Browsing by Author "Tunç, O."

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Advances in the Qualitative Theory of Integro-Differential Equations
(Nova Science Publishers, Inc., 2023) Tunç, O.; Sivasundaram, S.; Tunç, C.
This work investigates the advances from the past until now in the qualitative properties of solutions of linear and nonlinear integro-differential equations (IDEs). Here, we present an extensive literature on the qualitative properties of solutions, including asymptotic stability, uniform stability, instability and global uniform asymptotic stability of the zero solution, as well as boundedness, square integrability and existence of solutions to various linear and non-linear Volterra IDEs, without delay and with delay. We also present some applications of such equations in sciences and engineering. Some examples are given to illustrate the results of this work and show their applications. © 2023 Nova Science Publishers, Inc. All rights reserved.
Correction To: a Fractional Order Covid-19 Epidemic Model With Mittag-Leffler Kernel (Journal of Mathematical Sciences, (2023), 272, 2, (284-306), 10.1007/S10958-023-06417-x)
(Springer, 2023) Khan, H.; Ibrahim, M.; Khan, A.; Tunç, O.; Abdeljawad, T.
Dynamical and Sensitivity Analysis for Fractional Kundu–eckhaus System To Produce Solitary Wave Solutions Via New Mapping Approach
(Taylor and Francis Ltd., 2024) Rehman, A.U.; Riaz, M.B.; Tunç, O.
The fractional Kundu–Eckhaus (FKE) equation, a nonlinear mathematical model, holds significance in assessing optical fibre communication systems. It takes into account various factors, including dispersion, noise and nonlinearity, which can impact the quality of signal and rates of data transmission in the systems of optical fibre. Utilizing the FKE model can contribute to optimizing the features of optical fibre network. In this academic investigation, an innovative mapping approach is applied to the FKE model to unveil novel soliton solutions. This is achieved through the utilization of beta derivative by employing the new mapping method and computer algebraic system such as Maple. The derived results are expressed in terms of hyperbolic and trigonometric functions. Our study elucidates a variety of soliton patterns such as periodic, dark, kink, bright, singular, dark–bright soliton solutions. To facilitate comprehension, certain solutions are visually depicted through two-dimensional, three-dimensional, and phase plots depicting bifurcation characteristics utilizing Maple software. Furthermore, the sensitivity of the model is explored across diverse initial conditions. Our study establishes a connection between computer science and soliton physics, emphasizing the pivotal role of soliton phenomena in advancing simulations and computational modelling. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
Fractional Dynamics and Sensitivity Analysis of Measles Epidemic Model Through Vaccination
(Taylor and Francis Ltd., 2024) Riaz, M.B.; Raza, N.; Martinovic, J.; Bakar, A.; Kurkcu, H.; Tunç, O.
Measles is a highly contagious disease that mainly affects children worldwide. Even though a reliable and effective vaccination is available, there were 140,000 measles deaths worldwide in 2018, and most of them were children under the age five years. In this paper, we comprehensively investigate a novel fractional SVEIR (Susceptible-Vaccinated-Exposed-Infected-Recovered) model of the measles epidemic powered by nonlinear fractional differential equations to understand the epidemic’s dynamical behaviour. We use a non-singular Atangana-Baleanu fractional derivative to analyze the proposed model, taking advantage of non-locality. The existence, uniqueness, positivity and boundedness of the solutions are shown via concepts of fixed point theory, and we also perform the Ulam-Hyers stability of the considered model. The parameter sensitivity is discussed in the context of the variance with each parameter using 3-D graphics based on the basic reproduction number. Moreover, with the Atangana-Toufik numerical scheme, numerical findings are depicted for different fractional-order values. The presented approach produce results that are efficiently consistent and in excellent agreement with the theoretical results. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
A Fractional Order Covid-19 Epidemic Model With Mittag–leffler Kernel
(Springer, 2023) Khan, H.; Ibrahim, M.; Khan, A.; Tunç, O.; Abdeljawad, T.
