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13 pages, 5337 KB

Open AccessArticle

Asymptotic Convergence of Solutions for Singularly Perturbed Linear Impulsive Systems with Full Singularity

by Nauryzbay Aviltay and Muratkhan Dauylbayev

Symmetry 2025, 17(9), 1389; https://doi.org/10.3390/sym17091389 - 26 Aug 2025

Viewed by 270

This paper considers impulsive systems with singularities. The main novelty of this study is that the impulses (impulsive functions) and the initial value are singular. The asymptotic convergence of the solution to a singularly perturbed initial problem with an infinitely large initial value, [...] Read more.

This paper considers impulsive systems with singularities. The main novelty of this study is that the impulses (impulsive functions) and the initial value are singular. The asymptotic convergence of the solution to a singularly perturbed initial problem with an infinitely large initial value, as

ε \to 0,

to the solution to a corresponding modified degenerate initial problem is proved. It is established that the solution to the initial problem at point

t = 0

has an initial jump phenomenon, and the value of this initial jump is determined. The theoretical results are supported by illustrative examples with simulations. Singularly perturbed problems are characterized by the presence of a small parameter multiplying the highest derivatives in the differential equations. This leads to rapid changes in the solution near the boundary or at certain points inside the domain. In our problem, symmetry is violated due to the emergence of a boundary layer at the initial point and at the moments of discontinuity. As a result, the problem as a whole is asymmetric. Such asymmetry in the behavior of the solution is a main feature of singularly perturbed problems, setting them apart from regularly perturbed problems in which the solutions usually exhibit smoother changes. Full article

(This article belongs to the Section Mathematics)

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16 pages, 343 KB

Open AccessArticle

Structured Distance to Normality of Dirichlet–Neumann Tridiagonal Toeplitz Matrices

by Zhaolin Jiang, Hongxiao Chu, Qiaoyun Miao and Ziwu Jiang

Axioms 2025, 14(8), 609; https://doi.org/10.3390/axioms14080609 - 5 Aug 2025

Viewed by 218

Abstract

This paper conducts a rigorous study on the spectral properties and operator-space distances of perturbed Dirichlet–Neumann tridiagonal (PDNT) Toeplitz matrices, with emphasis on their asymptotic behaviors. We establish explicit closed-form solutions for the eigenvalues and associated eigenvectors, highlighting their fundamental importance for characterizing [...] Read more.

This paper conducts a rigorous study on the spectral properties and operator-space distances of perturbed Dirichlet–Neumann tridiagonal (PDNT) Toeplitz matrices, with emphasis on their asymptotic behaviors. We establish explicit closed-form solutions for the eigenvalues and associated eigenvectors, highlighting their fundamental importance for characterizing matrix stability in the presence of perturbations. By exploiting the structural characteristics of PDNT Toeplitz matrices, we obtain closed-form expressions quantifying the distance to normality, the deviation from normality. Full article

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18 pages, 288 KB

Open AccessArticle

Functional Differential Equations with Non-Canonical Operator: Oscillatory Features of Solutions

by Asma Al-Jaser, Faizah Alharbi, Dimplekumar Chalishajar and Belgees Qaraad

Axioms 2025, 14(8), 588; https://doi.org/10.3390/axioms14080588 - 29 Jul 2025

Viewed by 201

Abstract

This study focuses on investigating the asymptotic and oscillatory behavior of a new class of fourth-order nonlinear neutral differential equations. This research aims to achieve a qualitative advancement in the analysis and understanding of the relationships between the corresponding function and its derivatives. [...] Read more.

