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Öğe Analysis of the Fractal-Fractional Modelling of Immune-Tumor Problem(Springer, 2022) Partohaghighi, Mohammad; Rubasinghe, Kalani; Akgül, Ali; Akgül, Esra KaratasCancer is one of the biggest threats around the globe, albeit medical action has been prosperous, despite large challenges, at least for some diagnostics. A magnificent effort of personal and financial resources is dedicated, with flourishing results(but also with failures), to cancer analysis with special consideration to experimental and analytical immunology. Fractal-fractional operators have manifested the enigmatic performance of numerous natural phenoms, which ordinarily do not foretell in ordinary ones and fractional operators. In this study, we examine an Immune-Tumor dynamical system supporting the fractal-fractional frame. We authenticate the existence theory to guarantee the suggested system maintains at least one answer through Schauder’s fixed point theorem. Additionally, Banach’s fixed theory affirms the uniqueness of the answer to the aimed problem. A Non-linear functional examination was carried out to affirm that the introduced system is stable with respect to Ulam-Hyres’s theory supporting the fractal-fractional operator. Behavior of the offered problem is presented through the graphical representations, for the different amounts of fractional order and fractal orders successfully. © 2022, The Author(s), under exclusive licence to Springer Nature India Private Limited.Öğe Analysis of the Fractional Differential Equations Using Two Different Methods(Mdpi, 2023) Partohaghighi, Mohammad; Akgul, Ali; Akgul, Esra Karatas; Attia, Nourhane; De la Sen, Manuel; Bayram, MustafaNumerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.Öğe Comparative Study of the Fractional-Order Crime System as a Social Epidemic of the USA Scenario(Springer, 2022) Partohaghighi, Mohammad; Kumar, Vijay; Akgül, AliFractional derivatives are considered significant mathematical tools to design the fractional-order models of real phenomena. In this investigation, we are going to design and compare the non-integer models of the crime system by using three fractional-order operators called Atangana-Baleanu-Caputo, Caputo, and Caputo-Fabrizio derivatives for the first time. We use the real initial conditions for the subgroups of USA. To get the approximate solutions of the suggested models some numerical methods are derived. To see the performance of the numerical methods different values of the fractional orders are considered. The differences between the solutions under the used operators for each state variable are provided through some figures. © 2022, The Author(s), under exclusive licence to Springer Nature India Private Limited.Öğe Complex dynamics of multi strain TB model under nonlocal and nonsingular fractal fractional operator(Elsevier, 2021) Adnan; Ahmad, Shabir; Ullah, Aman; Riaz, Muhammad Bilal; Ali, Amir; Akgul, Ali; Partohaghighi, MohammadResearchers have recently begun to use fractal fractional operators in the Atangana-Baleanu sense to analyze complicated dynamics of various models in applied sciences, as the Atangana-Baleanu operator generalizes the integer and fractional order operators. To analyze the complex dynamics of the multi-strain TB model, we use the AB-fractal fractional operator. We use the Banach fixed point theorem to ensure that at most one solution exists to the model. Further, the Ulam-Hyers type stability of the model is investigated with the help of functional analysis. The Adams-Bashforth approach is used to get numerical results for the proposed model. The analysis of the chaotic behavior of the proposed TB model was missing in the literature. Therefore, for different values of fractional and fractal order, we study the nonlinear dynamics and chaotic behavior of the obtained results of the proposed model.Öğe Computational analysis of COVID-19 model outbreak with singular and nonlocal operator(Amer Inst Mathematical Sciences-Aims, 2022) Amin, Maryam; Farman, Muhammad; Akgul, Ali; Partohaghighi, Mohammad; Jarad, FahdThe SARS-CoV-2 virus pandemic remains a pressing issue with its unpredictable nature, and it spreads worldwide through human interaction. Current research focuses on the investigation and analysis of fractional epidemic models that discuss the temporal dynamics of the SARS-CoV-2 virus in the community. In this work, we choose a fractional-order mathematical model to examine the transmissibility in the community of several symptoms of COVID-19 in the sense of the Caputo operator. Sensitivity analysis of R0 and disease-free local stability of the system are checked. Also, with the assistance of fixed point theory, we demonstrate the existence and uniqueness of the system. In addition, numerically we solve the fractional model and presented some simulation results via actual estimation parameters. Graphically we displayed the effects of numerous model parameters and memory indexes. The numerical outcomes show the reliability, validation, and accuracy of the scheme.