Yazar "De la Sen, Manuel" seçeneğine göre listele

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    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, Manuel
    This 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.
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    A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel
    (Hindawi Ltd, 2021) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; De la Sen, Manuel
    In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.
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    A study of fractional order Ambartsumian equation involving exponential decay kernel
    (Amer Inst Mathematical Sciences-Aims, 2021) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; De la Sen, Manuel
    Recently, non-singular fractional operators have a significant role in the modeling of real-world problems. Specifically, the Caputo-Fabrizio operators are used to study better dynamics of memory processes. In this paper, under the non-singular fractional operator with exponential decay kernel, we analyze the Ambartsumian equation qualitatively and computationally. We deduce the result of the existence of at least one solution to the proposed equation through Krasnoselskii's fixed point theorem. Also, we utilize the Banach fixed point theorem to derive the result concerned with unique solution. We use the concept of functional analysis to show that the proposed equation is Ulam-Hyers and Ulam-Hyers-Rassias stable. We use an efficient analytical approach to compute a semi-analytical solution to the proposed problem. The convergence of the series solution to an exact solution is proved through non-linear analysis. Lastly, we present the solution for different fractional orders.
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    An Efficient Approach for Solving Differential Equations in the Frame of a New Fractional Derivative Operator
    (Mdpi, 2023) Attia, Nourhane; Akgul, Ali; Seba, Djamila; Nour, Abdelkader; De la Sen, Manuel; Bayram, Mustafa
    Recently, a new fractional derivative operator has been introduced so that it presents the combination of the Riemann-Liouville integral and Caputo derivative. This paper aims to enhance the reproducing kernel Hilbert space method (RKHSM, for short) for solving certain fractional differential equations involving this new derivative. This is the first time that the application of the RKHSM is employed for solving some differential equations with the new operator. We illustrate the convergence analysis of the applicability and reliability of the suggested approaches. The results confirm that the RKHSM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed method.
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    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, Mustafa
    Numerical 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.
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    Approximate and Exact Solutions in the Sense of Conformable Derivatives of Quantum Mechanics Models Using a Novel Algorithm
    (Mdpi, 2023) Liaqat, Muhammad Imran; Akgul, Ali; De la Sen, Manuel; Bayram, Mustafa
    The entirety of the information regarding a subatomic particle is encoded in a wave function. Solving quantum mechanical models (QMMs) means finding the quantum mechanical wave function. Therefore, great attention has been paid to finding solutions for QMMs. In this study, a novel algorithm that combines the conformable Shehu transform and the Adomian decomposition method is presented that establishes approximate and exact solutions to QMMs in the sense of conformable derivatives with zero and nonzero trapping potentials. This solution algorithm is known as the conformable Shehu transform decomposition method (CSTDM). To evaluate the efficiency of this algorithm, the numerical results in terms of absolute and relative errors were compared with the reduced differential transform and the two-dimensional differential transform methods. The comparison showed excellent agreement with these methods, which means that the CSTDM is a suitable alternative tool to the methods based on the Caputo derivative for the solutions of time-fractional QMMs. The advantage of employing this approach is that, due to the use of the conformable Shehu transform, the pattern between the coefficients of the series solutions makes it simple to obtain the exact solution of both linear and nonlinear problems. Consequently, our approach is quick, accurate, and easy to implement. The convergence, uniqueness, and error analysis of the solution were examined using Banach's fixed point theory.
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    Cattaneo-Christov heat flux model in radiative flow of (Fe3O4- TiO2/Transformer oil) and (Cu- TiO2/Transformer oil) magnetized hybrid nanofluids past through double rotating disks
    (Elsevier, 2023) Farooq, Umar; Imran, Muhammad; Fatima, Nahid; Noreen, Sobia; Tahir, Madeeha; Akgul, Ali; De la Sen, Manuel
    Recent progresses in nanotechnologies and nanoscience have led to the creation of hybrid nano-fluids, which are a complicated category of fluids with superior thermal features to regular nano-fluids. The current framework demonstrates the importance of a two-dimensional steady incompressible axisymmetric flow of Maxwell hybrid nanofluid over double disks with thermal radiation. This investigation analyzes a novel idea regarding the execution of the Cattaneo-Christov heat theory and melting phenomenon by considering transformer oil as a base fluid. Two dissimilar classes of hybrid nanofluid, Iron-Titanium oxide/Transformer oil (Fe3O4-TiO2/TO) and Cop-per-Titanium oxide/Transformer oil (Cu-TiO2/TO) have been taken into our research work. The main equations (PDEs)are translated into a present set of ODEs using the necessary similarity var-iables. In MATLAB, the shooting scheme is utilized to evaluate the numerical and graphical out-comes of physical flow parameters. The radial velocity rose as the volume fraction of nanoparticles increased. The radial velocity field is increased as the porosity parameter is enhanced. The tem-perature profile is decreased with increasing the values of the thermal redaction parameter. Furthermore, because of the higher compactness of the copper nanoparticles, the addition of the volume fraction of nanoparticles slows the flow profile, and because copper is an excellent conductor of heat, it raises the fluid temperature throughout the domain. The mathematical fallouts also tackle the idea of employing magnetized spinning discs in space engines and nuclear propulsion, and such a model carries useful applications in heat transfer enhancement in a wide range of in-dustrial thermal management devices and renewable energy generation systems.
