Spectral Methods for Non-Standard Eigenvalue Problems

Gheorghiu, Călin-Ioan

doi:10.1007/978-3-319-06230-3

Cited by 30 publications

(39 citation statements)

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“…For more details, see, e.g., [10,13,15]. The spectral collocation method is based on approximating the solution y(x) with a finite linear combination of a chosen set of orthogonal function as…”

Section: Spectral Methodsmentioning

confidence: 99%

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Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Plestenjak

Gheorghiu

Hochstenbach

2015

Journal of Computational Physics

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Section: Spectral Methodsmentioning

confidence: 99%

“…In case of the half line [0, ∞) we use the Laguerre collocation (LC) (see, e.g., [15]). The basis functions are products of Laguerre polynomials and the weight function e − 1 2 bx , where b > 0 is the scaling parameter.…”

Section: Spectral Methodsmentioning

confidence: 99%

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Plestenjak

Gheorghiu

Hochstenbach

2015

Journal of Computational Physics

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“…This and other general aspects of the method have been appropriately commented on in [3,4,6,7]. As a general and cornerstone reference see [2,Chapter 7], where a heuristic rule of thumb for the choice of M is established and spurious solutions are adequately treated, and also the recent monograph [21] for challenging nonstandard eigenvalue problems, where suitable eigenvalue solvers in finite dimension are also discussed, like, e.g., Jacobi-Davidson type methods [28].…”

Section: ν Counted With Multiplicities Satisfyingmentioning

confidence: 99%

Computing the Eigenvalues of Realistic Daphnia Models by Pseudospectral Methods

Breda¹,

Getto²,

Sanz³

et al. 2015

SIAM J. Sci. Comput.

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Abstract. This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled to the evolution of an unstructured resource described by a delay differential equation (DDE), resulting in dynamics over an infinite dimensional state space. As additional complexities, the right-hand sides are both of integral type (continuous age distribution) and given implicitly through external ordinary differential equations (ODEs). Moreover, discontinuities in the vital rates occur at a maturation age, also given implicitly through one of the above ODEs. With the aim at studying the local asymptotic stability of equilibria and relevant bifurcations, we revisit a pseudospectral approach recently proposed to compute the eigenvalues of the infinitesimal generator of linearized systems of coupled VFEs/DDEs. First, we modify it in view of extension to nonlinear problems for future developments. Then, we consider a suitable implementation to tackle all the computational difficulties mentioned above: a piecewise approach to handle discontinuities, numerical quadrature of integrals, and numerical solution of ODEs. Moreover, we rigorously prove the spectral accuracy of the method in approximating the eigenvalues and how this outstanding feature is influenced by the other unavoidable error sources. Implementation details and experimental computations on existing available data conclude the work.

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“…This approach, however, cannot be applied to other boundary conditions such as simply supported conditions. One of approaches used broadly is to introduce equations expressing boundary conditions at boundary nodes instead of original equations at nodes next to the boundary nodes [3,4]; the original equations at the nodes next to the boundary nodes are completely ignored. It is simple and easy to implement but not exact.…”

Section: Introductionmentioning

confidence: 99%

Fourier series expansion type of spectral collocation method for vibration analysis of cylindrical shells

Araki

Samejima

2016

Acoust. Sci. & Tech.

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An analysis method using a spectral collocation method for the vibration of cylindrical shells is proposed. Conventional spectral collocation methods have difficulty applying boundary conditions to fourth-order differential equations such as vibration equations of cylindrical shells. In this paper, an Hermite differentiation matrix is developed such that the proposed spectral collocation method can treat flexibly various boundary conditions. Since the vibration displacement of a cylindrical shell is periodic in the circumferential direction, it is solved semi-analytically using the Fourier series expansion. It is shown that the proposed method can offer more accurate solutions at a smaller number of unknowns, in less computation time and required memory than a finite element method.

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Spectral Methods for Non-Standard Eigenvalue Problems

Cited by 30 publications

References 0 publications

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Computing the Eigenvalues of Realistic Daphnia Models by Pseudospectral Methods

Fourier series expansion type of spectral collocation method for vibration analysis of cylindrical shells

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