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    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.
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    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, Dumitru
    In 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.
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    A new approach for one-dimensional sine-Gordon equation
    (Springer International Publishing Ag, 2016) Akgul, Ali; Inc, Mustafa; Kilicman, Adem; Baleanu, Dumitru
    In this work, we use a reproducing kernel method for investigating the sine-Gordon equation with initial and boundary conditions. Numerical experiments are studied to show the efficiency of the technique. The acquired results are compared with the exact solutions and results obtained by different methods. These results indicate that the reproducing kernel method is very effective.
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    A NEW NUMERICAL TECHNIQUE FOR SOLVING FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS
    (Univ Miskolc Inst Math, 2018) Acan, Omer; Baleanu, Dumitru
    We propose conformable Adomian decomposition method (CADM) for fractional partial differential equations (FPDEs). This method is a new Adomian decomposition method (ADM) based on conformable derivative operator (CDO) to solve FPDEs. At the same time, conformable reduced differential transform method (CRDTM) for FPDEs is briefly given and a numerical comparison is made between this method and the newly introduced CADM. In applied science, CADM can be used as an alternative method to obtain approximate and analytical solutions for FPDEs as CRDTM. In this study, linear and non-linear three problems are solved by these two methods. In these methods, the obtained solutions take the form of a convergent series with easily computable algorithms. For the applications, the obtained results by these methods are compared to each other and with the exact solutions. When applied to FPDEs, it is seem that CADM approach produces easy, fast and reliable solutions as CRDTM. 2010 Mathematics Subject Classification: 34A08; 34K28
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    A novel method for analysing the fractal fractional integrator circuit
    (Elsevier, 2021) Akgul, Ali; Ahmad, Shabir; Ullah, Aman; Baleanu, Dumitru; Akgul, Esra Karatas
    In this article, we propose the integrator circuit model by the fractal-fractional operator in which fractional-order has taken in the Atangana-Baleanu sense. Through Schauder's fixed point theorem, we establish existence theory to ensure that the model posses at least one solution and via Banach fixed theorem, we guarantee that the proposed model has a unique solution. We derive the results for Ulam-Hyres stability by mean of non-linear functional analysis which shows that the proposed model is Ulam-Hyres stable under the new fractal-fractional derivative. We establish the numerical results of the model under consideration through Atanaga-Toufik method. We simulate the numerical results for different sets of fractional order and fractal dimension.(C) 2021 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/).
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    Analysis and applications of the proportional Caputo derivative
    (Springer, 2021) Akgul, Ali; Baleanu, Dumitru
    In this paper, we investigate the analysis of the proportional Caputo derivative that recently has been constructed. We create some useful relations between this new derivative and beta function. We discretize the new derivative. We investigate the stability and obtain a stability condition for the new derivative.
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    Analysis of a fractional order Bovine Brucellosis disease model with discrete generalized Mittag-Leffler kernels
    (Elsevier, 2023) Farman, Muhammad; Shehzad, Aamir; Akgul, Ali; Baleanu, Dumitru; Attia, Nourhane; Hassan, Ahmed M.
    Bovine Brucellosis, a zoonotic disease, can infect cattle in tropical and subtropical areas. It remains a critical issue for both human and animal health in many parts of the world, especially those where livestock is an important source of food and income. An efficient method for monitoring the illness's increasing prevalence and developing low-cost prevention strategies for both its effects and recurrence is brucellosis disease modeling. We create a fractional-order model of Bovine Brucellosis using a discrete modified Atangana-Baleanu fractional difference operator of the Liouville-Caputo type. An analysis of the suggested system's well-posedness and a qualitative investigation are both conducted. The examination of the Volterra-type Lyapunov function for global stability is supported by the first and derivative tests. The Lipschitz condition is also used for the model in order to meet the criterion of the uniqueness of the exact solution. We created an endemic and disease-free equilibrium. Solutions are built in the discrete generalized form of the Mittag-Leffler kernel in order to analyze the effect of the fractional operator with numerical simulations and emphasize the effects of the sickness due to the many factors involved. The capacity of the suggested model to forecast an infectious disease like brucellosis can help researchers and decision-makers take preventive actions.
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    Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact
    (Mdpi, 2020) Farman, Muhammad; Akgul, Ali; Baleanu, Dumitru; Imtiaz, Sumaiyah; Ahmad, Aqeel
    The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest ratedparameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system's actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation.
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    ANALYSIS OF NEW TRANSFER FUNCTIONS WITH SUM INTEGRAL TRANSFORMATION
    (Wilmington Scientific Publisher, Llc, 2024) Akgul, Ali; Baleanu, Dumitru; Ulgul, Enver; Sakar, Necibullah; Attia, Nourhane
    We explore the novel SUM integral transform method for solving ordinary and partial differential equations, offering an effective approach beyond conventional Laplace and Sumudu transforms. Using this method, we address various differential equations, deriving transfer functions for classical and fractional derivatives. The resultant transfer functions provide valuable insights into diverse mathematical models.