This study focuses on investigating the asymptotic and oscillatory behavior of a new class of fourth-order nonlinear neutral differential equations. This research aims to achieve a qualitative advancement in the analysis and understanding of the relationships between the corresponding function and its derivatives. By utilizing various techniques, innovative criteria have been developed to ensure the oscillation of all solutions of the studied equations without resorting to additional constraints. Effective analytical tools are provided, contributing to a deeper theoretical understanding and expanding their application scope. The paper concludes by presenting examples that illustrate the practical impact of the results, highlighting the theoretical value of the research in the field of functional differential equations. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations, 2nd Edition)

11 pages, 284 KB

Open AccessArticle

Oscillation Theorems of Fourth-Order Differential Equations with a Variable Argument Using the Comparison Technique

by Osama Moaaz, Wedad Albalawi and Refah Alotaibi

Axioms 2025, 14(8), 587; https://doi.org/10.3390/axioms14080587 - 29 Jul 2025

Viewed by 196

Abstract

In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation

(a

ϕ (u)

u^{‴})

+q

(u \circ h) = 0

, which is of a functional type with a delay. The oscillation [...] Read more.

In this study, we establish new oscillation criteria for solutions of the fourth-order differential equation

(a

ϕ (u)

u^{‴})

+q

(u \circ h) = 0

, which is of a functional type with a delay. The oscillation behavior of solutions of fourth-order delay equations has been studied using many techniques, but previous results did not take into account the existence of the function

ϕ

except in second-order studies. The existence of

ϕ

increases the difficulty of obtaining monotonic and asymptotic properties of the solutions and also increases the possibility of applying the results to a larger area of special cases. We present two criteria to ensure the oscillation of the solutions of the studied equation for two different cases of

ϕ

. Our approach is based on using the comparison principle with equations of the first or second order to benefit from recent developments in studying the oscillation of these orders. We also provide several examples and compare our results with previous ones to illustrate the novelty and effectiveness. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

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19 pages, 349 KB

Open AccessArticle

Normalized Ground States for the Sobolev Critical Fractional Kirchhoff Equation with at Least Mass Critical Growth

by Peng Ji and Fangqi Chen

Fractal Fract. 2025, 9(8), 482; https://doi.org/10.3390/fractalfract9080482 - 24 Jul 2025

Viewed by 268

Abstract

In this paper, we delve into the following nonlinear fractional Kirchhoff-type problem [...] Read more.

In this paper, we delve into the following nonlinear fractional Kirchhoff-type problem

{(a + b | | (- Δ)}^{\frac{s}{2}} {u | |}_{2}^{2} {) (- Δ)}^{s} u + λ u = g (u) + {| u |}^{2_{s}^{*} - 2} u

in

R^{3}

with prescribed mass

\int_{R^{3}} {| u |}^{2} d x = ρ > 0,

where

s \in (\frac{3}{4}, 1), λ \in R, 2_{s}^{*} = \frac{6}{3 - 2 s}

. Under some general growth assumptions imposed on g, we employ minimization of the energy functional on the linear combination of Nehari and Poho

\overset{˘}{z}

aev constraints intersected with the closed ball in the

L^{2} (R^{3})

of radius

ρ

to prove the existence of normalized ground state solutions to the equation. Moreover, we provide a detailed description for the asymptotic behavior of the ground state energy map. Full article

23 pages, 1065 KB

Open AccessArticle

Modeling and Neural Network Approximation of Asymptotic Behavior for Delta Fractional Difference Equations with Mittag-Leffler Kernels

by Pshtiwan Othman Mohammed, Muteb R. Alharthi, Majeed Ahmad Yousif, Alina Alb Lupas and Shrooq Mohammed Azzo

Fractal Fract. 2025, 9(7), 452; https://doi.org/10.3390/fractalfract9070452 - 9 Jul 2025

Viewed by 425

Abstract

The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order

0.5

subject to a given initial condition. We establish [...] Read more.

