Browsing by Author "Tunc, Osman"

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The Analysis of Some Special Results of a Lasota-Wazewska Model With Mixed Variable Delays
(Springer, 2021) Yazgan, Ramazan; Tunc, Osman
This study is about getting some conditions that guarantee the existence and uniqueness of the weighted pseudo almost periodic (WPAP) solutions of a Lasota-Wazewska model with time-varying delays. Some adequate conditions have been obtained for the existence and uniqueness of the WPAP solutions of the Lasota-Wazewska model, which we dealt with using some differential inequalities, the WPAP theory, and the Banach fixed point theorem. Besides, an application is given to demonstrate the accuracy of the conditions of our main results.
Analysis of the Integro-Differential Jaulent-Miodek Evolution Equation by Nonlinear Self-Adjointness and Lie Theory
(Springer, 2025) Junaid-U-Rehman, Muhammad; Riaz, Muhammad Bilal; Tunc, Osman; Martinovic, Jan
The current exploration is related to the extraction of the exact explicit solutions of the Integrodifferential Jaulent-Miodek evolution (IDJME) equation. In general, the Jaulent-Miodek equation has many applications in many divisions of physics, for example optics and fluid dynamics. The symmetries of this (2 + 1)-dimensional (IDJME) equation are derived, and it admits the 8th Lie algebra. The similarity transformation method is considered to convert the NLPDE to the nonlinear ODE by using the translational symmetries. Then, we calculated the traveling-wave solutions by an analytical method, namely the extended direct algebraic method. Some obtained solutions are represented by giving suitable parameter values to understand their physical interpretation. The results obtained from this method involve various types of functions, for example, exponential, logarithmic, hyperbolic, and trigonometric. The self-adjointness theory is employed to classify and help us to compute the conserved quantities of the assumed model. A similar work related to this does not exist in the literature.
Analytical Optical Soliton Solution & Stability Analysis of the Dimensionless Time-Dependent Paraxial Equation
(Indian Assoc Cultivation Science, 2025) Abdullah, Ghaus ur; Rahman, Ghaus U.; Tunc, Osman
This study investigates the analytical solutions of optical solitons governed by the a time-dependent Paraxial Equation incorporating Kerr law nonlinearity. A rigorous analytical method is developed to obtain these solutions, offering insights into the behavior of optical solitons in nonlinear media. Here, we have applied the eMETEM approach for the first time to create robust solutions to the time-dependent Paraxial Equation. Building an efficient plan to solve the governing model has been our main goal. The Kerr law nonlinearity, which describes the intensity-dependent refractive index, profoundly affects the propagation characteristics of optical pulses. Through systematic analysis, key properties such as soliton formation, stability, and evolution are elucidated. The derived analytical solutions provide a valuable framework for understanding the intricate dynamics of optical solitons in nonlinear media, facilitating advancements in various applications including optical communication and signal processing.
Analytical Study of Fractional Dna Dynamics in the Peyrard-Bishop Oscillator-Chain Model
(Elsevier, 2024) Riaz, Muhammad Bilal; Fayyaz, Marriam; Rahman, Riaz Ur; Martinovic, Jan; Tunc, Osman
In this research, we present a new auxiliary equation approach, which uses two distinct fractional derivatives: /3- and M-truncated fractional derivatives to explore the space-time fractional Peyrard-Bishop DNA dynamic model equation. This examines the nonlinear interplay between neighboring displacements and hydrogen bonds through mathematical modeling of DNA vibration dynamics. The solutions are tasked with examining the nonlinear interaction among neighboring displacements of the DNA strand. The generated solutions exhibit various wave patterns under varying fractional values and parametric conditions: w-shape, bright, combined periodic wave solutions, dark-bright, bell shaped, m-shaped, w-shaped with two bright solutions, and m-shape with two dark solutions. Graphical representations provide a complete analysis of these physical features. The results demonstrate the successful implementation of the proposed approach, which will be advantageous for locating analytical remedies to more nonlinear challenges.
Application of Commutator Calculus To the Study of Linear Impulsive Systems
(Elsevier Science Bv, 2019) Slyn'ko, V. I.; Tunc, Osman; Bivziuk, V. O.
In this paper, the formulas of commutator calculus are applied to the investigation of the stability of linear impulsive differential equations. It is assumed that the moments of impulse action satisfy the average dwell-time (ADT) condition. Sufficient conditions for the asymptotic stability of linear impulsive differential equations in a Banach space are obtained. In the Hilbert space, the stability of the original linear differential equation is reduced to the investigation of a linear differential equation with equidistant moments of impulse action and perturbed discrete dynamics. This reduction simplifies the application of Lyapunov's direct method and the construction of Lyapunov functions. We give examples in the spaces R-2 and X = C[0, l] to illustrate the effectiveness of results obtained. Finally, a sufficient generality of the obtained results on the dynamic properties of linear operators of the linear impulsive differential equation is established. (C) 2018 Elsevier B.V. All rights reserved.