Öğe Fractal-fractional Klein-Gordon equation: A numerical study(Elsevier, 2022) Partohaghighi, Mohammad; Mirtalebi, Zahrasadat; Akgul, Ali; Riaz, Muhammad BilalIn this work, we solve a new kind of the fractional Klein-Gordon problem numerically. In fact, we study the mentioned problem under fractal-fractional operator with the Riemann-Liouville frame with Mittag-Leffler kernel. We use an efficient operational matrix (OM) technique employing the shifted Chebyshev cardinal functions (CCFs) to get the approximate solutions of the considered equation. Moreover, an OM for the considered derivative is gained using the basic functions. To get the approximate solutions of the presented equation we change the principal model into an algebraic system. To see the numerical results of the problem, we provide the related graphs of the exact and approximate solutions along with the absolute errors of each example. The accuracy and reliability of the numerical solutions can be found form the figures. Also, for each example Tables displaying the values of solutions and errors are reported.Öğe Fractional study of a novel hyper-chaotic model involving single non-linearity(Elsevier, 2022) Partohaghighi, Mohammad; Veeresha, P.; Akguel, Ali; Inc, Mustafa; Riaz, Muhamamad BilalThe applications of hyperchaotic systems (HCSs) can be widely seen in diverse fields associated with engineering due to their complicated dynamics, randomness, and high delicacy and sensibility. In the present work, we aim to investigate a new hyper-chaotic system involving a single non-linearity under the fractional Caputo-Fabrizio (CF) derivative for the first time. In fact, there is no previous study using fractional derivatives in this system. A new mathematical system using a fractional-order operator will be designed with the novel operator. The Caputo-Fabrizio non-integer operator is aimed to be employed to capture complex nature. In order to solve the extracted dynamical system, a quadratic numerical scheme is applied. This study contains stability and convergence sections for the considered method. Moreover, numerical results of the problem under various values of fractional orders and different values of initial conditions (ICs) are provided to show the performance of the suggested scheme. Figures of solutions for each dependent variable can be observed.Öğe Fractional study of the Covid-19 model with different types of transmissions(Academic Publication Council, 2023) Partohaghighi, Mohammad; Akgul, AliWe investigate a mathematical system of the recent COVID-19 disease focusing particularly on the transmissibility of individuals with different types of signs under the Caputo fractional derivative. To get the approximate solutions of the fractional order system we employ the fractional-order Alpert multiwavelet(FAM). The fractional operational integration matrix of Riemann-Liouville (RLFOMI) employing the FAM functions is considered. The origin system will be transformed into a system of algebraic equations. Also, an error estimation of the supposed scheme is considered. Satisfactory results are gained under various values of fractional order with the chosen initial conditions (ICs).Öğe Modelling and simulations of the SEIR and Blood Coagulation systems using Atangana-Baleanu-Caputo derivative(Pergamon-Elsevier Science Ltd, 2021) Partohaghighi, Mohammad; Akgul, AliIn this work, we investigate the SEIR and Blood Coagulation systems using a specific type of fractional derivative. SEIR epidemic model which outlines the close communication of contagious disease is estimated to dominate the measles epidemic for infected groups. Moreover, Blood coagulation is a protective tool that restricts the loss of blood upon the rupture of endothelial tissues. This process is a complicated one that is managed by various mechanical and biochemical mechanisms. Indeed, the fractional Atangana-Baleau-Caputo derivative operator is exercised to achieve the new models of fractional equations of the SEIR epidemic and Blood Coagulation. Moreover, the existence and uniqueness of the considered systems are checked. Also, simulations are provided under selecting different amounts of fractional orders using Atangana-Toufik method. Additionally, chaotic behaviors of the proposed models by adopting different values of orders are presented, clearly to show the robustness and reliability of the recommended scheme. During graphs of simulations which are obtained under applying various values of orders, show that the used algorithm is highly effective to solve such fractional systems employing various initial conditions(ICs)compared to the other methods. (c) 2021 Elsevier Ltd. All rights reserved.