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    Controllability of Impulsive Neutral Fractional Stochastic Systems
    (Mdpi, 2022) Ain, Qura Tul; Nadeem, Muhammad; Akgul, Ali; De la Sen, Manuel
    The study of dynamic systems appears in various aspects of dynamical structures such as decomposition, decoupling, observability, and controllability. In the present research, we study the controllability of fractional stochastic systems (FSF) and examine the Poisson jumps in finite dimensional space where the fractional impulsive neutral stochastic system is controllable. Sufficient conditions are demonstrated with the aid of fixed point theory. The Mittag-Leffler (ML) matrix function defines the controllability of the Grammian matrix (GM). The relation to symmetry is clear since the controllability Grammian is a hermitian matrix (since the integrand in its definition is hermitian) and this is the complex version of a symmetric matrix. In fact, such a Grammian becomes a symmetric matrix in the specific scenario where the controllability Grammian is a real matrix. Some examples are provided to demonstrate the feasibility of the present theory.
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    Crossover Dynamics of Rotavirus Disease under Fractional Piecewise Derivative with Vaccination Effects: Simulations with Real Data from Thailand, West Africa, and the US
    (Mdpi, 2022) Naowarat, Surapol; Ahmad, Shabir; Saifullah, Sayed; De la Sen, Manuel; Akguel, Ali
    Many diseases are caused by viruses of different symmetrical shapes. Rotavirus particles are approximately 75 nm in diameter. They have icosahedral symmetry and particles that possess two concentric protein shells, or capsids. In this research, using a piecewise derivative framework with singular and non-singular kernels, we investigate the evolution of rotavirus with regard to the effect of vaccination. For the considered model, the existence of a solution of the piecewise rotavirus model is investigated via fixed-point results. The Adam-Bashforth numerical method along with the Newton polynomial is implemented to deduce the numerical solution of the considered model. Various versions of the stability of the solution of the piecewise rotavirus model are presented using the Ulam-Hyres concept and nonlinear analysis. We use MATLAB to perform the numerical simulation for a few fractional orders to study the crossover dynamics and evolution and effect of vaccination on rotavirus disease. To check the validity of the proposed approach, we compared our simulated results with real data from various countries.
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    Dynamical study of groundwater systems using the new auxiliary equation method
    (Elsevier, 2024) Shahid, Naveed; Baber, Muhammad Zafarullah; Shaikh, Tahira Sumbal; Iqbal, Gulshan; Ahmed, Nauman; Akgul, Ali; De la Sen, Manuel
    In this research, the exact solitary wave solutions to the non-linear problem of underground water levels are found. This study examines the transport of solutes in groundwater systems with variable density flow. The mathematical equation that is used to explain how groundwater moves through an aquifer is known as the groundwater flow equation and is used in hydrogeology. The auxiliary equation method is used to gain the analytical solutions to the underlying model equation. These solutions are gained in the form of hyperbolic, trigonometric, exponential, and rational function solutions. Mathematica generates two-dimensional and threedimensional graphs with suitable parameter values. The resulting solutions are also useful for researching wave interactions in several novel structures.