We consider a nonlinear fractional-order Covid-19 model in a sense of the Atagana–Baleanu fractional derivative used for the analytic and computational studies. The model consists of six classes of persons, including susceptible, protected susceptible, asymptomatic infected, symptomatic infected, quarantined, and recovered individuals. The model is studied for the existence of solution with the help of a successive iterative technique with limit point as the solution of the model. The Hyers–Ulam stability is also studied. A numerical scheme is proposed and tested on the basis of the available literature. The graphical results predict the curtail of spread within the next 5000 days. Moreover, there is a gradual increase in the population of protected susceptible individuals. © 2023, Springer Nature Switzerland AG.
New Qualitative Criteria for Solutions of Volterra Integro-Differential Equations
(Taylor and Francis Ltd., 2018) Tunç, C.; Tunç, O.
In this paper, we consider certain non-linear scalar Volterra integro-differential equations and Volterra integro-differential systems of first order. We investigate the boundedness, stability, uniformly asymptotic stability, integrability and square integrability of solutions to the scalar equations and the system considered. The technique used to prove the results of the paper is based on the second method of Lyapunov. From the obtained results, we extend and improve some related results that can be found in the literature. © 2018, © 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
A Note on Certain Qualitative Properties of a Second Order Linear Differential System
(Natural Sciences Publishing USA, 2015) Tunç, C.; Tunç, O.
In this note, two theorems are presented concerning the well known second order linear differential system X+a(t)X =P(t). While the results are not new, the proofs presented simplify previous works since the Gronwall inequality is avoided which is the usual case. The technique of proof involves the integral test and examples are included to illustrate the results. © 2015 NSP.
A Note on the Stability and Boundedness of Solutions To Non-Linear Differential Systems of Second Order
(University of Bahrain, 2017) Tunç, C.; Tunç, O.
In this work, we are concerned with the investigation of the qualitative behaviors of certain systems of non-linear differential equations of second order. We make a comparison between applications of the integral test and Lyapunov's function approach on some recent stability and boundedness results in the literature. An example is furnished to illustrate the hypotheses and main results in this paper. © 2017 University of Bahrain
On System of Variable Order Nonlinear P-Laplacian Fractional Differential Equations With Biological Application
(MDPI, 2023) Khan, H.; Alzabut, J.; Gulzar, H.; Tunç, O.; Pinelas, S.
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. © 2023 by the authors.
On the Fixed Point Theorem for Large Contraction Mappings With Applications To Delay Fractional Differential Equations
(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Mesmouli, M.B.; Akın, E.; Iambor, L.F.; Tunç, O.; Hassan, T.S.
This paper explores a new class of mappings and presents several fixed-point results for these mappings. We define these mappings by combining well-known mappings in the literature, specifically the large contraction mapping and Chatterjea’s mapping. This combination allows us to achieve significant fixed-point results in complete metric spaces, both in a continuous and a non-continuous sense. Additionally, we provide an explicit example to validate our findings. Furthermore, we discuss a general model for fractional differential equations using the Caputo derivative. Finally, we outline the benefits of our study and suggest potential areas for future research. © 2024 by the authors.
On the Qualitative Analyses of Integro-Differential Equations With Constant Time Lag
(Natural Sciences Publishing, 2020) Tunç, O.
In this article, some new investigations are done on stability, asymptotically stability and instability of the zero solution and boundedness, integrability of solutions and integrability of derivatives of solutions of certain nonlinear Volterra delayed integrodifferential equations (DIDEs) by using the Lyapunov's functional method. To fulfill the aim of this work, three meaningful Lyapunov functionals are defined as main tools. Then, five new results are given on the mentioned qualitative properties of solutions of the considered DIDEs. Our results have contributions to the relevant literature and they have sufficient conditions. They include and improve some former results that can be found in [1]. ©2020 NSP Natural Sciences Publishing Cor.
On the Qualitative Analysis of Solutions of a Class of Nonlinear Differential Equations of the Second Order With Constant Delay
(Springer, 2023) Tunç, C.; Tunç, O.
We consider a class of nonlinear second-order differential equations with constant delay and investigate qualitative properties of the solutions, namely, global stability of the zero solution, eventually uniform boundedness of solutions, existence of periodic solutions, and existence of a unique stationary oscillation of the considered equations. As far as the technique of the proofs is concerned, we use the Lyapunov–Krasovskii functional method and the second Lyapunov method to prove our main results. We also improve and correct some former results available from the literature. Finally, in some particular cases, we provide three examples as illustrations and applications of the obtained new results. Hence, we make some contributions to the topic of the paper. © 2023, Springer Nature Switzerland AG.