Öğe New fractional modelling and control analysis of the circumscribed self-excited spherical strange attractor(Pergamon-Elsevier Science Ltd, 2022) Akgul, Ali; Partohaghighi, MohammadThe purpose of this study is to present and examine a novel non-integer model of the circumscribed selfexcited spherical strange attractor, which has not been worked yet. To design the fractional-order model, we use the Caputo-Fabrizio derivative. In order to ensure the existence of the solution Picard-Lindel and fixed-point theories are provided. Moreover, the stability of the considered fractional-order model is shown using the Picard iteration and fixed point theory approach. To get the approximate solutions of the proposed fractional model an efficient numerical scheme called the fractional Euler method(FEM) is used. To see the performance of the used method, the behavior of the numerical solutions of the model is examined under various initial conditions(ICs) and fractional orders. Considerable chaotic behaviors of the solutions are obtained which prove the accuracy and reliability of FEM. (c) 2022 Elsevier Ltd. All rights reserved.Öğe New Fractional Modelling and Simulations of Prey–Predator System with Mittag–Leffler Kernel(Springer, 2023) Partohaghighi, Mohammad; Akgül, AliPredator–prey models are regarded as the structural blocks of the bio- and ecosystems as biomasses are headed by their resource masses. During the current investigation, we examine the impact of a contagious disease on the growth of ecological varieties. We study a non-integer-order predator–prey system by applying the Atangana–Baleanu–Caputo derivative. We use an effective techniqueto get the numerical solutions and to discover the system’s dynamical behavior using different values of fractional order which indicates that how how the proposed scheme is suitable to solve the dynamical systems containing the derivatives with non-singular kernels. Moreover, the existence of the results is given utilizing the fixed-point theorem. Also, diagrams via numerical simulations of the approximate solutions are shown in different dimensions. © 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited.Öğe New numerical simulation of the oscillatory phenomena occurring in the bioethanol production process(Springer Heidelberg, 2023) Partohaghighi, Mohammad; Akgul, Ali; Akgul, Esra Karatas; Asad, Jihad; Safdar, Rabia; Yao, GuangmingThe process of bioethanol production has been characterized with a structured and nonsegregated form of yeast growth dynamics. In this work, a geometric numerical method is applied to obtain the approximate solution of the oscillatory phenomena transpiring in the process of bioethanol production. This method is called group preserving scheme which is based on Lie group, proper for solving ordinary differential equations. In this regard, The Minkowski Cayley transformation is used to create this numerical method to get the approximate solutions of the problems. Moreover, figures are provided to show the reliability and accuracy of the proposed method.Öğe New Type Modelling of the Circumscribed Self-Excited Spherical Attractor(Mdpi, 2022) Partohaghighi, Mohammad; Akgul, Ali; Alqahtani, Rubayyi T.The fractal-fractional derivative with the Mittag-Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover, the theorems of Schauder's fixed point and Banach fixed existence theory are used to guarantee that there are solutions to the model. Approximate solutions to the problem are presented by an effective method. To prove the efficiency of the given technique, different values of fractal and fractional orders as well as initial conditions are selected. Figures of the approximate solutions are provided for each case in different dimensions.