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Öğe A Comparative Analysis of the Fractional-Order Coupled Korteweg-De Vries Equations with the Mittag-Leffler Law(Hindawi Ltd, 2022) Aljahdaly, Noufe H.; Akgul, Ali; Shah, Rasool; Mahariq, Ibrahim; Kafle, JeevanThis article applies efficient methods, namely, modified decomposition method and new iterative transformation method, to analyze a nonlinear system of Korteweg-de Vries equations with the Atangana-Baleanu fractional derivative. The nonlinear fractional coupled systems investigated in this current analysis are the system of Korteweg-de Vries and the modified system of Korteweg-de Vries equations applied as a model in nonlinear physical phenomena arising in chemistry, biology, physics, and applied sciences. Approximate analytical results are represented in the form of a series with straightforward components, and some aspects showed an appropriate dependence on the values of the fractional-order derivatives. The convergence and uniqueness analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. The series result achieved applying this technique is proved to be accurate and reliable with minimal calculations. The numerical simulations for obtained solutions are discussed for different values of the fractional order.Öğe A Comparative Study of Time Fractional Nonlinear Drinfeld-Sokolov-Wilson System via Modified Auxiliary Equation Method(Mdpi, 2023) Akram, Ghazala; Sadaf, Maasoomah; Zainab, Iqra; Abbas, Muhammad; Akgul, AliThe time-fractional nonlinear Drinfeld-Sokolov-Wilson system, which has significance in the study of traveling waves, shallow water waves, water dispersion, and fluid mechanics, is examined in the presented work. Analytic exact solutions of the system are produced using the modified auxiliary equation method. The fractional implications on the model are examined under b-fractional derivative and a new fractional local derivative. Extracted solutions include rational, trigonometric, and hyperbolic functions with dark, periodic, and kink solitons. Additionally, by specifying values for fractional parameters, graphs are utilized to comprehend the fractional effects on the obtained solutions.Öğe A comparative study on non-Newtonian fractional-order Brinkman type fluid with two different kernels(Wiley, 2024) Sarwar, Shahzad; Aleem, Maryam; Imran, Muhammad Asjad; Akgul, AliThis study carried out the free convective non-Newtonian fluid of Brinkman type flow near an upright plate moving with velocity f(t). A fractional order model for non-Newtonian fluid of Brinkman type flow is proposed. The time derivative in the proposed fractional flow model is considered by using the two types of fractional derivatives namely Caputo fractional derivative and Atangana-Baleanu fractional derivative. The system of conjugated fractional partial differential equations for the temperature theta, velocity u and concentration C are worked out by applying optimal homotopy asymptotic technique. The effectuate of tangible and fractional variables on the domains of velocity u, temperature theta and concentration C are envisioned graphically. The rate of heat and mass transfer in the form of Nu and Sh is also calculated for both fractional derivatives. The numerical results reveal the efficiency, reliability, significant features, and simple in computation with high accuracy of consider method for non-Newtonian fractional order fluid of Brinkman type flow. We ascertained that our results are in excellent agreement with the exact results.Öğe A comprehensive mathematical structuring of magnetically effected Sutterby fluid flow immersed in dually stratified medium under boundary layer approximations over a linearly stretched surface(Elsevier, 2022) Bilal, Sardar; Shah, Imtiaz Ali; Akgul, Ali; Tekin, Merve Tastan; Botmart, Thongchai; Yousef, El SayedOn the implication of tension force, viscoelastic materials deform in accordance with viscous and elastic characterization. Dilute polymer solutions portray the examples of viscoelastic liquids and Sutterby model successfully represent it due to high degree polymerization distributions. In addition, polymer aqueous solutions behave as shear thinning and thickening liquids in response to infinite shear stress so Sutterby fluid is considered as best model to depict the features of liquids at high stress magnitude. Diverse utilities of diluted polymeric solutions are encountered in industrial, biological and technological practices, for instance, agricultural sprayers, cleansing products, clay coaters, polymerized melts and many more. So, this research communicates theoretical and computational thermal assessment of Sutterby fluid containing radiation aspects over a linearly moving sheet embedded in stratified medium which exposes the novelty of work. Moreover, the impacts of magnetic field and chemical reactions are also obliged. So, the principal objective pertains to adumbrate flow behavior of Sutterby liquid in the attendance