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    Fractional Order Operator for Symmetric Analysis of Cancer Model on Stem Cells with Chemotherapy
    (Mdpi, 2023) Azeem, Muhammad; Farman, Muhammad; Akgul, Ali; De la Sen, Manuel
    Cancer is dangerous and one of the major diseases affecting normal human life. In this paper, a fractional-order cancer model with stem cells and chemotherapy is analyzed to check the effects of infection in individuals. The model is investigated by the Sumudu transform and a very effective numerical method. The positivity of solutions with the ABC operator of the proposed technique is verified. Fixed point theory is used to derive the existence and uniqueness of the solutions for the fractional order cancer system. Our derived solutions analyze the actual behavior and effect of cancer disease in the human body using different fractional values. Modern mathematical control with the fractional operator has many applications including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions of a given fractional differential equation in a fairly systematic way. Such an analysis will provide a better understanding to control the of cancer disease in the human body.
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    Imaging Ultrasound Propagation Using the Westervelt Equation by the Generalized Kudryashov and Modified Kudryashov Methods
    (Mdpi, 2022) Ghazanfar, Sidra; Ahmed, Nauman; Iqbal, Muhammad Sajid; Akgul, Ali; Bayram, Mustafa; De la Sen, Manuel
    This article deals with the study of ultrasound propagation, which propagates the mechanical vibration of the molecules or of the particles of a material. It measures the speed of sound in air. For this reason, the third-order non-linear model of the Westervelt equation was chosen to be studied, as the solutions to such problems have much importance for physical purposes. In this article, we discuss the exact solitary wave solutions of the third-order non-linear model of the Westervelt equation for an acoustic pressure p representing the equation of ultrasound with high intensity, as used in acoustic tomography. Moreover, the non-linear coefficient B / A (being a part of space-dependent coefficient K), has also been investigated in this literature. This problem is solved using the Generalized Kudryashov method along with a comparison of the Modified Kudryashov method. All of the solutions have been discussed with both surface and contour plots, which shows the behavior of the solution. The images are prepared in a well-established way, showing the production of tissues inside the human body.
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    Modelling and Analysis of a Measles Epidemic Model with the Constant Proportional Caputo Operator
    (Mdpi, 2023) Farman, Muhammad; Shehzad, Aamir; Akgul, Ali; Baleanu, Dumitru; De la Sen, Manuel
    Despite the existence of a secure and reliable immunization, measles, also known as rubeola, continues to be a leading cause of fatalities globally, especially in underdeveloped nations. For investigation and observation of the dynamical transmission of the disease with the influence of vaccination, we proposed a novel fractional order measles model with a constant proportional (CP) Caputo operator. We analysed the proposed model's positivity, boundedness, well-posedness, and biological viability. Reproductive and strength numbers were also verified to examine how the illness dynamically behaves in society. For local and global stability analysis, we introduced the Lyapunov function with first and second derivatives. In order to evaluate the fractional integral operator, we used different techniques to invert the PC and CPC operators. We also used our suggested model's fractional differential equations to derive the eigenfunctions of the CPC operator. There is a detailed discussion of additional analysis on the CPC and Hilfer generalised proportional operators. Employing the Laplace with the Adomian decomposition technique, we simulated a system of fractional differential equations numerically. Finally, numerical results and simulations were derived with the proposed measles model. The intricate and vital study of systems with symmetry is one of the many applications of contemporary fractional mathematical control. A strong tool that makes it possible to create numerical answers to a given fractional differential equation methodically is symmetry analysis. It is discovered that the proposed fractional order model provides a more realistic way of understanding the dynamics of a measles epidemic.
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    On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique
    (Amer Inst Mathematical Sciences-Aims, 2023) Bouazza, Zoubida; Souhila, Sabit; Etemad, Sina; Souid, Mohammed Said; Akguel, Ali; Rezapour, Shahram; De la Sen, Manuel
    This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.
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    Study of HIV Disease and Its Association with Immune Cells under Nonsingular and Nonlocal Fractal-Fractional Operator
    (Wiley-Hindawi, 2021) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; De la Sen, Manuel
    HIV, like many other infections, is a severe and lethal infection. Fractal-fractional operators are frequently used in modeling numerous physical processes in the current decade. These operators provide better dynamics of a mathematical model because these are the generalization of integer and fractional-order operators. This paper aims to study the dynamics of the HIV model during primary infection by fractal-fractional Atangana-Baleanu (AB) operators. The sufficient conditions for the existence and uniqueness of the solution of the proposed model under the AB operator are derived via fixed point theory. The numerical scheme is presented by using the Adams-Bashforth method. Numerical results are demonstrated for different fractal and fractional orders to see the effect of fractional order and fractal dimension on the dynamics of HIV and CD4+ T-cells during primary infection.