Öğe Novel Mathematical Modelling of Platelet-Poor Plasma Arising in a Blood Coagulation System with the Fractional Caputo-Fabrizio Derivative(Mdpi, 2022) Partohaghighi, Mohammad; Akgul, Ali; Guran, Liliana; Bota, Monica-FeliciaThis study develops a fractional model using the Caputo-Fabrizio derivative with order a for platelet-poor plasma arising in a blood coagulation system. The existence of solutions ensures that there are solutions to the considered system of equations. Approximate solutions to the recommended model are presented by selecting different numbers of fractional orders and initial conditions (ICs). For each case, graphs of solutions are supplied through different dimensions.Öğe Numerical analysis of the fractal-fractional diffusion model of ignition in the combustion process(Elsevier, 2024) Partohaghighi, Mohammad; Mortezaee, Marzieh; Akguel, Ali; Hassan, Ahmed M.; Sakar, NecibullahThe study employs the fractal-fractional operator to derive a distinct variant of the fractal-fractional diffusion equation. To address this challenge, a novel operational matrix technique (OM) is introduced, utilizing shifted Chebyshev cardinal functions (CCFs). Additionally, fundamental functions are employed to establish an OM tailored to the specific derivative in question. Through the application of these operational matrix techniques, the core equation is transformed into an algebraic system, paving the way for the resolution of the presented issue. The study showcases graphical representations of both exact and approximated solutions, accompanied by corresponding error graphs. Furthermore, comprehensive tables present the values of solutions and errors across various examples. For each test case, a comparative analysis of solutions at specific time points is also presented.Öğe Numerical estimation of the fractional advection-dispersion equation under the modified Atangana-Baleanu-Caputo derivative(Elsevier, 2023) Partohaghighi, Mohammad; Mortezaee, Marzieh; Akgul, Ali; Eldin, Sayed M.The transport of contaminants is a crucial environmental issue, and accurate modeling of this phenomenon is vital for developing effective strategies for its management. In this study, We introduce a non-integer model of the advection-dispersion problem arising in the transport of contaminants. The used derivative is described in the modified Atangana-Baleanu-Caputo (MABC) sense which is a new definition based on an extension of the Atangana and Baleanu derivatives. We employ discrete Chebyshev polynomials to gain the numerical solution of the considered equation. First, we generate a new operational matrix through discrete Chebyshev polynomials properties and proposed derivative. Next, via discrete Chebyshev polynomials and the operational matrix, we gain an algebraic system whose solutions are easily obtained. Finally, we solve some examples and compare the results with those obtained from other numerical methods to confirm the practicality and accuracy of the suggested scheme.Öğe Numerical estimation of the fractional Klein-Gordon equation with Discrete(Elsevier, 2024) Partohaghighi, Mohammad; Mortezaee, Marzieh; Akgul, AliWe embark on a thorough analysis of a fractional model, concentrating our efforts on exploring the intricacies of the Klein -Gordon equation, within the framework of a specialized fractional operator. Our methodology is defined by the incorporation of the Modified Atangana Baleanu Caputo (MABC) derivative, representing an enhanced evolution of the original Atangana-Baleanu derivative. The core objective of our intricate investigation is to uncover the approximate solutions of the meticulously crafted model, employing the refined computational capabilities of Discrete Chebyshev Polynomials (DCPs) to facilitate our analytical endeavors. Laying the analytical groundwork, we meticulously derive a sequence of cutting -edge operational matrices, which are synthesized by amalgamating the intrinsic attributes of DCPs and the pertinent derivatives. This sophisticated development directs us towards an elaborate algebraic system, demanding precise and accurate resolution. The structured operational matrices play a crucial role in transforming the intricate fractional differential equations into more manageable algebraic equations, allowing the application of versatile numerical techniques to find solutions and making the entire process more approachable and conclusive. Our commitment to methodological rigor and computational precision is unwavering, ensuring the reliability and validity of the proposed methodology through extensive testing on diverse examples, revealing minimal errors in the outcomes. These