Yazar "Abdeljawad, Thabet" seçeneğine göre listele

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    Analysis of fractal-fractional model of tumor-immune interaction
    (Elsevier, 2021) Ahmad, Shabir; Ullah, Aman; Abdeljawad, Thabet; Akgul, Ali; Mlaiki, Nabil
    Recently, Atangana proposed new operators by combining the fractional and fractal calculus. These recently proposed operators, referred to as fractal-fractional operators, have been widely used to study the complex dynamics of a problem. Cancer is a prevalent disease today and is difficult to cure. The immune system tends to fight it as cancer sets up in the body. In this manuscript, the novel operators have been used to analyze the relationship between the immune system and cancer cells. The tumor-immune model has been studied qualitatively and quantitatively via Atangana-Baleanu fractal-fractional operator. The existence and uniqueness results of the model under Atangana-Baleanu fractal-fractional operator have proved through fixed point theorems. The Ulam-Hyres stability for the model has derived through non-linear analysis. Numerical results have developed through Lagrangian-piece wise interpolation for the different fractal-fractional operators. To visualize the relationship between immune cells and cancers cells under novel operators in a various sense, we simulate the numerical results for the different sets of fractional and fractal orders.
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    Analysis of Nonlinear Mathematical Model of COVID-19 via Fractional-Order Piecewise Derivative
    (Akif AKGUL, 2023) Sinan, Muhammad; Shah, Kamal; Abdeljawad, Thabet; Akgül, Ali
    Short memory and long memory terms are excellently explained using the concept of piecewise fractional order derivatives. In this research work, we investigate dynamical systems addressing COVID-19 under piecewise equations with fractional order derivative (FOD). Here, we study the sensitivity of the proposed model by using some tools from the nonlinear analysis. Additionally, we develop a numerical scheme to simulate the model against various fractional orders by using Matlab 2016. All the results are presented graphically. © 2023 Chaos Theory and Applications. All rights reserved.
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    CHAOTIC BEHAVIOR OF BHALEKAR-GEJJI DYNAMICAL SYSTEM UNDER ATANGANA-BALEANU FRACTAL FRACTIONAL OPERATOR
    (World Scientific Publ Co Pte Ltd, 2022) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Abdeljawad, Thabet
    In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal-fractional derivatives in which fractional derivative is taken in Atangana-Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder's fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam-Hyers stable under the new fractal-fractional derivative. We establish the numerical results of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we study the chaos behavior of the proposed model via simulation at 2D and 3D phase. We show the effect of fractal dimension on integer and fractional order through simulations.
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    Computational analysis of fuzzy fractional order non-dimensional Fisher equation
    (Iop Publishing Ltd, 2021) Ahmad, Shabir; Ullah, Aman; Ullah, Abd; Akgul, Ali; Abdeljawad, Thabet
    In recent decades, fuzzy differential equations of integer and arbitrary order are extensively used for analyzing the dynamics of a mathematical model of the physical process because crisp operators of integer and arbitrary order are not able to study the model being studied when there is uncertainty in values used in modeling. In this article, we have considered the time-fractional Fisher equation in a fuzzy environment. The basic aim of this article is to deduce a semi-analytical solution to the fuzzy fractional-order non-dimensional model of the Fisher equation. Since the Laplace-Adomian method has a good convergence rate. We use the Laplace- Adomian decomposition method (LADM) to determine a solution under a fuzzy concept in parametric form. We discuss the convergence and error analysis of the proposed method. For the validity of the proposed scheme, we provide few examples with detailed solutions. We provide comparisons between exact and approximate solutions through graphs. In the end, the conclusion of the paper is provided.
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    EXISTENCE AND STABILITY RESULTS FOR COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING AB-CAPUTO DERIVATIVE
    (World Scientific Publ Co Pte Ltd, 2023) Mehmood, Nayyar; Abbas, Ahsan; Akgul, Ali; Abdeljawad, Thabet; Alqudah, Manara A.