The asymptotic behavior of discrete Riemann–Liouville fractional difference equations is a fundamental problem with important mathematical and physical implications. In this paper, we investigate a particular case of such an equation of the order

0.5

subject to a given initial condition. We establish the existence of a unique solution expressed via a Mittag-Leffler-type function. The delta-asymptotic behavior of the solution is examined, and its convergence properties are rigorously analyzed. Numerical experiments are conducted to illustrate the qualitative features of the solution. Furthermore, a neural network-based approximation is employed to validate and compare with the analytical results, confirming the accuracy, stability, and sensitivity of the proposed method. Full article

(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)

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19 pages, 1286 KB

Open AccessArticle

Adsorption–Desorption at Anomalous Diffusion: Fractional Calculus Approach

by Ivan Bazhlekov and Emilia Bazhlekova

Fractal Fract. 2025, 9(7), 408; https://doi.org/10.3390/fractalfract9070408 - 24 Jun 2025

Viewed by 658

Abstract

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of [...] Read more.

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed. Full article

(This article belongs to the Special Issue Analysis of Heat Conduction and Anomalous Diffusion in Fractional Calculus)

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19 pages, 540 KB

Open AccessFeature PaperArticle

Exact Parametric and Semi-Analytical Solutions for the Rucklidge-Type Dynamical System

by Remus-Daniel Ene, Nicolina Pop and Rodica Badarau

Mathematics 2025, 13(13), 2052; https://doi.org/10.3390/math13132052 - 20 Jun 2025

Cited by 1 | Viewed by 240

Abstract

The behavior of the Rucklidge-type dynamical system was investigated, providing some semi-analytical solutions, in this paper. This system was analytically investigated by means of the Optimal Auxiliary Functions Method (OAFM) for two cases. An exact parametric solution was obtained. The effect of the [...] Read more.

The behavior of the Rucklidge-type dynamical system was investigated, providing some semi-analytical solutions, in this paper. This system was analytically investigated by means of the Optimal Auxiliary Functions Method (OAFM) for two cases. An exact parametric solution was obtained. The effect of the physical parameters was investigated on the asymptotic behaviors and damped oscillations of the solutions. Damped oscillations are essential for analyzing and designing various mechanical, biological, and electrical systems. Many of the applications involving these systems represent the main reason of this work. A comparison between the obtained results via the OAFM, the analytical solution obtained with the iterative method, and the corresponding numerical solution was performed. The accuracy of the analytical and corresponding numerical results is illustrated by graphical and tabular representations. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems Interacting in Complex Networks)

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15 pages, 310 KB

Open AccessArticle

Asymptotical Behavior of Impulsive Linearly Implicit Euler Method for the SIR Epidemic Model with Nonlinear Incidence Rates and Proportional Impulsive Vaccination

by Zhi-Wei Xu and Gui-Lai Zhang

Axioms 2025, 14(6), 470; https://doi.org/10.3390/axioms14060470 - 16 Jun 2025

Viewed by 279

Abstract

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive [...] Read more.

This paper is concerned with the asymptotical behavior of the impulsive linearly implicit Euler method for the SIR epidemic model with nonlinear incidence rates and proportional impulsive vaccination. We point out the solution of the impulsive linearly implicit Euler method for the impulsive SIR system is positive for arbitrary step size when the initial values are positive. By applying discrete Floquet’s theorem and small-amplitude perturbation skills, we proved that the disease-free periodic solution of the impulsive system is locally stable. Additionally, in conjunction with the discrete impulsive comparison theorem, we show that the impulsive linearly implicit Euler method maintains the global asymptotical stability of the exact solution of the impulsive system. Two numerical examples are provided to illustrate the correctness of the results. Full article

(This article belongs to the Special Issue Differential Equations and Inverse Problems, 2nd Edition)

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17 pages, 4274 KB

Open AccessArticle

On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations

by Wenjin Li, Jiaxuan Sun and Yanni Pang

Mathematics 2025, 13(12), 1926; https://doi.org/10.3390/math13121926 - 10 Jun 2025

Viewed by 302

Abstract

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term [...] Read more.

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term

{({[x^{″} (t)]}^{α})}^{'} + q (t) x^{α} (σ (t)) + f (t) = 0, t \geq t_{0} .