of aforementioned physical parameters. Mathematical formulation in view of governing relations are changed into nondimensionalized form through transformation approach. Convergent and accurate solution is accessed through renowned numerical shooting procedure along with integrated Runge-Kutta scheme. The computed results of emerged parameters on velocity, concentration and thermal fields are revealed by means of snapshots. Magnitude of associated wall drag coefficient and reduced heat and mass fluxes are explained in graphical and tabular formats. The salient outcomes are as follows. Consequence of the proposed research investigation infers that augmenting power index momentum distribution decays whereas skin friction uplifts. Furthermore, it is inferred that concentration distribution amplifies against upsurging magnitude of solutal stratification parameter while opposite nature is noticed in case of temperature profile against associated stratification parameter. Additionally, it is concluded that escalating magnitude of radiation parameter tends to elevates the dimensionless temperature profile. Subsequently, rise in concentration profile against Schmidt number is observed whereas against Prandtl number temperature of fluid expressing declining aptitude. Also, declining response in momentum of fluid is manifested against Reynold and Deborah numbers.(c) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).Öğe A comprehensive study of subdivision collocation method for Burgers' equation(Taylor & Francis Inc, 2024) Ejaz, Syeda Tehmina; Bibi, Saima; Akgul, Ali; Hassani, Murad KhanThis study explores the use of subdivision schemes to efficiently solve Burgers' equation. Burgers' equation is a fundamental fluid dynamics equation that describes the nonlinear behavior of fluid flow. This type of nonlinear equation is difficult to solve analytically, which makes the numerical solution an important tool. The subdivision collocation method (SCM) converts Burgers' equation into a system of algebraic linear equations using the quasilinearization technique. The results of this study demonstrate that the proposed approach yields accurate numerical solutions for Burgers' equation. Additionally, the subdivision approach is computationally efficient and requires fewer computational resources than existing numerical methods, making it a promising tool for solving Burgers' equation in practical applications. Overall, this study provides valuable insights into the approximate solution of Burgers' equation by implementing subdivision schemes.Öğe A computational fluid dynamics analysis on Fe3O4-H2O based nanofluid axisymmetric flow over a rotating disk with heat transfer enhancement(Nature Portfolio, 2023) Farooq, Umar; Hassan, Ali; Fatima, Nahid; Imran, Muhammad; Alqurashi, M. S.; Noreen, Sobia; Akgul, AliIn present times modern electronic devices often come across thermal difficulties as an outcome of excessive heat production or reduction in surface area for heat exclusion. The current study is aimed to inspect the role of iron (III) oxide in heat transfer enhancement over the rotating disk in an axisymmetric flow. Water is utilized as base fluid conveying nano-particle over the revolving axisymmetric flow mechanism. Additionally, the computational fluid dynamics (CFD) approach is taken into consideration to design and compute the present problem. For our convenience, two-dimensional axisymmetric flow configurations are considered to illustrate the different flow profiles. For radial, axial, and tangential velocity profiles, the magnitude of the velocity, streamlines, and surface graphs are evaluated with the similarity solution in the computational fluid dynamics module. The solution of dimensionless equations and the outcomes of direct simulations in the CFD module show a comparable solution of the velocity profile. It is observed that with an increment in nanoparticle volumetric concentration the radial velocity decline where a tangential motion of flow enhances. Streamlines stretch around the circular surface with the passage of time. The high magnetization force 0 = m(1) = 6 resist the free motion of the nanofluid around the rotating disk. Such research has never been done, to the best of the researchers' knowledge. The outcomes of this numerical analysis could be used for the design, control, and optimization of numerous thermal engineering systems, as described above, due to the intricate physics of nanofluid under the influences of magnetic field and the inclusion of complex geometry. Ferrofluids are metallic nanoparticle colloidal solutions. These kinds of fluids do not exist in nature. Depending on their purpose, ferrofluids are produced using a variety of processes. One of the most essential characteristics of ferrofluids is that they operate in a zero-gravity environment. Ferrofluids have a wide range of uses in engineering and medicine. Ferrofluids have several uses, including heat control loudspeakers and frictionless sealing. In the sphere of medicine, however, ferrofluid is employed in the treatment of cancer via magneto hyperthermia.