An Application of Lyapunov-Razumikhin Method To Behaviors of Volterra Integro-Differential Equations
(Springer-verlag Italia Srl, 2021) Nieto, Juan J.; Tunc, Osman
This work presents some extensions and improvements of former results that allow proving asymptotic stability, uniform stability and global uniform asymptotic stability of zero solution to a class of non-linear Volterra integro-differential equations (VIDEs). Via the Lyapunov-Krasovskii and the Lyapunov-Razumikhin methods, three new results are proved on the mentioned concepts. These results are proved using Lyapunov functional and quadratic Lyapunov function. The results of this paper improve and extend the known ones in the literature. Some examples are given to validate these results and the concepts introduced.
Asymptotic Behavior of Solutions of Volterra Integro-Differential Equations With and Without Retardation
(Rocky Mt Math Consortium, 2021) Graef, John R.; Tunc, Osman
Asymptotic stability, uniform stability, integrability, and boundedness of solutions of Volterra integro- differential equations with and without constant retardation are investigated using a new type of Lyapunov- Krasovskii functionals. An advantage of the new functionals used here is that they eliminate using Gronwall's inequality. Compared to related results in the literature, the conditions here are more general, simple, and convenient to apply. Examples to show the application of the theorems are included.
Bifurcation Dynamics and Control Mechanism of a Fractional-Order Delayed Brusselator Chemical Reaction Model
(Univ Kragujevac, Fac Science, 2023) Xu, Changjin; Mu, Dan; Liu, Zixin; Pang, Yicheng; Aouiti, Chaouki; Tunc, Osman; Zeb, Anwar
Building differential dynamical systems to describe the changing relationship among chemical components is a vital aspect in chemistry. In this present manuscript, we put forward a new fractionalorder delayed Brusselator chemical reaction model. By virtue of contraction mapping principle, we investigate the existence and uniqueness of the solution of fractional-order delayed Brusselator chemical reaction model. With the aid of mathematical analysis technique, we consider the non-negativeness of the solution of the fractionalorder delayed Brusselator chemical reaction model. Making use of the theory of fractional-order dynamical system, we explore the stability and Hopf bifurcation issue of the fractional-order delayed Brusselator chemical reaction model. By designing a reasonable PD. controller, we have availably controlled the time of emergence of Hopf bifurcation of the fractional-order delayed Brusselator chemical reaction model. A sufficient criterion guaranteeing the stability and the onset of Hopf bifurcation of the fractional-order controlled delayed Brusselator chemical reaction model is set up. Computer simulations are implemented to validate the theoretical findings. The derived fruits of this manuscript have great theoretical significance in controlling the concentrations of chemical substances.
A Coupled Nonlinear System of Integro-Differential Equations Using Modified Abc Operator
(World Scientific Publ Co Pte Ltd, 2025) Khan, Hasib; Alzabut, Jehad; Almutairi, D. K.; Alqurashi, Wafa khalaf; Pinelas, Sandra; Tunc, Osman; Azim, Mohammad athar
This paper explores the necessary conditions required for the solutions of an integro-differential system of n-fractional differential equations (n-FDEs) in the modified-ABC case of derivative with initial conditions. The presumed problem is a linearly perturbed system. Some classical fixed point theorems are utilized to derive the solution existence criteria. Additionally, a numerical methodology utilizing Lagrange's interpolation polynomial is developed and implemented in a dynamical framework of a power system for practical applications. In addition, we investigate the properties of Hyers-Ulam's stability and uniqueness. The findings are evaluated using graphical methods to assess the precision and suitability of the approaches
Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems
(Mdpi, 2021) Tunc, Osman; Tunc, Cemil; Wang, Yuanheng
This paper deals with non-perturbed and perturbed systems of nonlinear differential systems of first order with multiple time-varying delays. Here, for the considered systems, easily verifiable and applicable uniformly asymptotic stability, integrability, and boundedness criteria are obtained via defining an appropriate Lyapunov-Krasovskii functional (LKF) and using the Lyapunov-Krasovskii method (LKM). Comparisons with a former result that can be found in the literature illustrate the novelty of the stability theorem and show new contributions to the qualitative theory of solutions. A discussion of two illustrative examples and the obtained results are presented.