    In this paper, we use Krasnoselskii's fixed point theorem to find existence results for the solution of the following nonlinear fractional differential equations (FDEs) for a coupled system involving AB-Caputo fractional derivative ABC(0)D(alpha)??(l) = zeta(l,??(l),P(l)), 1 < alpha & <= 2, (SIC) AB( )C(0)D(sigma)P(l) = xi(l,??(l),P(l)), 1 < sigma <= 2,f or alll is an element of [0, 1], with boundary conditions (SIC) ??(0) = 0, lambda??'(eta) = gamma??'(1), P(0) = 0,lambda'(eta) = gamma'(1).We discuss uniqueness with the help of the Banach contraction principle. The criteria for Hyers-Ulam stability of given AB-Caputo fractional-coupled boundary value problem (BVP) is also discussed. Some examples are provided to validate our results. In Example 1, we find a unique and stable solution of AB-Caputo fractional-coupled BVP. In Example 2, the analysis of approximate and exact solutions with errors of nonlinear integral equations is elaborated with graphs.
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    EXISTENCE AND STABILITY RESULTS OF FRACTIONAL DIFFERENTIAL EQUATIONS MITTAG-LEFFLER KERNEL
    (World Scientific Publ Co Pte Ltd, 2024) Abbas, Ahsan; Mehmood, Nayyar; Akgul, Ali; Amacha, Inas; Abdeljawad, Thabet
    This paper presents the following AB-Caputo fractional boundary value problem (ABC)(0)D(alpha)u(sigma) = G(sigma, u(sigma)), sigma is an element of[0, 1] with integral-type boundary conditions u(0) = 0 = u ''(0), gamma u(1) = lambda integral(1)(0) g(1)(kappa)u(kappa)d kappa, of order 2 < alpha <= 3. Schauder and Krasnoselskii's fixed point theorems are used to find existence results. Uniqueness is obtained via the Banach contraction principle. To investigate the stability of a given problem, Hyers-Ulam stability is discussed. An example is provided to validate our results.
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    EXISTENCE RESULTS FOR ABC-FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED AND INTEGRAL TYPE OF BOUNDARY CONDITIONS
    (World Scientific Publ Co Pte Ltd, 2021) Mehmood, Nayyar; Abbas, Ahsan; Abdeljawad, Thabet; Akgul, Ali
    This paper presents a study on the existence theory of fractional differential equations involving Atangana-Baleanu (AB) derivative of order 1 < alpha <= 2, with non-separated and integral type boundary conditions. An existence result for the solutions of given AB-fractional differential equation is proved using Krasnoselskii's fixed point theorem, while the uniqueness of the solution is obtained using Banach contraction principle. Some conditions are proposed under which the given boundary value problem is Hyers-Ulam stable. Examples are given to validate our results.
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    EXISTENCE RESULTS FOR MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS INVOLVING ATANGANA-BALEANU DERIVATIVE
    (World Scientific Publ Co Pte Ltd, 2023) Abbas, Ahsan; Mehmood, Nayyar; Akgul, Ali; Abdeljawad, Thabet; Alqudah, Manar A.
    In this paper, the existence results for the solutions of the multi-term ABC-fractional differential boundary value problem (BVP) (delta(2)0(ABC)D(alpha+2) + delta( 1)0(ABC)D(alpha+1) + delta (0)0(ABC)D(alpha))x(t) = zeta(t,x(t))of order 0 < alpha < 1 with nonlocal boundary conditions have been derived by using Krasnoselskii's fixed point theorem. The uniqueness of the solution is obtained with the help of Banach contraction principle. Examples are provided to confirm our obtained results.
