Mathematics

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29 pages, 1223 KiB

Open AccessArticle

Nonlinear SIRS Fractional-Order Model: Analysing the Impact of Public Attitudes towards Vaccination, Government Actions, and Social Behavior on Disease Spread

by Protyusha Dutta, Nirapada Santra, Guruprasad Samanta and Manuel De la Sen

Mathematics 2024, 12(14), 2232; https://doi.org/10.3390/math12142232 - 17 Jul 2024

Cited by 3 | Viewed by 956

Abstract

This present work develops a nonlinear SIRS fractional-order model with a system of four equations in the Caputo sense. This study examines the impact of positive and negative attitudes towards vaccination, as well as the role of government actions, social behavior and public [...] Read more.

This present work develops a nonlinear SIRS fractional-order model with a system of four equations in the Caputo sense. This study examines the impact of positive and negative attitudes towards vaccination, as well as the role of government actions, social behavior and public reaction on the spread of infectious diseases. The local stability of the equilibrium points is analyzed. Sensitivity analysis is conducted to calculate and discuss the sensitivity index of various parameters. It has been established that the illness would spread across this system when the basic reproduction number is larger than 1, the system becomes infection-free when the reproduction number lies below its threshold value of 1. Numerical figures depict the effects of positive and negative attitudes towards vaccination to make the system disease-free sooner. A comprehensive study regarding various values of the order of fractional derivatives together with integer-order derivatives has been discussed in the numerical section to obtain some useful insights into the intricate dynamics of the proposed system. The Pontryagin principle is used in the formulation and subsequent discussion of an optimum control issue. The study also reveals the significant role of government actions in controlling the epidemic. A numerical analysis has been conducted to compare the system’s behavior under optimal control and without optimal control, aiming to discern their differences. The policies implemented by the government are regarded as the most adequate control strategy, and it is determined that the execution of control mechanisms considerably diminishes the ailment burden. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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15 pages, 290 KiB

Open AccessArticle

New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

by Osman Tunç

Mathematics 2024, 12(9), 1342; https://doi.org/10.3390/math12091342 - 28 Apr 2024

Cited by 4 | Viewed by 1346

Abstract

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability [...] Read more.

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

11 pages, 281 KiB

Open AccessArticle

Boundedness of Vector Linéard Equation with Multiple Variable Delays

by Melek Gözen

Mathematics 2024, 12(5), 769; https://doi.org/10.3390/math12050769 - 4 Mar 2024

Viewed by 943

Abstract

In this article, we consider a system of ordinary differential equations (ODEs) of second order with two variable time delays. We obtain new conditions for uniform ultimate bounded (UUB) solutions of the considered system. The technique of the proof is based on the [...] Read more.

In this article, we consider a system of ordinary differential equations (ODEs) of second order with two variable time delays. We obtain new conditions for uniform ultimate bounded (UUB) solutions of the considered system. The technique of the proof is based on the Lyapunov–Krasovskii functional (LKF) method using a new LKF. The main result of this article extends and improves a recent result for ODEs of second order with a constant delay to a more general system of ODEs of second order with two variable time delays. In this particular case, we also give a numerical example to verify the application of the main result of this article. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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13 pages, 269 KiB

Open AccessArticle

New Results on Ulam Stabilities of Nonlinear Integral Equations

by Osman Tunç, Cemil Tunç and Jen-Chih Yao

Mathematics 2024, 12(5), 682; https://doi.org/10.3390/math12050682 - 26 Feb 2024

Cited by 10 | Viewed by 1237

Abstract

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and [...] Read more.

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

16 pages, 290 KiB

Open AccessArticle

Global Existence and Uniqueness of Solutions of Integral Equations with Multiple Variable Delays and Integro Differential Equations: Progressive Contractions

by Osman Tunç, Cemil Tunç and Jen-Chih Yao

Mathematics 2024, 12(2), 171; https://doi.org/10.3390/math12020171 - 5 Jan 2024

Cited by 13 | Viewed by 1348

Abstract

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated [...] Read more.