Existence and Stabilization for Impulsive Differential Equations of Second Order With Multiple Delays
(Texas State Univ, 2024) Pinelas, Sandra; Tunc, Osman; Korkmaz, Erdal; Tunc, Cemil
Existence and stability of solutions are important parts in the qualitative study of delay differential equations. The stabilizing by imposing proper impulse controls are used in many areas of natural sciences and engineering. This article provides sufficient conditions for the existence and exponential stabilization of solutions to delay impulsive differential equations of second-order with multiple delays. The main tools in this article are the Schaefer fixed point theorem, fixed impulse effects, and Lyapunov-Krasovskii functionals. The outcomes extend earlier results in the literature.
Existence of Solutions for Nonlinear Impulsive Multiple Retarded Differential and Impulsive Integro-Differential Equations of Second Order
(Yokohama Publ, 2024) Bohner, Martin; Tunc, Osman; Tunc, Cemil
. In this paper, we deal with the existence of solutions of certain impulsive multiple retarded differential equations (ImMRDEs) and impulsive multiple retarded integro-differential equations (ImMRIDEs) of second order. We prove two new results on the existence of solutions of the considered ImMRDE and ImMRIDE of second order. The technique of the proofs depends on the Schaefer fixed point theorem (Schaefer FPT) and fixed moments of impulse effects. The outcomes of this paper have more general forms and improve the known results in the relevant literature, and they have new contributions to the theory of impulsive retarded differential equations (ImMRDEs). The outcomes of this paper have also new complementary properties for the works in relation to the symmetry of impulsive differential equations (ImDEs) of second order with or without delay, impulsive retarded integro-differential equations (ImMRIDEs) of second order, and some others.
Exploring Analytical Solutions and Modulation Instability for the Nonlinear Fractional Gilson-Pickering Equation
(Elsevier, 2024) Rahman, Riaz Ur; Riaz, Muhammad Bilal; Martinovic, Jan; Tunc, Osman
The primary goal of this research is to explore the complex dynamics of wave propagation as described by the nonlinear fractional Gilson-Pickering equation (fGPE), a pivotal model in plasma physics and crystal lattice theory. Two alternative fractional derivatives, termed fi and M -truncated, are employed in the analysis. The new auxiliary equation method (NAEM) is applied to create diverse explicit solutions for surface waves in the given equation. This study includes a comparative evaluation of these solutions using different types of fractional derivatives. The derived solutions of the nonlinear fGPE, which include unique forms like dark, bright, and periodic solitary waves, are visually represented through 3D and 2D graphs. These visualizations highlight the shapes and behaviors of the solutions, indicating significant implications for industry and innovation. The proposed method's ability to provide analytical solutions demonstrates its effectiveness and reliability in analyzing nonlinear models across various scientific and technical domains. A comprehensive sensitivity analysis is conducted on the dynamical system of the f GPE. Additionally, modulation instability analysis is used to assess the model's stability, confirming its robustness. This analysis verifies the stability and accuracy of all derived solutions.
Exponential Stability in the L P-Norm of Nonlinear Coupled Hyperbolic Spatially Inhomogeneous Systems
(Elsevier Science inc, 2024) Slynko, Vitalii; Tunc, Osman; Atamas, Ivan
We study exponential stability of equilibrium in the L (P) -norm (1 < P <= infinity, P not equal 2) of nonlinear 1D systems of hyperbolic equations. A method of construction of Lyapunov functions based on the W. Magnus representation of fundamental solutions of ordinary differential equation (ODE) linear systems is proposed. Sufficient conditions for exponential L (P) -stability (1 < P < infinity, P not equal 2) are obtained and sufficient conditions for exponential L (infinity) -stability are derived by passing to the limit. The obtained results are compared with the well-known results.
A Fractal-Fractional Covid-19 Model With a Negative Impact of Quarantine on the Diabetic Patients
(Elsevier, 2023) Khan, Hasib; Alzabut, Jehad; Tunc, Osman; Kaabar, Mohammed K. A.
In this article, we consider a Covid-19 model for a population involving diabetics as a subclass in the fractal-fractional (FF) sense of derivative. The study includes: existence results, uniqueness, stability and numerical simulations. Existence results are studied with the help of fixed-point theory and applications. The numerical scheme of this paper is based upon the Lagrange's interpolation polynomial and is tested for a particular case with numerical values from available open sources. The results are getting closer to the classical case for the orders reaching to 1 while all other solutions are different with the same behavior. As a result, the fractional order model gives more significant information about the case study.
