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Search Results (4)

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Keywords = Rouché’s theorem

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29 pages, 592 KB  
Article
Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay
by Manal Alotaibi
Mathematics 2025, 13(17), 2714; https://doi.org/10.3390/math13172714 - 23 Aug 2025
Viewed by 794
Abstract
This paper presents a fully implicit finite difference scheme for the numerical approximation of a wave equation featuring strong damping and a distributed delay term. The discretization employs second-order accurate approximations in both time and space. Although implicit, the scheme does not ensure [...] Read more.
This paper presents a fully implicit finite difference scheme for the numerical approximation of a wave equation featuring strong damping and a distributed delay term. The discretization employs second-order accurate approximations in both time and space. Although implicit, the scheme does not ensure unconditional stability due to the nonlocal nature of the delayed damping. To address this, we perform a stability analysis based on Rouché’s theorem from complex analysis and derive a sufficient condition for asymptotic stability of the discrete system. The resulting criterion highlights the interplay among the discretization parameters, the damping coefficient, and the delay kernel. Two quadrature techniques, the composite trapezoidal rule (CTR) and the Gaussian quadrature rule (GQR), are employed to approximate the convolution integral. Numerical experiments validate the theoretical predictions and illustrate both stable and unstable dynamics across different parameter regimes. Full article
(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)
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29 pages, 452 KB  
Article
Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems
by Manuel De la Sen
Mathematics 2024, 12(19), 3023; https://doi.org/10.3390/math12193023 - 27 Sep 2024
Viewed by 1224
Abstract
This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which [...] Read more.
This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays. Full article
12 pages, 548 KB  
Article
An Application of Rouché’s Theorem to Delimit the Zeros of a Certain Class of Robustly Stable Polynomials
by Noé Martínez, Luis E. Garza and Gerardo Romero
Mathematics 2023, 11(20), 4244; https://doi.org/10.3390/math11204244 - 11 Oct 2023
Cited by 1 | Viewed by 1574
Abstract
An important problem related to the study of the robust stability of a linear system that presents variation in terms of an uncertain parameter consists of understanding the variation in the roots of a system’s characteristic polynomial in terms of the uncertain parameter. [...] Read more.
An important problem related to the study of the robust stability of a linear system that presents variation in terms of an uncertain parameter consists of understanding the variation in the roots of a system’s characteristic polynomial in terms of the uncertain parameter. In this contribution, we propose an algorithm to provide sufficient conditions on the uncertain parameter in such a way that a robustly stable family of polynomials has all of its zeros inside a specific subset of its stability region. Our method is based on the Rouché’s theorem and uses robustly stable polynomials constructed by using basic properties of orthogonal polynomials. Full article
(This article belongs to the Special Issue Analysis and Control of Dynamical Systems)
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14 pages, 446 KB  
Article
Zeros of Convex Combinations of Elementary Families of Harmonic Functions
by Jennifer Brooks, Megan Dixon, Michael Dorff, Alexander Lee and Rebekah Ottinger
Mathematics 2023, 11(19), 4057; https://doi.org/10.3390/math11194057 - 25 Sep 2023
Cited by 1 | Viewed by 1522
Abstract
Brilleslyper et al. investigated how the number of zeros of a one-parameter family of harmonic trinomials varies with a real parameter. Brooks and Lee obtained a similar theorem for an analogous family of harmonic trinomials with poles. In this paper, we investigate the [...] Read more.
Brilleslyper et al. investigated how the number of zeros of a one-parameter family of harmonic trinomials varies with a real parameter. Brooks and Lee obtained a similar theorem for an analogous family of harmonic trinomials with poles. In this paper, we investigate the number of zeros of convex combinations of members of these families and show that it is possible for a convex combination of two members of a family to have more zeros than either of its constituent parts. Our main tool to prove these results is the harmonic analog of Rouché’s theorem. Full article
(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory, 2nd Edition)
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