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

17 pages, 374 KiB

Open AccessArticle

Dynamics of a Higher-Order Three-Dimensional Nonlinear System of Difference Equations

by Murad Khan Hassani, Yasin Yazlik, Nouressadat Touafek, Mohammed Salah Abdelouahab, Mouataz Billah Mesmouli and Fatma E. Mansour

Mathematics 2024, 12(1), 16; https://doi.org/10.3390/math12010016 - 20 Dec 2023

Cited by 1 | Viewed by 1172

Abstract

In this paper, we study the semi-cycle analysis of positive solutions and the asymptotic behavior of positive solutions of three-dimensional system of difference equations with a higher order under certain parametric conditions. Furthermore, we show the boundedness and persistence, the rate of convergence [...] Read more.

In this paper, we study the semi-cycle analysis of positive solutions and the asymptotic behavior of positive solutions of three-dimensional system of difference equations with a higher order under certain parametric conditions. Furthermore, we show the boundedness and persistence, the rate of convergence of the solutions and the global asymptotic stability of the unique equilibrium point of the proposed system under certain parametric conditions. Finally, for this system, we offer some numerical examples which support our analytical results. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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9 pages, 254 KiB

Open AccessArticle

Conditions for the Oscillation of Solutions to Neutral Differential Equations of Higher Order

by Maryam Al-Kandari

Mathematics 2023, 11(24), 4909; https://doi.org/10.3390/math11244909 - 9 Dec 2023

Viewed by 1096

Abstract

In this research, we applied three techniques—the comparison technique, the Riccati technique, and the integral averages technique to analyze and establish various conditions and properties associated with the oscillatory behavior of even-order neutral differential equations. These findings contribute to a better understanding of [...] Read more.

In this research, we applied three techniques—the comparison technique, the Riccati technique, and the integral averages technique to analyze and establish various conditions and properties associated with the oscillatory behavior of even-order neutral differential equations. These findings contribute to a better understanding of the dynamics of such equations. To demonstrate the efficacy of these new conditions and properties, we present illustrative examples. This study offers valuable insights into the behavior of neutral differential equations, advancing our knowledge in this field. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

10 pages, 282 KiB

Open AccessArticle

A Time-Fractional Parabolic Inequality on a Bounded Interval

by Amal Alshabanat, Eman Almoalim, Mohamed Jleli and Bessem Samet

Mathematics 2023, 11(24), 4892; https://doi.org/10.3390/math11244892 - 6 Dec 2023

Viewed by 1190

Abstract

We study a time-fractional parabolic inequality posed on a bounded interval and involving a wight function W, where the fractional derivative is considered in the Caputo sense. We establish a general condition ensuring that the set of weak solutions is empty. Next, [...] Read more.

We study a time-fractional parabolic inequality posed on a bounded interval and involving a wight function W, where the fractional derivative is considered in the Caputo sense. We establish a general condition ensuring that the set of weak solutions is empty. Next, some particular cases of the weight function W are discussed. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

23 pages, 859 KiB

Open AccessArticle

Bifurcation Behavior and Hybrid Controller Design of a 2D Lotka–Volterra Commensal Symbiosis System Accompanying Delay

by Qingyi Cui, Changjin Xu, Wei Ou, Yicheng Pang, Zixin Liu, Peiluan Li and Lingyun Yao

Mathematics 2023, 11(23), 4808; https://doi.org/10.3390/math11234808 - 28 Nov 2023

Cited by 42 | Viewed by 1596

Abstract

All the time, differential dynamical models with delay has witness a tremendous application value in characterizing the internal law among diverse biological populations in biology. In the current article, on the basis of the previous publications, we formulate a new Lotka–Volterra commensal symbiosis [...] Read more.