Under the oscillation criteria, we propose a sufficient condition to ensure that all solutions for the equation exhibit oscillatory behavior when

α

is the quotient of two positive odd integers, supported by concrete examples to verify the accuracy of these conditions. Furthermore, for the case

α = 1,

a sufficient condition is established to guarantee that the solutions either oscillate or asymptotically converge to zero. Moreover, under these criteria, we demonstrate that the global oscillatory behavior of solutions remains unaffected by time-delay functions, nonhomogeneous terms, or nonlinear perturbations when

α = 1 .

Finally, numerical simulations are provided to validate the effectiveness of the derived conclusions. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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16 pages, 793 KB

Open AccessArticle

Note on Iterations of Nonlinear Rational Functions

by Michal Fečkan, Amira Khelifa, Yacine Halim and Ibraheem M. Alsulami

Axioms 2025, 14(6), 450; https://doi.org/10.3390/axioms14060450 - 7 Jun 2025

Viewed by 386

Abstract

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By [...] Read more.

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By establishing a connection with generalized Balancing sequences, we derive explicit formulas that describe the system’s asymptotic behavior. Our main contribution is proving the existence of a unique globally asymptotically stable equilibrium point for all trajectories, regardless of initial conditions. We also provide analytical expressions for the solutions and support our findings with numerical examples. These results offer valuable insights into the dynamics of nonlinear rational systems and form a theoretical basis for further exploration in this area. Full article

(This article belongs to the Section Mathematical Analysis)

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18 pages, 25291 KB

Open AccessArticle

Theoretical and Computational Insights into a System of Time-Fractional Nonlinear Schrödinger Delay Equations

by Mai N. Elhamaky, Mohamed A. Abd Elgawad, Zhanwen Yang and Ahmed S. Rahby

Axioms 2025, 14(6), 432; https://doi.org/10.3390/axioms14060432 - 1 Jun 2025

Viewed by 460

Abstract

This research focuses on the theoretical asymptotic stability and long-time decay of the zero solution for a system of time-fractional nonlinear Schrödinger delay equations (NSDEs) in the context of the Caputo fractional derivative. Using the fractional Halanay inequality, we demonstrate theoretically when the [...] Read more.

This research focuses on the theoretical asymptotic stability and long-time decay of the zero solution for a system of time-fractional nonlinear Schrödinger delay equations (NSDEs) in the context of the Caputo fractional derivative. Using the fractional Halanay inequality, we demonstrate theoretically when the considered system decays and behaves asymptotically, employing an energy function in the sense of the L₂ norm. Together with utilizing the finite difference method for the spatial variables, we investigate the long-time stability for the semi-discrete system. Furthermore, we operate the L1 scheme to approximate the Caputo fractional derivative and analyze the long-time stability of the fully discrete system through the discrete energy of the system. Moreover, we demonstrate that the proposed numerical technique energetically captures the long-time behavior of the original system of NSDEs. Finally, we provide numerical examples to validate the theoretical results. Full article

(This article belongs to the Section Mathematical Analysis)

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13 pages, 281 KB

Open AccessArticle

Decay Estimates for a Lamé Inverse Problem Involving Source and Damping Term with Variable-Exponent Nonlinearities

by Zülal Mısır and Metin Yaman

Axioms 2025, 14(6), 424; https://doi.org/10.3390/axioms14060424 - 30 May 2025

Viewed by 291

Abstract

We investigate an inverse problem involving source and damping term with variable-exponent nonlinearities. We establish adequate conditions on the initial data for the decay of solutions as the integral overdetermination approaches zero over time within an acceptable range of variable exponents. This class [...] Read more.