Öğe A Computational Scheme for the Numerical Results of Time-Fractional Degasperis-Procesi and Camassa-Holm Models(Mdpi, 2022) Nadeem, Muhammad; Jafari, Hossein; Akgul, Ali; De la Sen, ManuelThis article presents an idea of a new approach for the solitary wave solution of the modified Degasperis-Procesi (mDP) and modified Camassa-Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (LT) and homotopy perturbation method (HPM) to formulate the idea of the Laplace transform homotopy perturbation method (LHPTM). This study is considered under the Caputo sense. This proposed strategy does not depend on any assumption and restriction of variables, such as in the classical perturbation method. Some numerical examples are demonstrated and their results are compared graphically in 2D and 3D distribution. This approach presents the iterations in the form of a series solutions. We also compute the absolute error to show the effective performance of this proposed scheme.Öğe A cotangent fractional Gronwall inequality with applications(Amer Inst Mathematical Sciences-Aims, 2024) Sadek, Lakhlifa; Akgul, Ali; Bataineh, Ahmad Sami; Hashim, IshakThis article presents the cotangent fractional Gronwall inequality, a novel understanding of the Gronwall inequality within the context of the cotangent fractional derivative. We furnish an explanation of the cotangent fractional derivative and emphasize a selection of its distinct characteristics before delving into the primary findings. We present the cotangent fractional Gronwall inequality (Lemma 3.1) and a Corollary 3.2 using the Mittag-Leffler function, we establish singularity and compute an upper limit employing the Mittag-Leffler function for solutions in a nonlinear delayed cotangent fractional system, illustrating its practical utility. To underscore the real -world relevance of the theory, a tangible instance is given.Öğe A fast iterative method to find the matrix geometric mean of two HPD matrices(Wiley, 2019) Bin Jebreen, Haifa; Akgul, AliThe purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three-step root-solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well-known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.Öğe A finite difference scheme to solve a fractional order epidemic model of computer virus(Amer Inst Mathematical Sciences-Aims, 2023) Iqbal, Zafar; Rehman, Muhammad Aziz-ur; Imran, Muhammad; Ahmed, Nauman; Fatima, Umbreen; Akgul, Ali; Rafiq, MuhammadIn this article, an analytical and numerical analysis of a computer virus epidemic model is presented. To more thoroughly examine the dynamics of the virus, the classical model is transformed into a fractional order model. The Caputo differential operator is applied to achieve this. The Jacobian approach is employed to investigate the model's stability. To investigate the model's numerical solution, a hybridized numerical scheme called the Grunwald Letnikov nonstandard finite difference (GL-NSFD) scheme is created. Some essential characteristics of the population model are scrutinized, including positivity boundedness and scheme stability. The aforementioned features are validated using test cases and computer simulations. The mathematical graphs are all detailed. It is also investigated how the fundamental reproduction number R0 functions in stability analysis and illness dynamics.Öğe A fractal fractional model for computer virus dynamics(Pergamon-Elsevier Science Ltd, 2021) Akgul, Ali; Fatima, Umbreen; Iqbal, Muhammad Sajid; Ahmed, Nauman; Raza, Ali; Iqbal, Zafar; Rafiq, MuhammadThe gist behind this study is to extend the classical computer virus model into fractal fractional model and subsequently to solve the model by Atangana-Toufik method. This method solve nonlinear model under consideration very efficiently. We use the Mittag-Leffler kernels on the proposed model. Atangana-Baleanu integral operator is used to solve the set of fractal-fractional expressions. In this model, three types of equilibrium points are described i.e trivial, virus free and virus existing points. These fixed points are used to establish some standard results to discuss the stability of the system by calculating the Jacobian matrices at these points. Routh-Hurwitz criteria is used to verify that the system is locally asymptotically stable at all the steady states. The emphatic role of the basic reproduction number R-0 is also brought into lime light for stability analysis. Sensitivity analysis of R-0 is also discussed. Optimal existence and uniqueness of the solution is the nucleus of this study. Computer simulations and patterns and graphical patterns illustrate reliability and productiveness of the proposed method. (C) 2021 Elsevier Ltd. All rights reserved.