary. A few problems are worked out to illustrate the effectiveness of the method and show by the same token that we have indeed a general method capable of handling with. Oct 04, 2012 a regular fourth order differential equation with. Research article double discontinuous inverse problems for. In this work we study a sturmliouville operator with a piecewise continuous coe cient and a spectral parameter in the boundary condition. Following are summaries of some of the studies describing boundary eigenvalue problems with dependent boundary conditions.
Finally, we combine the ideas used for addressing matrix equations and parameterdependent eigenvalue problems, and propose a lowrank reduced basis approach for solving parameter dependent lyapunov equations. Research article eigenvalues of vectorial sturmliouville. This requires the solution of a large number of pde eigenvalue problems. Eigenvalues and eigenvectors of a matrix dependent on. This implies that matrix perturbations are restricted to a speci. Box, dhahran, saudi arabia correspondence should be addressed to bilal chanane.
The book by parlett 148 is an excellent treatise of the problem. Calculating eigenvalues and eigenvectors of parameter dependent. Gao, eigenvalues of discrete linear secondorder periodic and antiperiodic eigenvalue problems with signchanging weight, linear algebra appl. Meirovitch 14 introduced a method of solving an eigenvalue problem for gyroscopic system by transforming it to a standard.
Spectral properties of a fourthorder eigenvalue problem. Asymptotics of the eigenvalues of a selfadjoint fourth order boundary value. Parameterdependent eigenvalue problem occurs in a host of engineering. Direct computation yields that all the eigenvalues are positive and each eigenvalue satisfies the equation cot. Spectral properties of a fourthorder eigenvalue problem with. Pdf every physical system contains parameters, and one goal of the present contribution is to study the movements of eigenvalues in the. Keywords partial differential equations eigenvalue problem parameter dependent. Half inverse problems for the impulsive operator with eigenvalue dependent boundary conditions yasser khalili, milad yadollahzadeh, mohsen khaleghi moghadam communicated by ira w. Thenthere are exactly m discrete eigenvalues of 18. We investigate the spectral proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst four terms of the eigenvalue. Pdf parameterdependent matrix eigenvalue problems and. These tools had been used to prove that the problems. Approximation of pde eigenvalue problems involving parameter dependent matrices published in.
We consider in this research an eigenvalue problem with general dependent boundary conditions and investigate classes of boundary conditions which are selfadjoint. We discuss the solution of eigenvalue problems associated with partial. Eigenvalue stability of radial basis function discretizations. Computation of multiple eigenvalues and generalized. Dependence of eigenvalues of sturmliouville problems with. Parameter dependent matrices occur frequently when stabilized formulations are used for the. For this numerical example, the values of the model parameters are taken as. Aliyev, on basis properties of root functions of a boundary value problem containing a spectral parameter in the boundary conditions, dokl. In particular, we are interested in detecting, and accurately approximating, parameters values where the coalescing occurs. A few problems are worked out to illustrate the effectiveness of the method and show by the same token that we have indeed. Parameterdependent matrix eigenvalue problems and their. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial differential. On eigenvalues of matrices dependent on a parameter.
Boundary value problems with the eigenparameter in the boundary con ditions have a long. While the relationship for group velocity is correct for a general class of wave phenomena, we restrict ourselves to more familiar cases when the phase velocity and the group velocity are positive. The aim is to determine the wave number which leads to the maximum growth rate of the amplitude of the wave. Numerical solution of linear and nonlinear eigenvalue problems. Countably in nite eigenvalues the following essential result characterizes the behavior of the entire set of eigenvalues of sturmliouville problems. Eigenvalueshave theirgreatest importance in dynamic problems. We first analyze the properties on the analytic perturbation of the invariant pair of a nonlinear eigenvalue problem and provide a method to compute the first. Gaul institut a fur mechanik, universitat stuttgart, germany. Example 1 chanane 1, 1dversion taken from fom binding and browne 19. Spectral asymptotics of selfadjoint fourth order boundary value problems with eigenvalue parameter dependent boundary conditions. One can also ask the same question for objects such as the schrodinger. Since we are only interested in a smooth eigenfunction corresponding. Second, we tackle the computation of the eigenvalues of sturmliouville problems with quite general separated param eter dependent boundary conditions although the known theory is for boundary conditions which are a. The analysis of contour integrals the analysis of contour integrals.
Numerical methods for large scale structured parameter dependent polynomial eigenvalue problems compute eigenvalues in trapezoidal region around 0 determine projectors on important spectral subspaces for model reduction model reduction for parameterized model optimization of frequencies implementation of parallel solver in sfe concept. Feb 01, 2012 consider the following parameter dependent nonlinear eigenvalue problem. These of the methods approximately predict the interval range for eigenvalues. On the one hand, such boundary conditions hinder the construction of test and trial space functions which could incorporate them and thus providing wellconditioned galerkin. Jan 31, 2021 a new method is proposed to compute the eigenpairs of a parameter. Several approaches for approximation of interval eigenvalue problems in terms of symmetric 25 and asymmetric matrices 68 have been suggested. Parameter dependent matrices occur frequently when stabilized formulations are. Reduced order modeling of parameter dependent nonlinear. However, if c, is nonsingular it is more efficient to convert 16 to the algebraic eigenvalue problem of finding the eigenvalues of the matrix a, given by a c,c, c,c, i 0 1. For more on singular problems see 3 and 14 for example. On the numerical treatment of the eigenparameter dependent. A vector bundle view of parameterdependent boundaryvalue. In this paper, the three parameter problem considered for numerical treatment is in the form of a linear ordinary differential equation given by 1. Such problems have an operator representation of the form l 2m i k a 1.
Differential eigenvalue problems in which the parameter. Pdf eigenvalues of vectorial sturmliouville problems. We investigate the spectral proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst four terms of the eigenvalue asymptotics. Spectral theory of selfadjoint higher order di erential. The simulation of drift instabilities in the plasma edge leads to cubic polynomial pde eigenvalue problems with parameter dependent coefficients. Properties of sturmliouville eigenfunctions and eigenvalues. Inverse problems according to various spectral data for eigenparameter linearly dependent sturmliouville. Another famous example of a boundary eigenvalue problem for a differential.
Calculating eigenvalues and eigenvectors of parameterdependent. Eigenvalues of vectorial sturmliouville problems with parameter dependent boundary conditions bilalchanane department of mathematics and statistics, king fahd university of petroleum and minerals, p. This example makes it clear that the sturmliouville problems are of broad. Spectral properties of the differential operators of the. We study the eigenvalue problem auu subject to boundary conditions b0. A multilevel jacobidavidson method for polynomial pde. Selfadjoint fourth order differential operators with eigenvalue. The eigenvalue problem associated with gyroscopic systems has several mathematical features that are di erent from that for nongyroscopic problems 12.
Assuming that we can nd the eigenvalues i, nding x i has been reduced to nding the nullspace na ii. Problems of this type occur, for example, in the stability analysis of spatially discretized and. Numerical methods for parametric model reduction in the simulation. Analysis and numerical solution of eigenvalue problems. If a square matrix at depends smoothly on a parameter t, are its eigenvalues. An eigenvalue approximation for parameter dependent undamped. Approximation of pde eigenvalue problems involving parameter. Spectral asymptotics of eigenvalue problems with nonlinear. Several books dealing with numerical methods for solving eigenvalue problems involving symmetric or hermitian matrices have been written and there are a few software packages both public and commercial available. Mehraliyev, basis properties in l p of root functions of sturmliouville problem with spectral parameter dependent boundary conditions, mediterr.
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