All the time, differential dynamical models with delay has witness a tremendous application value in characterizing the internal law among diverse biological populations in biology. In the current article, on the basis of the previous publications, we formulate a new Lotka–Volterra commensal symbiosis system accompanying delay. Utilizing fixed point theorem, inequality tactics and an appropriate function, we gain the sufficient criteria on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated delayed Lotka–Volterra commensal symbiosis system. Making use of stability and bifurcation theory of delayed differential equation, we focus on the emergence of bifurcation behavior and stability nature of the formulated delayed Lotka–Volterra commensal symbiosis system. A new delay-independent stability and bifurcation conditions on the model are presented. By constructing a positive definite function, we explore the global stability. By constructing two diverse hybrid delayed feedback controllers, we can adjusted the domain of stability and time of appearance of Hopf bifurcation of the delayed Lotka–Volterra commensal symbiosis system. The effect of time delay on the domain of stability and time of appearance of Hopf bifurcation of the model is given. Matlab experiment diagrams are provided to sustain the acquired key outcomes. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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28 pages, 1189 KiB

Open AccessArticle

Mathematical and Stability Analysis of Dengue–Malaria Co-Infection with Disease Control Strategies

by Azhar Iqbal Kashif Butt, Muhammad Imran, Brett A. McKinney, Saira Batool and Hassan Aftab

Mathematics 2023, 11(22), 4600; https://doi.org/10.3390/math11224600 - 9 Nov 2023

Cited by 8 | Viewed by 2556

Abstract

Historically, humans have been infected by mosquito-borne diseases, including dengue fever and malaria fever. There is an urgent need for comprehensive methods in the prevention, control, and awareness of the hazards posed by dengue and malaria fever to public health. We propose a [...] Read more.

Historically, humans have been infected by mosquito-borne diseases, including dengue fever and malaria fever. There is an urgent need for comprehensive methods in the prevention, control, and awareness of the hazards posed by dengue and malaria fever to public health. We propose a new mathematical model for dengue and malaria co-infection with the aim of comprehending disease dynamics better and developing more efficient control strategies in light of the threat posed to public health by co-infection. The proposed mathematical model comprises four time-dependent vector population classes (

S E I_{d} I_{m}

) and seven host population classes (

S E I_{d} I_{m} I_{d m} T R

). First, we show that the proposed model is well defined by proving that it is bounded and positive in a feasible region. We further identify the equilibrium states of the model, including disease-free and endemic equilibrium points, where we perform stability analysis at equilibrium points. Then, we determine the reproduction number

R_{0}

to measure the level of disease containment. We perform a sensitivity analysis of the model’s parameters to identify the most critical ones for potential control strategies. We also prove that the proposed model is well posed. Finally, the article examines three distinct co-infection control measures, including spraying or killing vectors, taking precautions for one’s own safety, and reducing the infectious contact between the host and vector populations. The control analysis of the proposed model reveals that all control parameters are effective in disease control. However, self-precaution is the most effective and accessible method, and the reduction of the vector population through spraying is the second most effective strategy to implement. Disease eradication is attainable as the vector population decreases. The effectiveness of the implemented strategies is also illustrated with the help of graphs. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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12 pages, 282 KiB

Open AccessArticle

Existence and Uniqueness of Positive Solutions for Semipositone Lane-Emden Equations on the Half-Axis

by Imed Bachar

Mathematics 2023, 11(19), 4184; https://doi.org/10.3390/math11194184 - 6 Oct 2023

Viewed by 909

Abstract

Semipositone Lane–Emden type equations are considered on the half-axis. Such equations have been used in modelling several phenomena in astrophysics and mathematical physics and are often difficult to solve analytically. We provide sufficient conditions for the existence of a positive continuous solution and [...] Read more.

Semipositone Lane–Emden type equations are considered on the half-axis. Such equations have been used in modelling several phenomena in astrophysics and mathematical physics and are often difficult to solve analytically. We provide sufficient conditions for the existence of a positive continuous solution and we describe its global behavior. Our approach is based on a perturbed operator technique and fixed point theorems. Some examples are presented to illustrate the main results. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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15 pages, 320 KiB

Open AccessArticle

An Improved Approach to Investigate the Oscillatory Properties of Third-Order Neutral Differential Equations

by Osama Moaaz and Yousef Alnafisah

Mathematics 2023, 11(10), 2290; https://doi.org/10.3390/math11102290 - 15 May 2023

Cited by 3 | Viewed by 1382

Abstract

In this work, by considering a third-order differential equation with delay-neutral arguments, we investigate the oscillatory behavior of solutions. It is known that the relationships between the solution and its derivatives of different orders, as well as between the solution and its corresponding [...] Read more.

