New Conditions for Testing the Asymptotic Behavior of Solutions of Odd-Order Neutral Differential Equations with Multiple Delays

Masood, Fahd; Moaaz, Osama; Askar, Sameh; Alshamrani, Ahmad M.

doi:10.3390/axioms12070658

Cited by 5 publications

(1 citation statement)

References 31 publications

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“…The reader is directed to consult the references [11][12][13][14][15] for comprehensive insights into the methodologies, techniques, and findings relating to the investigation of oscillatory behavior in third-order NDDEs. Furthermore, the aforementioned studies [16][17][18][19][20] primarily center their attention on the examination of DDEs with odd orders.…”

Section: Introductionmentioning

confidence: 99%

New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays

Omar,

Moaaz,

AlNemer

et al. 2023

Symmetry

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This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing findings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The significance of the findings was illustrated through the application of these findings to specific cases of the investigated equation.

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Section: Introductionmentioning

confidence: 99%

New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays

Omar,

Moaaz,

AlNemer

et al. 2023

Symmetry

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Analytical solutions of systems of linear retarded and neutral delay differential equations by the Laplace transform: featuring limit cycles

Kerr,

Lopez,

Gonzalez-Parra

2023

Preprint

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In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution approaches asymptotically to a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases becomes more accurate as the independent variable increases.

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Third-Order Neutral Differential Equations with Non-Canonical Forms: Novel Oscillation Theorems

Almarri,

Batiha,

Bazighifan

et al. 2024

Axioms

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This paper explores the asymptotic and oscillatory properties of a class of third-order neutral differential equations with multiple delays in a non-canonical form. The main objective is to simplify the non-canonical form by converting it to a canonical form, which reduces the complexity of the possible cases of positive solutions and their derivatives from four cases in the non-canonical form to only two cases in the canonical form, which facilitates the process of inference and development of results. New criteria are provided that exclude the existence of positive solutions or Kneser-type solutions for this class of equations. New criteria that guarantee the oscillatory behavior of all solutions that satisfy the conditions imposed on the studied equation are also derived. This work makes a qualitative contribution to the development of previous studies in the field of neutral differential equations, as it provides new insights into the oscillatory behavior of neutral equations with multiple delays. To confirm the strength and effectiveness of the results, three examples are included that highlight the accuracy of the derived criteria and their practical applicability, which enhances the value of this research and expands the scope of its use in the field.

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New Conditions for Testing the Asymptotic Behavior of Solutions of Odd-Order Neutral Differential Equations with Multiple Delays

Cited by 5 publications

References 31 publications

New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays

New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays

Analytical solutions of systems of linear retarded and neutral delay differential equations by the Laplace transform: featuring limit cycles

Third-Order Neutral Differential Equations with Non-Canonical Forms: Novel Oscillation Theorems

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