On the Qualitative Behaviors of Volterra-Fredholm Integro Differential Equations With Multiple Time-Varying Delays
(Taylor and Francis Ltd., 2024) Tunç, C.; Tunç, O.
This article considers a Volterra-Fredholm integro-differential equation including multiple time-varying delays. The aim of this article is to study the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the Volterra-Fredholm integro-differential equation including multiple time-varying delays. We prove four new results in connection with the uniqueness of solution, the Ulam–Hyers–Rassias stability and the Ulam–Hyers stability of the considered Volterra-Fredholm integro-differential equation, respectively. The new results of this article involve sufficient conditions. The techniques of the proofs depend on the fixed point method according to the definitions of a suitable metric, operators and the related calculations. In particular case of the considered Volterra-Fredholm integro-differential equation, two illustrative examples are presented to verify the applications of the results. This article also involves some new complementary outcomes in connection with qualitative theory of Volterra-Fredholm integro-differential equations with delays. © 2024 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group on behalf of the University of Bahrain.
On the Ulam Stabilities of Nonlinear Integral Equations and Integro-Differential Equations
(John Wiley and Sons Ltd, 2024) Tunç, O.; Tunç, C.; Petruşel, G.; Yao, J.-C.
In this research, two systems of nonlinear Volterra integral equation and Volterra integro-differential equation were considered. New results in sense of Ulam stabilities in relation to these two systems were proved on a finite interval. The proof of the results on the Ulam stabilities of that classes of the equations are based on the nonlinear alternative related to Banach's contraction principle. The outcomes of this research give new contribution to the theory of Ulam stabilities. © 2024 John Wiley & Sons, Ltd.
Signature of Conservation Laws and Solitary Wave Solution With Different Dynamics in Thomas–fermi Plasma: Lie Theory
(Elsevier B.V., 2024) Fayyaz, M.; Riaz, M.B.; Rehman, M.J.U.; Tunç, O.
We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved. © 2024 The Authors
The Stability of Nonlinear Delay Integro-Differential Equations in the Sense of Hyers-Ulam
(Walter de Gruyter GmbH, 2023) Graef, J.R.; Tunç, C.; Sengun, M.; Tunç, O.
In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach's fixed point theorem, the construction of a Picard operator, and an application of Pachpatte's inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions. © 2023 the author(s), published by De Gruyter.
Thermal and Flow Properties of Jeffrey Fluid Through Prabhakar Fractional Approach: Investigating Heat and Mass Transfer With Emphasis on Special Functions
(Springer, 2024) Riaz, M.B.; Rehman, A.U.; Chan, C.K.; Zafar, A.A.; Tunç, O.
The core purpose of this work is the investigation, formulation and develop a mathematical model by dint of a new fractional modeling approach namely Prabhakar fractional operator to study the dynamics of Jeffrey fluid flow and heat transfer phenomena. Exact analysis of natural convective flow of the Jeffrey fluid to derive analytical solutions with the non-integer order derivative Prabhakar fractional operator with non-singular type kernel alongwith application of generalized laws namely Fick’s and Fourier’s are reported here. This fractional operator has multi parameter generalized Mittag-Leffler kernel. The fluid flow is elaborated near an infinitely vertical plate with characteristics velocity u0. The modeling of the considered problem is done in terms of the partial differential equations together with generalized boundary conditions. Introduced the appropriate set of variables to transform the governing equations into dimensionless form. Laplace transform is operated on the fractional system of equations and results are presented in series form and also presented the solution in the form of special functions. The pertinent parameter’s influence such as α, Pr, β, Gm, Sc, γ, Gr on the fluid flow is brought under consideration to reveal the interesting results. In comparison, we noticed the Prabhakar-like non integer approach shows better results than the existing operators in the literature, and graphs are drawn to show the results. Also, obtained the results in a limiting sense such as second grade fluid, Newtonian fluid in fractionalized form as well as Jeffrey and viscous fluid models for classical form from Prabhakar-like non integer Jeffrey fluid model. © The Author(s), under exclusive licence to Springer Nature India Private Limited 2024.