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    Analysis of the fractional tumour-immune-vitamins model with Mittag-Leffler kernel
    (Elsevier, 2020) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Baleanu, Dumitru
    Recently, Atangana-Baleanu fractional derivative has got much attention of the researchers due to its nonlocality and non-singularity. This operator contains an accurate kernel that describes the better dynamics of systems with a memory effect. In this paper, we investigate the fractional-order tumour-immune-vitamin model (TIVM) under Mittag-Leffler derivative. The existence of at least one solution and a unique solution has discussed through fixed point results. We established the Hyres-Ulam stability of the proposed model under the Mittag-Leffler derivative. The fractional Adams-Bashforth method has used to achieve numerical results. Finally, we simulate the obtained numerical results for different fractional orders to show the effect of vitamin intervention on decreased tumour cell growth and cancer risk. At the end of the paper, the conclusion has provided.
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    Analysis of the proportional Caputo-Fabrizio derivative
    (Int Scientific Research Publications, 2024) Akgul, Ali; Baleanu, Dumitru
    This article examines a recently created proportional Caputo-Fabrizio derivative. We find multiple significant relationships between the beta function and this new derivative. Discreteization is applied to the new derivative. We consider stability analysis to define a stability requirement for the new derivative.
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    Analytical Approximate Solutions of (n+1)-Dimensional Fractal Heat-Like and Wave-Like Equations
    (Mdpi, 2017) Acan, Omer; Baleanu, Dumitru; Al Qurashi, Maysaa Mohamed; Sakar, Mehmet Giyas
    In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems.
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    Analytical solutions for free convection flow of Casson nanofluid over an infinite vertical plate
    (Amer Inst Mathematical Sciences-Aims, 2021) Ahmad, Mushtaq; Asjad, Muhammad Imran; Akgul, Ali; Baleanu, Dumitru
    This research article is design to elaborate the rule and significance of fractional derivative for heat transport in drilling of nanofluid. The respective nanofluid formed by the suspension of clay nanoparticles in the base fluids namely Casson fluid. The physical flow phenomenon is demonstrated with the help of partial differential equations by utilizing the respective thermophysical properties of nanoparticles. Also the geometric and thermal conditions are imposed in flow domain. In the governing equations, the partial derivative with respect to time replaced by new hybrid fractional derivative and then solved analytically for temperature and velocity field with the help of Laplace transformed. The obtained solutions for temperature and velocity are presented geometrically by Mathcad software to see the effectiveness of potent parameters. The temperature and velocity present a significant increasing trend for increasing volume fraction parameter. The obtained results for temperature as well as velocity are also compared with the existing literature and it is concluded that field variables with new hybrid fractional derivative, show more decaying trend as compare to the results with Caputo and Caputo-Fabrizio fractional derivatives.
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    Analytical study of a Hepatitis B epidemic model using a discrete generalized nonsingular kernel
    (Amer Inst Mathematical Sciences-Aims, 2024) Farman, Muhammad; Akgul, Ali; Conejero, J. Alberto; Shehzad, Aamir; Nisar, Kottakkaran Sooppy; Baleanu, Dumitru
    Hepatitis B is a worldwide viral infection that causes cirrhosis, hepatocellular cancer, the need for liver transplantation, and death. This work proposed a mathematical representation of Hepatitis B Virus (HBV) transmission traits emphasizing the significance of applied mathematics in comprehending how the disease spreads. The work used an updated Atangana-Baleanu fractional difference operator to create a fractional -order model of HBV. The qualitative assessment and wellposedness of the mathematical framework were looked at, and the global stability of equilibrium states as measured by the Volterra -type Lyapunov function was summarized. The exact answer was guaranteed to be unique using the Lipschitz condition. Additionally, there were various analyses of this new type of operator to support the operator's efficacy. We observe that the explored discrete fractional operators will be x 2 -increasing or decreasing in certain domains of the time scale N j : = j , j + 1 ,... by looking at the fundamental characteristics of the proposed discrete fractional operators along with x -monotonicity descriptions. For numerical simulations, solutions were constructed in the discrete generalized form of the Mittag-Leffler kernel, highlighting the impacts of the illness caused by numerous causes. The order of the fractional derivative had a significant influence on the dynamical process utilized to construct the HBV model. Researchers and policymakers can benefit from the suggested model's ability to forecast infectious diseases such as HBV and take preventive action.
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    Approximate solutions to the conformable Rosenau-Hyman equation using the two-step Adomian decomposition method with Pade approximation
    (Wiley, 2020) Akgul, Ali; Aliyu, Aliyu Isa; Inc, Mustafa; Yusuf, Abdullahi; Baleanu, Dumitru
    This paper adopts the Adomian decomposition method and the Pade approximation techniques to derive the approximate solutions of a conformable Rosenau-Hyman equation by considering the new definition of the Adomian polynomials. The Pade approximate solutions are derived along with interesting figures showing both the analytic and approximate solutions.