In this work, by considering a third-order differential equation with delay-neutral arguments, we investigate the oscillatory behavior of solutions. It is known that the relationships between the solution and its derivatives of different orders, as well as between the solution and its corresponding function, can help to obtain more efficient oscillation criteria for differential equations of neutral type. So, we deduce some new relationships of an iterative nature. Then, we test the effect of these relationships on the criteria that exclude positive solutions to the studied equation. By comparing our results with previous results in the literature, we show the importance and novelty of the new results. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

29 pages, 1545 KiB

Open AccessArticle

Design and Analysis of a New COVID-19 Model with Comparative Study of Control Strategies

by Azhar Iqbal Kashif Butt, Saira Batool, Muhammad Imran and Muneerah Al Nuwairan

Mathematics 2023, 11(9), 1978; https://doi.org/10.3390/math11091978 - 22 Apr 2023

Cited by 12 | Viewed by 1807

Abstract

The COVID-19 pandemic has become a worldwide concern and has caused great frustration in the human community. Governments all over the world are struggling to combat the disease. In an effort to understand and address the situation, we conduct a thorough study of [...] Read more.

The COVID-19 pandemic has become a worldwide concern and has caused great frustration in the human community. Governments all over the world are struggling to combat the disease. In an effort to understand and address the situation, we conduct a thorough study of a COVID-19 model that provides insights into the dynamics of the disease. For this, we propose a new

L_{S} H_{S} E A I H R

COVID-19 model, where susceptible populations are divided into two sub-classes: low-risk susceptible populations,

L_{S}

, and high-risk susceptible populations,

H_{S}

. The aim of the subdivision of susceptible populations is to construct a model that is more reliable and realistic for disease control. We first prove the existence of a unique solution to the purposed model with the help of fundamental theorems of functional analysis and show that the solution lies in an invariant region. We compute the basic reproduction number and describe constraints that ensure the local and global asymptotic stability at equilibrium points. A sensitivity analysis is also carried out to identify the model’s most influential parameters. Next, as a disease transmission control technique, a class of isolation is added to the intended

L_{S} H_{S} E A I H R

model. We suggest simple fixed controls through the adjustment of quarantine rates as a first control technique. To reduce the spread of COVID-19 as well as to minimize the cost functional, we constitute an optimal control problem and develop necessary conditions using Pontryagin’s maximum principle. Finally, numerical simulations with and without controls are presented to demonstrate the efficiency and efficacy of the optimal control approach. The optimal control approach is also compared with an approach where the state model is solved numerically with different time-independent controls. The numerical results, which exhibit dynamical behavior of the COVID-19 system under the influence of various parameters, suggest that the implemented strategies, particularly the quarantine of infectious individuals, are effective in significantly reducing the number of infected individuals and achieving herd immunity. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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18 pages, 993 KiB

Open AccessArticle

Finite-Element Method for the Simulation of Lipid Vesicle/Fluid Interactions in a Quasi–Newtonian Fluid Flow

by Aymen Laadhari

Mathematics 2023, 11(8), 1950; https://doi.org/10.3390/math11081950 - 20 Apr 2023

Cited by 2 | Viewed by 1508

Abstract

We present a computational framework for modeling an inextensible single vesicle driven by the Helfrich force in an incompressible, non-Newtonian extracellular Carreau fluid. The vesicle membrane is captured with a level set strategy. The local inextensibility constraint is relaxed by introducing a penalty [...] Read more.