A Fractional Order Zika Virus Model With Mittag?leffler Kernel
(Pergamon-elsevier Science Ltd, 2021) Begum, Razia; Tunc, Osman; Khan, Hasib; Gulzar, Haseena; Khan, Aziz
Zika virus is one of the lethal virus which is a threat to humans health. It can be transmitted from human to human, from mosquitos to human, from human to mosquitos. Since there is no vaccine or complete treatment of the Zika viral infection. Therefore, scientists are working on the optimal control strategies. One of the control strategies is the awareness about the spread. In this article, we have presented and analyzed a mathematical model for the Zika virus and have checked the results on long time. The model has closer results to the classical based on our numerical scheme by the help of Lagrange's interpolation polynomial. (c) 2021 Elsevier Ltd. All rights reserved.
Global Existence and Uniqueness of Solutions of Integral Equations With Multiple Variable Delays and Integro Differential Equations: Progressive Contractions
(Mdpi, 2024) Tunc, Osman; Tunc, Cemil; Yao, Jen-Chih
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.
Mathematical Analysis of Stochastic Epidemic Model of Mers-Corona & Application of Ergodic Theory
(Elsevier, 2023) Hussain, Shah; Tunc, Osman; Rahman, Ghaus Ur; Khan, Hasib; Nadia, Elissa
The "Middle East Respiratory" (MERS-Cov) is among the world's dangerous diseases that still exist. Presently it is a threat to Arab countries, but it is a horrible prediction that it may propagate like COVID-19. In this article, a stochastic version of the epidemic model, MERS-Cov, is presented. Initially, a mathematical form is given to the dynamics of the disease while incorporating some unpredictable factors. The study of the underlying model shows the existence of positive global solution. Formulating appropriate Lyapunov functionals, the paper will also explore parametric conditions which will lead to the extinction of the disease from a community. Moreover, to reveal that the infection will persist, ergodic stationary distribution will be carried out. It will also be shown that a threshold quantity exists, which will determine some essential parameters for exploring other dynamical aspects of the main model. With the addition of some examples, the underlying stochastic model of MERS-Cov will be studied graphically for more illustration.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
Modeling and Simulations for the Mitigation of Atmospheric Carbon Dioxide Through Forest Management Programs
(Amer inst Mathematical Sciences-aims, 2024) Riaz, Muhammad Bilal; Raza, Nauman; Martinovic, Jan; Bakar, Abu; Tunc, Osman
The growing global population causes more anthropogenic carbon dioxide (CO2) 2 ) emissions and raises the need for forest products, which in turn causes deforestation and elevated CO2 2 levels. A rise in the concentration of carbon dioxide in the atmosphere is the major reason for global warming. Carbon dioxide concentrations must be reduced soon to achieve the mitigation of climate change. Forest management programs accommodate a way to manage atmospheric CO2 2 levels. For this purpose, we considered a nonlinear fractional model to analyze the impact of forest management policies on mitigating atmospheric CO2 2 concentration. In this investigation, fractional differential equations were solved by utilizing the Atangana Baleanu Caputo derivative operator. It captures memory effects and shows resilience and efficiency in collecting system dynamics with less processing power. This model consists of four compartments, the concentration of carbon dioxide C (t), human population N (t), forest biomass B (t), and forest management programs P (t) at any time t. The existence and uniqueness of the solution for the fractional model are shown. Physical properties of the solution, non-negativity, and boundedness are also proven. The equilibrium points of the model were computed and further analyzed for local and global asymptotic stability. For the numerical solution of the suggested model, the Atangana-Toufik numerical scheme was employed. The acquired results validate analytical results and show the significance of arbitrary order delta . The effect of deforestation activities and forest management strategies were also analyzed on the dynamics of atmospheric carbon dioxide and forest biomass under the suggested technique. The illustrated results describe that the concentration of CO2 2 can be minimized if deforestation activities are controlled and proper forest management policies are developed and implemented. Furthermore, it is determined that switching to low-carbon energy sources, and developing and implementing more effective mitigation measures will result in a decrease in the mitigation of CO 2 .
New and Improved Criteria on Fundamental Properties of Solutions of Integro-Delay Differential Equations With Constant Delay
(Mdpi, 2021) Tunc, Cemil; Wang, Yuanheng; Tunc, Osman; Yao, Jen-Chih
This paper is concerned with certain non-linear unperturbed and perturbed systems of integro-delay differential equations (IDDEs). We investigate fundamental properties of solutions such as uniformly stability (US), uniformly asymptotically stability (UAS), integrability and instability of the un-perturbed system of the IDDEs as well as the boundedness of the perturbed system of IDDEs. In this paper, five new and improved fundamental qualitative results, which have less conservative conditions, are obtained on the mentioned fundamental properties of solutions. The technique used in the proofs depends on Lyapunov-Krasovski functionals (LKFs). In particular cases, three examples and their numerical simulations are provided as numerical applications of this paper. This paper provides new, extensive and improved contributions to the theory of IDDEs.