Öğe A fractal-fractional sex structured syphilis model with three stages of infection and loss of immunity with analysis and modeling(Elsevier, 2023) Farman, Muhammad; Shehzad, Aamir; Akgul, Ali; Hincal, Evren; Baleanu, Dumitru; El Din, Sayed M.Treponema pallidum, a spiral-shaped bacterium, is responsible for the sexually transmitted disease syphilis. Millions of people in less developed countries are getting the disease despite the accessibility of effective preventative methods like condom use and effective and affordable treatment choices. The disease can be fatal if the patient does not have access to adequate treatment. Prevalence has hovered between endemic levels in industrialized countries for decades and is currently rising. Using the Mittag-Leffler kernel, we develop a fractal-fractional model for the syphilis disease. Qualitative as well as quantitative analysis of the fractional order system are performed. Also, fixed point theory and the Lipschitz condition are used to fulfill the criteria for the existence and uniqueness of the exact solution. We illustrate the system's Ulam-Hyers stability for disease-free and endemic equilibrium. The analytical solution is supported by numerical simulations that show how the dynamics of the spread of syphilis within the population are influenced by fractional-order derivatives. The outcomes show that the suggested methods are effective in delivering better results. Overall, this research helps to develop more precise and comprehensive approaches to understanding and regulating syphilis disease transmission and progression.Öğe A fractional Newton method with 2?th-order of convergence and its stability(Pergamon-Elsevier Science Ltd, 2019) Akgul, Ali; Cordero, Alicia; Torregrosa, Juan R.The use of fractional calculus in many branches of Science and Engineering is wide in the last years. There are different kinds of derivatives that can be useful in different problems. In this manuscript, we put the focus in the effect of this kind of fractional derivatives in the search of roots of nonlinear equations and its dependence on the initial estimations. (C) 2019 Elsevier Ltd. All rights reserved.Öğe A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs(Cell Press, 2024) Ali, Nasir; Siddiqui, Hafiz Muhammad Afzal; Riaz, Muhammad Bilal; Qureshi, Muhammad Imran; Akgul, AliThis article investigates the concept of dominant metric dimensions in zero divisor graphs (ZDgraphs) associated with rings. Consider a finite commutative ring with unity, denoted as R, where nonzero elements x and y are identified as zero divisors if their product results in zero (x . y = 0). The set of zero divisors in ring R is referred to as L(R). To analyze various algebraic properties of R, a graph known as the zero-divisor graph is constructed using L(R). This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of R. To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo m, denoted as Zm[i], the ring of integers modulo n, denoted as Zn, and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.Öğe A greedy algorithm for partition of unity collocation method in pricing American options(Wiley, 2019) Fadaei, Yasin; Khan, Zareen A.; Akgul, AliA greedy algorithm in combination with radial basis functions partition of unity collocation (GRBF-PUC) scheme is used as a locally meshless method for American option pricing. The radial basis function partition of unity method (RBF-PUM) is a localization technique. Because of having interpolation matrices with large condition numbers, global approximants and some local ones suffer from instability. To overcome this, a greedy algorithm is added to RBF-PUM. The greedy algorithm furnishes a subset of best nodes among the points X. Such nodes are then used as points of trial in a locally supported RBF approximant for each partition. Using of greedy selected points leads to decreasing the condition number of interpolation matrices and reducing the burdensome in pricing American options.Öğe A Homotopy Perturbation Solution for Solving Highly Nonlinear Fluid Flow Problem Arising in Mechanical Engineering(Amer Inst Physics, 2018) Khan, Yasir; Akgul, Ali; Faraz, Naeem; Inc, Mustafa; Akgul, Esra Karatas; Baleanu, DumitruIn this paper, a highly nonlinear equations are treated analytically via homotopy perturbation method for fluid mechanics problem. The non-linear differential equations are transformed to a coupled non-linear ordinary, differential equations via similarity transformations. Graphical results are presented and discussed for various physical parameters.Öğe A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations(Amer Inst Mathematical Sciences-Aims, 2022) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Jarad, FahdIt is important to deal with the exact solution of nonlinear PDEs of non-integer orders. Integral transforms play a vital role in solving differential equations of integer and fractional orders. 