We present a computational framework for modeling an inextensible single vesicle driven by the Helfrich force in an incompressible, non-Newtonian extracellular Carreau fluid. The vesicle membrane is captured with a level set strategy. The local inextensibility constraint is relaxed by introducing a penalty which allows computational savings and facilitates implementation. A high-order Galerkin finite element approximation allows accurate calculations of the membrane force with high-order derivatives. The time discretization is based on the double composition of the one-step backward Euler scheme, while the time step size is flexibly controlled using a time integration error estimation. Numerical examples are presented with particular attention paid to the validation and assessment of the model’s relevance in terms of physiological significance. Optimal convergence rates of the time discretization are obtained. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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17 pages, 1069 KiB

Open AccessArticle

On System of Variable Order Nonlinear p-Laplacian Fractional Differential Equations with Biological Application

by Hasib Khan, Jehad Alzabut, Haseena Gulzar, Osman Tunç and Sandra Pinelas

Mathematics 2023, 11(8), 1913; https://doi.org/10.3390/math11081913 - 18 Apr 2023

Cited by 25 | Viewed by 2419

Abstract

The study of variable order differential equations is important in science and engineering for a better representation and analysis of dynamical problems. In the literature, there are several fractional order operators involving variable orders. In this article, we construct a nonlinear variable order [...] Read more.

The study of variable order differential equations is important in science and engineering for a better representation and analysis of dynamical problems. In the literature, there are several fractional order operators involving variable orders. In this article, we construct a nonlinear variable order fractional differential system with a p-Laplacian operator. The presumed problem is a general class of the nonlinear equations of variable orders in the ABC sense of derivatives in combination with Caputo’s fractional derivative. We investigate the existence of solutions and the Hyers–Ulam stability of the considered equation. The presumed problem is a hybrid in nature and has a lot of applications. We have given its particular example as a waterborne disease model of variable order which is analysed for the numerical computations for different variable orders. The results obtained for the variable orders have an advantage over the constant orders in that the variable order simulations present the fluctuation of the real dynamics throughout our observations of the simulations. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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21 pages, 674 KiB

Open AccessArticle

Global Dynamics for Competition between Two Wolbachia Strains with Bidirectional Cytoplasmic Incompatibility

by Qiming Huang, Lijie Chang, Zhaowang Zhang and Bo Zheng

Mathematics 2023, 11(7), 1691; https://doi.org/10.3390/math11071691 - 1 Apr 2023

Cited by 1 | Viewed by 1792

Abstract

Releasing Wolbachia-infected mosquitoes to suppress or replace wild vector mosquitoes has been carried out in 24 countries worldwide, showing great promise in controlling mosquitoes and mosquito-borne diseases. To face the instability of Wolbachia infection in different environments during the area-wide application, we [...] Read more.

Releasing Wolbachia-infected mosquitoes to suppress or replace wild vector mosquitoes has been carried out in 24 countries worldwide, showing great promise in controlling mosquitoes and mosquito-borne diseases. To face the instability of Wolbachia infection in different environments during the area-wide application, we should consider the overlapping of two Wolbachia strains. In this case, bidirectional cytoplasmic incompatibility occurs, which results in mating partners infected with exclusive Wolbachia strains producing inviable offspring. To determine the better Wolbachia candidate for release, we develop an ordinary differential equation model to study the global dynamics for competition between two Wolbachia strains. Our theoretical results on the sharp estimate of stable curves completely determine the fate of the two Wolbachia strains, which help choose appropriate strains for release. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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11 pages, 289 KiB

Open AccessArticle

Oscillatory Properties of Fourth-Order Advanced Differential Equations

by Alanoud Almutairi, Ali Hasan Ali, Omar Bazighifan and Loredana Florentina Iambor

Mathematics 2023, 11(6), 1391; https://doi.org/10.3390/math11061391 - 13 Mar 2023

Cited by 6 | Viewed by 1564

Abstract

This paper presents a study on the oscillatory behavior of solutions to fourth-order advanced differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first and second-order delay differential equations. The results of this work contribute to a deeper understanding [...] Read more.

This paper presents a study on the oscillatory behavior of solutions to fourth-order advanced differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first and second-order delay differential equations. The results of this work contribute to a deeper understanding of fourth-order differential equations and their connections to various branches of mathematics and practical sciences. The findings emphasize the importance of continued research in this area. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

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