results underscore the robust reliability and substantial effectiveness of the presented approach, thereby confirming its promising applicability in addressing similar fractional models of differential equations. The negligible discrepancies observed in the results serve as a testament to the potential widespread applicability of this methodology, offering substantial contributions to the existing scientific discourse and providing fertile ground for future research in the realm of fractional calculus and its associated fields. The minimal discrepancies detected in the outcomes exemplify the expansive applicability of this methodology, marking significant advancements in the scientific narrative and fostering opportunities for ensuing research in fractional calculus and related domains. This methodology can potentially be employed in areas such as quantum mechanics and signal processing, allowing for enhanced analysis and solutions of complex systems, thereby contributing to advancements in the development of more accurate models and simulations in these fields.Öğe On the fractal-fractional modelling of the smoking problem(Inderscience Enterprises Ltd, 2022) Partohaghighi, Mohammad; Akgul, AliThe tobacco epidemic is an example of the most significant common health peril the world has ever suffered. Cancer-related lung throat and heart disease are the principal warnings because of smoking. The importance of this problem motivated us to model it in the frame of fractal fractional derivative. We use Atangana-Baleanu-Caputo operator. Moreover, to confirm that there is a solution for the proposed model, we apply the theorems of Schauder fixed point and Banach fixed. Also, to obtain the numerical solutions of the offered model Atangana-Toufik technique is applied. To show the performance of the used method for the considered problem, different values of fractal and fractional orders are chosen. Additionally, successful graphs of solutions are provided for each case.Öğe Oscillatory and complex behaviour of Caputo-Fabrizio fractional order HIV-1 infection model(Amer Inst Mathematical Sciences-Aims, 2022) Ahmad, Shabir; Ullah, Aman; Partohaghighi, Mohammad; Saifullah, Sayed; Akgul, Ali; Jarad, FahdHIV-1 infection is a dangerous diseases like Cancer, AIDS, etc. Many mathematical models have been introduced in the literature, which are investigated with different approaches. In this article, we generalize the HIV-1 model through nonsingular fractional operator. The non-integer mathematical model of HIV-1 infection under the Caputo-Fabrizio derivative is presented in this paper. The concept of Picard-Lindelof and fixed-point theory are used to address the existence of a unique solution to the HIV-1 model under the suggested operator. Also, the stability of the suggested model is proved through the Picard iteration and fixed point theory approach. The model's approximate solution is constructed through three steps Adams-Bashforth numerical method. Numerical simulations are provided for different values of fractional-order to study the complex dynamics of the model. Lastly, we provide the oscillatory and chaotic behavior of the proposed model for various fractional orders.Öğe Solving the time-fractional inverse Burger equation involving fractional Heydari-Hosseininia derivative(Amer Inst Mathematical Sciences-Aims, 2022) Partohaghighi, Mohammad; Akgul, Ali; Asad, Jihad; Wannan, RaniaHeydari-Hosseininia (HH) fractional derivative is a newly introduced concept of fractional calculus which conquers the restrictions of non-singular fractional derivatives in the Caputo-Fabrizio (CF) and Atangana-Baleanu senses. For instance, it is not easy to get the closed-form of the fractional derivative of functions using CF because of the construction of its kernel function. In this paper, we present a powerful numerical scheme based on energy boundary functions to get the approximate solutions of the time-fractional inverse Burger equation containing HH-derivative: (HH)D(tau)(alpha)h(z, tau) - h(z, tau)h(z)(z, tau) = h(zz)(z, tau) + H(z, tau), which (HH)D(alpha)(tau )is the HH-derivative with regard to alpha-order. This problem has never been investigated earlier so, this is our motivation to work on this important problem. Some numerical examples are presented to verify the efficiency of the presented technique. Graphs of the exact and numerical solutions along with the plot of absolute error are provided for each example. Tables are given to see and compare the results point by point for each example.