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    Explicit solitary wave structures for the fractional-order Sobolev-type equations and their stability analysis
    (Elsevier, 2024) Shahzad, Tahir; Ahmed, Muhammad Ozair; Baber, Muhammad Zafarullah; Ahmed, Nauman; Akgul, Ali; Abdeljawad, Thabet; Amacha, Inas
    The current research is concerned with solitary wave structures to the time fractional -order Sobolev-type equations. The special types of Sobolev-type equations are under consideration such as the generalized hyperelastic-rod wave (HRW) equation, and Camassa-Holm (CH) equation. These equations occur in several fields, including particularly in quantum field theory, plasma theory, ecology, consolidation of clay and fluid dynamics. The underlying models are investigated analytically by applying two techniques, such as the generalized projective Riccati equation (GPRE) and the modified auxiliary equation (MAE). The gained results are obtained from the different families of solutions such as, including a periodic wave, kink -type wave peakon, a singular wave, and dark solutions. The gained results are denoted as hyperbolic and trigonometric functions. Furthermore, we check that the underlying models are stable using the concept of linearized stability. The propagation behavior of the gained results is displayed in 3D, 2D, and contour visualizations to investigate the influence of various relevant parameters. These results will help the researchers to understand the physical situations.
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    Modeling and analysis of fractional order Ebola virus model with Mittag-Leffler kernel
    (Elsevier, 2022) Farman, Muhammad; Akgul, Ali; Abdeljawad, Thabet; Naik, Parvaiz Ahmad; Bukhari, Nabila; Ahmad, Aqeel
    In this article, we presented a nonlinear time-fractional mathematical model of the Ebola Virus in order to understand the outbreak of this epidemic disease. Ebola virus is a highly contagious disease that can be spread in the population depending upon the number of individuals and their dynamics in the community. The Caputo and Atangana Baleanu fractional derivative operators are employed to get the solution of the system of fractional differential equations. The qualitative analysis has been made for the fractional-order model. Fixed-point theorem and an iterative schemes are used to get the existence and uniqueness. The actual behavior of the time-fractional model has been obtained by employing Laplace Adomian Decomposition technique. Finally, numerical results have been established for the system of fractional differential equations with simulations to demonstrate the impacts of the fractional-order parameters on the proposed system to achieve the theoretical outcomes and a comparison has been made with the Caputo for better analysis. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.
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    Numerical analysis of fractional human liver model in fuzzy environment
    (Taylor & Francis Ltd, 2021) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Abdeljawad, Thabet
    Many papers have shown that fractional derivatives are preferable to other operators when the data or information is exact, but this is not the case in practice because we live in an uncertain environment. Fuzzy operators are the best option for modelling in this situation. In this paper, we use the fuzzy fractional Caputo's derivative to generalize the fractional-order human liver model. We consider both types of H-differentiability (type 1 and type 2). We establish a general procedure of solution under the concept of H-differentiability through fuzzy Laplace transform. We implement the proposed scheme to derive the numerical results of the model. We present the archived theoretical solution via two- and three-dimensional graphs at different values of fractional orders and specific fuzzy triangular initial conditions. We present the evolution of the proposed model for some values of phi(0) is an element of[0, 1] to see the effect of uncertainty on the secretion of Bromsulphthalein in the blood and liver.
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    Semi-analytical solutions of the 3rd order fuzzy dispersive partial differential equations under fractional operators
    (Elsevier, 2021) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Abdeljawad, Thabet
    The purpose of this article is to extend the fractional third order dispersive PDE under singular and non-singular fractional operators via the notion of fuzziness. We investigate the fuzzy dispersive PDE in one and higher dimension under Caputo, Caputo-Fabrizio, and AtanganaBaleanu fractional operators and provide two examples to each derivative. We derive the general algorithm and numerical results in series of the models and test problems with the help of fuzzy Laplace transform. The numerical results confirm that solutions obtained in the fuzzy sense are more generalized than the fractional-order solution. We mention in remarks following each example that we recover the solutions of the fractional-order equations by putting the lower and upper functions of the fuzzy number ge equals to 1 in the fuzzy solutions of the proposed dispersive PDEs. We demonstrate the numerical results through 2D and 3D plots for different fractional-order and uncertainty k is an element of[0,1]. We provide a comparison between Caputo, Caputo-Fabrizio and Atangana-Baleanu fuzzy fractional dispersive PDE. In the end, we give the conclusion of the article and future work. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.