We investigate an inverse problem involving source and damping term with variable-exponent nonlinearities. We establish adequate conditions on the initial data for the decay of solutions as the integral overdetermination approaches zero over time within an acceptable range of variable exponents. This class of inverse problems, where internal terms such as source and damping are to be determined from indirect measurements, has significant relevance in real-world applications—ranging from geophysical prospecting to biomedical engineering and materials science. The accurate identification of these internal mechanisms plays a crucial role in optimizing system performance, improving diagnostic accuracy, and constructing predictive models. Therefore, the results obtained in this study not only contribute to the theoretical understanding of nonlinear dynamic systems but also provide practical insights for reconstructive analysis and control in applied settings. The asymptotic behavior and decay conditions we derive are expected to be of particular interest to researchers dealing with stability, uniqueness, and identifiability in inverse problems governed by nonstandard growth conditions. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Numerical Modeling)

12 pages, 256 KB

Open AccessArticle

Large-Time Behavior of Solutions to Darcy–Boussinesq Equations with Non-Vanishing Scalar Acceleration Coefficient

by Huichao Wang, Zhibo Hou and Quan Wang

Mathematics 2025, 13(10), 1570; https://doi.org/10.3390/math13101570 - 10 May 2025

Viewed by 295

Abstract

We study the large-time behavior of solutions to Darcy–Boussinesq equations with a non-vanishing scalar acceleration coefficient, which model buoyancy-driven flows in porous media with spatially varying gravity. First, we show that the system admits steady-state solutions of the form [...] Read more.

We study the large-time behavior of solutions to Darcy–Boussinesq equations with a non-vanishing scalar acceleration coefficient, which model buoyancy-driven flows in porous media with spatially varying gravity. First, we show that the system admits steady-state solutions of the form

(u, ρ, p) = (0, ρ_{s}, p_{s})

, where

ρ_{s}

is characterised by the hydrostatic balance

\nabla p_{s} = - ρ_{s} \nabla Ψ

. Second, we prove that the steady-state solution satisfying

\nabla ρ_{s} = δ (x, y) \nabla Ψ

is linearly stable provided that

δ (x, y) < δ_{0} < 0

, while the system exhibits Rayleigh–Taylor instability if

Ψ = g y

,

ρ_{s} = δ_{0} g

and

δ_{0} > 0

. Finally, despite the inherent Rayleigh–Taylor instability that may trigger exponential growth in time, we prove that for any sufficiently regular initial data, the solutions of the system asymptotically converge towards the vicinity of a steady-state solution, where the velocity field is zero, and the new state is determined by hydrostatic balance. This work advances porous media modeling for geophysical and engineering applications, emphasizing the critical interplay of gravity, inertia, and boundary conditions. Full article

(This article belongs to the Special Issue Recent Studies on Partial Differential Equations and Its Applications)

24 pages, 2098 KB

Open AccessArticle

Quasiparticle Solutions to the 1D Nonlocal Fisher–KPP Equation with a Fractal Time Derivative in the Weak Diffusion Approximation

by Alexander V. Shapovalov and Sergey A. Siniukov

Fractal Fract. 2025, 9(5), 279; https://doi.org/10.3390/fractalfract9050279 - 25 Apr 2025

Cited by 1 | Viewed by 444

Abstract

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order (

α

, where [...] Read more.

In this paper, we propose an approach for constructing quasiparticle-like asymptotic solutions within the weak diffusion approximation for the generalized population Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP) equation, which incorporates nonlocal quadratic competitive losses and a fractal time derivative of non-integer order (

α

, where

0 < α \leq 1

). This approach is based on the semiclassical approximation and the principles of the Maslov method. The fractal time derivative is introduced in the framework of

F^{α}

calculus. The Fisher–KPP equation is decomposed into a system of nonlinear equations that describe the dynamics of interacting quasiparticles within classes of trajectory-concentrated functions. A key element in constructing approximate quasiparticle solutions is the interplay between the dynamical system of quasiparticle moments and an auxiliary linear system of equations, which is coupled with the nonlinear system. General constructions are illustrated through examples that examine the effect of the fractal parameter (

α

) on quasiparticle behavior. Full article

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