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    Comparison on Solving a Class of Nonlinear Systems of Partial Differential Equations and Multiple Solutions of Second Order Differential Equations
    (Springer International Publishing Ag, 2019) Akgul, Ali; Akgul, Esra Karatas; Khan, Yasir; Baleanu, Dumitru
    [Abstract Not Available]
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    Dynamical Transmission of Coronavirus Model with Analysis and Simulation
    (Tech Science Press, 2021) Farman, Muhammad; Akgul, Ali; Ahmad, Aqeel; Baleanu, Dumitru; Saleem, Muhammad Umer
    COVID-19 acts as a serious challenge to the whole world. Epidemiological data of COVID-19 is collected through media and web sources to analyze and investigate a system of nonlinear ordinary differential equation to understand the outbreaks of this epidemic disease. We analyze the diseases free and endemic equilibrium point including stability of the model. The certain threshold value of the basic reproduction number R-0 is found to observe whether population is in disease free state or endemic state. Moreover, the epidemic peak has been obtained and we expect a considerable number of cases. Finally, some numerical results are presented which show the effect of parameters estimation and different step size on our obtained solutions at the real data of some countries to check the actual behavior of the COVID-19 at different countries.
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    Dynamics exploration for a fractional-order delayed zooplankton-phytoplankton system
    (Pergamon-Elsevier Science Ltd, 2023) Li, Peiluan; Gao, Rong; Xu, Changjin; Li, Ying; Akgul, Ali; Baleanu, Dumitru
    In this study, we are concerned with the dynamics of a new established fractional-order delayed zooplankton- phytoplankton system. The existence and uniqueness of the solution are proved via Banach fixed point theorem. Non-negativeness of the solution is studied by mathematical inequality technique. The boundedness of the solution is analyzed by virtue of constructing an appropriate function. A novel delay-independent sufficient condition ensuring the stability and the onset of Hopf bifurcation for the established fractional -order delayed zooplankton-phytoplankton system is derived by means of Laplace transform, stability criterion and bifurcation knowledge of fractional-order differential equation. The global stability condition for the involved fractional-order delayed zooplankton-phytoplankton system is built by using a suitable positive definite function. Taking advantage of hybrid control tactics, we effectively control the time of occurrence of Hopf bifurcation for the established fractional-order delayed zooplankton-phytoplankton system. The study manifests that delay plays a vital role in controlling the stability and the time of occurrence of Hopf bifurcation for the involved fractional-order delayed zooplankton-phytoplankton system and the fractional -order controlled zooplankton-phytoplankton system involving delays. To verify the correctness of established chief results, computer simulation figures are distinctly displayed. The derived conclusions of this research are entirely new and possess potential theoretical value in preserving the balance of biological population. Up to now, there are few publications on detailed and comprehensive dynamic analysis on fractional-order delayed zooplankton-phytoplankton system via various exploration ways.
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    Dynamics of HIV-TB coinfection model using classical and Caputo piecewise operator: A dynamic approach with real data from South-East Asia, European and American regions
    (Pergamon-Elsevier Science Ltd, 2022) Xu, Changjin; Liu, Zixin; Pang, Yicheng; Akgul, Ali; Baleanu, Dumitru
    In this study, we analyse the behaviour of the coinfection of the HIV-TB model using a piecewise operator in the classical-Caputo sense. For the aforementioned disease model, we present the existence as well as the uniqueness of a solution having a piecewise derivative. We also study the different versions of stability using Ulam-Hyers stability in nonlinear analysis. We use the piecewise Newton polynomial technique to obtain an approximation of the solution to the proposed problem. The simulations for the suggested coinfection model are presented. The simulations are carried out for the disease-free as well as endemic equilibrium. Additionally, the comparison between the simulated and real data is presented, where we obtain the best-fitted dynamics of the infected class with TB.
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    Effects of hybrid nanofluid on novel fractional model of heat transfer flow between two parallel plates
    (Elsevier, 2021) Ikram, Muhammad Danish; Asjad, Muhammad Imran; Akgul, Ali; Baleanu, Dumitru
    In this paper, it has been discussed the fractional model of Brinkman type fluid (BTF) holding hybrid nanoparticles. Titanium dioxide (TiO2) and silver (Ag) nanoparticles were liquefied in water (H2O) (base fluid) to make a hybrid nanofluid. The magnetohydrodynamic (MHD) free convection flow of the nanofluid (Ag - TiO2 - H2O)was measured in a bounded microchannel. The BTF model was generalized using constant proportional Caputo fractional operator (CPC) with effective thermophysical properties. By introducing dimensionless variables, the governing equations of the model were solved by Laplace transform method. The testified outcomes are stated as M-function. The impact of associated parameters were measured graphically using Mathcad and offered a comparison with the existing results from the literature. The effect of related parameters was physically discussed. It was concluded that constant proportional Caputo fractional operator (CPC) showed better memory effect than Caputo-Fabrizio fractional operator (CF) (Saqib et al., 2020). (C) 2021 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/).