'o obtain analytical solutions to integer and fractional-order DEs, a few transforms, such as Laplace transforms, Sumudu transforms, and Elzaki transforms, have been widely used by researchers. We propose the Yang transform homotopy perturbation (YTHP) technique in this paper. We present the relation of Yang transform (YT) with the Laplace transform. We find a formula for the YT of fractional derivative in Caputo sense. We deduce a procedure for computing the solution of fractional-order nonlinear PDEs involving the power-law kernel. We show the convergence and error estimate of the suggested method. We give some examples to illustrate the novel method. We provide a comparison between the approximate solution and exact solution through tables and graphs.Öğe A mathematical analysis and simulation for Zika virus model with time fractional derivative(Wiley, 2020) Farman, Muhammad; Ahmad, Aqeel; Akgul, Ali; Saleem, Muhammad Umer; Rizwan, Muhammad; Ahmad, Muhammad OzairZika is a flavivirus that is transmitted to humans either through the bites of infected Aedes mosquitoes or through sexual transmission. Zika has been associated with congenital anomalies, like microcephalus. We developed and analyzed the fractional-order Zika virus model in this paper, considering the vector transmission route with human influence. The model consists of four compartments: susceptible individuals arex(1)(t), infected individuals arex(2)(t),x(3)(t)shows susceptible mosquitos, andx(4)(t)shows the infected mosquitos. The fractional parameter is used to develop the system of complex nonlinear differential equations by using Caputo and Atangana-Baleanu derivative. The stability analysis as well as qualitative analysis of the fractional-order model has been made and verify the non-negative unique solution. Finally, numerical simulations of the model with Caputo and Atangana Baleanu are discussed to present the graphical results for different fractional-order values as well as for the classical model. A comparison has been made to check the accuracy and effectiveness of the developed technique for our obtained results. This investigative research leads to the latest information sector included in the evolution of the Zika virus with the application of fractional analysis in population dynamics.Öğe A mathematical fractal-fractional model to control tuberculosis prevalence with sensitivity, stability, and simulation under feasible circumstances(Elsevier Ltd, 2024) Farman, Muhammad; Shehzad, Aamir; Nisar, Kottakkaran Sooppy; Hincal, Evren; Akgul, AliBackground: Tuberculosis, a global health concern, was anticipated to grow to 10.6 million new cases by 2021, with an increase in multidrug-resistant tuberculosis. Despite 1.6 million deaths in 2021, present treatments save millions of lives, and tuberculosis may overtake COVID-19 as the greatest cause of mortality. This study provides a six-compartmental deterministic model that employs a fractal–fractional operator with a power law kernel to investigate the impact of vaccination on tuberculosis dynamics in a population. Methods: Some important characteristics, such as vaccination and infection rate, are considered. We first show that the suggested model has positive bounded solutions and a positive invariant area. We evaluate the equation for the most important threshold parameter, the basic reproduction number, and investigate the model's equilibria. We perform sensitivity analysis to determine the elements that influence tuberculosis dynamics. Fixed-point concepts show the presence and uniqueness of a solution to the suggested model. We use the two-step Newton polynomial technique to investigate the effect of the fractional operator on the generalized form of the power law kernel. Results: The stability analysis of the fractal–fractional model has been confirmed for both Ulam–Hyers and generalized Ulam–Hyers types. Numerical simulations show the effects of different fractional order values on tuberculosis infection dynamics in society. According to numerical simulations, limiting contact with infected patients and enhancing vaccine efficacy can help reduce the tuberculosis burden. The fractal–fractional operator produces better results than the ordinary integer order in the sense of memory effect at diffract fractal and fractional order values. Conclusion: According to our findings, fractional modeling offers important insights into the dynamic behavior of tuberculosis disease, facilitating a more thorough comprehension of their epidemiology and possible means of control. © 2024Öğe A new application of the Legendre reproducing kernel method(Amer Inst Mathematical Sciences-Aims, 2022) Foroutan, Mohammad Reza; Hashemi, Mir Sajjad; Gholizadeh, Leila; Akgul, Ali; Jarad, FahdIn this work, we apply the reproducing kernel method to coupled system of second and fourth order boundary value problems. We construct a novel algorithm to acquire the numerical results of the nonlinear boundary-value problems. We also use the Legendre polynomials. Additionally, we discuss the convergence analysis and error estimates. We demonstrate the numerical simulations to prove the efficiency of the presented method.
