The method of external excitation for solving generalized Sturm–Liouville problems

Reutskiy, S.Yu.

doi:10.1016/j.cam.2009.10.022

Cited by 14 publications

(7 citation statements)

References 12 publications

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“…Solving the generalized SLP is more difficult than that to solve the SLP. Some numerical methods were developed to tackle the eigen-parameter dependent boundary conditions problems (Liu, 2010(Liu, , 2012Aliyev & Kerimov, 2008;Binding & Browne, 1997;Chanane, 2005Chanane, , 2007Chanane, , 2008Reutskiy, 2008Reutskiy, , 2010Annaby & Tharwat, 2006;El-Gamel, 2014).…”

Section: Generalized Sturm-liouville Problemsmentioning

confidence: 99%

Accurate Eigenvalues for the Sturm-Liouville Problems, Involving Generalized and Periodic Ones

Liu¹

2022

JMR

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In the paper, the eigenvalues of Sturm-Liouville problems (SLPs), generalized SLPs and periodic SLPs are solved. First, we propose a new method to transform the SLP with mixed boundary conditions to a {generalized} SLP for a transformed variable, for which the Dirichlet boundary conditions occur on two-side, but the coefficients are nonlinear functions of eigenvalue. To computing the eigenvalue and eigenfunction, we further recast the transformed system to an initial value problem for a new variable. In terms of the relative residual of two consecutive terminal values of the new variable a nonlinear equation is solved for seeking the eigenvalue by the fictitious time integration method (FTIM), which monotonically converges to the exact eigenvalue. We solve a numerically characteristic equation by the half-interval method (HIM) and a derivative-free iterative scheme LHL {(Liu, Hong $\&$ Li, 2021)} to achieve high precision eigenvalues. Next, the generalized SLP is transformed to a {new} one, so that the Dirichlet boundary condition happens on the right-end. By using the boundary shape function method and the uniqueness condition of the transformed variable, a definite initial value problem is derived for the new variable. To match the right-end Dirichlet boundary condition a numerically characteristic equation is {obtained and} solved by the HIM and LHL. Finally, new techniques for solving the periodic SLPs with three types periodic boundary conditions are proposed, which preserve the periodic boundary conditions with the aids of boundary shape functions. Three iterative algorithms are developed, which converge quickly.  All the proposed iterative algorithms are identified by testing some examples.

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Section: Generalized Sturm-liouville Problemsmentioning

confidence: 99%

Accurate Eigenvalues for the Sturm-Liouville Problems, Involving Generalized and Periodic Ones

Liu¹

2022

JMR

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“…Also, it is not necessarily required to observe the voltage across the port resistors. It is rather possible to work with some arbitrary observable of the field solution such as the electric field within the solution domain or some integral measure of it (as for instance suggested in [11,12]). A particular strength of this method is that the excitation can be chosen in a way that a desired eigenmode is more excited than others.…”

Section: Eigenproblem-resonator Equivalence and Excitation Formulationmentioning

confidence: 99%

“…In [10][11][12], so-called methods of external and internal excitation are proposed for the solution of eigenproblems. These methods apply an excitation to the eigenproblem under consideration and observe a norm of the solution dependent on the varying eigenvalue.…”

Section: Introductionmentioning

confidence: 99%

“…The method is implemented together with two different regularization schemes. In [10,11] homogeneous waveguide problems and in [12] generalized Sturm-Liouville problems are considered. The particular property of these methods is that they do not require the solution of an algebraic eigenproblem, but the repeated solution of corresponding excitation problems.…”

Section: Introductionmentioning

confidence: 99%

“…Such methods can never focus the entire excitation energy into an isolated eigenfrequency and they are restricted by the always limited observation time, where even several eigenmodes have to be resolved simultaneously. In contrast, our method and the methods presented in [11,12] directly work in the domain of the eigenvalue and have thus the possibility to direct the entire excitation engergy into one isolated eigenmode. Moreover, not only eigenfrequencies are considered in this paper, but also complex propagation constants.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving Periodic Eigenproblems by Solving Corresponding Excitation Problems in the Domain of the Eigenvalue

Eibert¹,

Weitsch²,

Chen³

et al. 2012

PIER

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Abstract-Periodic eigenproblems describing the dispersion behavior of periodically loaded waveguiding structures are considered as resonating systems. In analogy to resonators, their eigenvalues and eigensolutions are determined by solving corresponding excitation problems directly in the domain of the eigenvalue. Arbitrary excitations can be chosen in order to excite the desired modal solutions, where in particular lumped ports and volumetric current distributions are considered. The method is employed together with a doubly periodic hybrid finite element boundary integral technique, which is able to consider complex propagation constants in the periodic boundary conditions and the Green's functions. Other numerical solvers such as commercial simulation packages can also be employed with the proposed procedure, where complex propagation constants are typically not directly supported. However, for propagating waves with relatively small attenuation, it is shown that the attenuation constant can be determined by perturbation methods. Numerical results for composite right/left-handed waveguides and for the leaky modes of a grounded dielectric slab are presented.

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Computing the eigenvalues of the generalized Sturm–Liouville problems based on the Lie-group SL(2,R)

Liu

2012

Journal of Computational and Applied Mathematics

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The method of external excitation for solving generalized Sturm–Liouville problems

Cited by 14 publications

References 12 publications

Accurate Eigenvalues for the Sturm-Liouville Problems, Involving Generalized and Periodic Ones

Accurate Eigenvalues for the Sturm-Liouville Problems, Involving Generalized and Periodic Ones

Solving Periodic Eigenproblems by Solving Corresponding Excitation Problems in the Domain of the Eigenvalue

Computing the eigenvalues of the generalized Sturm–Liouville problems based on the Lie-group SL(2,R)

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