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Öğe A method for solving the generalized Camassa-Choi problem with the Mittag-Leffler function and temporal local derivative(Elsevier, 2023) Hashemi, Mir Sajjad; Akguel, Ali; Hassan, Ahmed; Bayram, MustafaThis paper focuses on a reduction technique to discover exact solutions for the generalized Camassa-Choi equation with temporal local M-derivative. The paper presents various types of exact solutions along with their corresponding first integrals. Furthermore, the interactions between the orders of alpha and beta in the M-derivative are taken into account and depicted graphically for the derived solutions. Remarkably, the paper demonstrates that in certain situations, exact solutions can be obtained for any value of n, which holds significant mathematical intrigue. The authors note that Nucci's reduction technique has not previously been employed for differential equations with M-derivative, to the best of their knowledge.Öğe 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, MustafaRecently, 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.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 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, MustafaThe 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.Öğe Construction of Novel Bright-Dark Solitons and Breather Waves of Unstable Nonlinear Schrodinger Equations with Applications(Mdpi, 2023) Sarwar, Ambreen; Arshad, Muhammad; Farman, Muhammad; Akgul, Ali; Ahmed, Iftikhar; Bayram, Mustafa; Rezapour, ShahramThe unstable nonlinear Schrodinger equations (UNLSEs) are universal equations of the class of nonlinear integrable systems, which reveal the temporal changing of disruption in slightly stable and unstable media. In current paper, an improved auxiliary equation technique is proposed to obtain the wave results of UNLSE and modified UNLSE. Numerous varieties of results are generated in the mode of some special Jacobi elliptic functions and trigonometric and hyperbolic functions, many of which are distinctive and have significant applications such as pulse propagation in optical fibers. The exact soliton solutions also give information on the soliton interaction in unstable media. Furthermore, with the assistance of the suitable parameter values, various kinds of structures such as bright-dark, multi-wave structures, breather and kink-type solitons, and several periodic solitary waves are depicted that aid in the understanding of the physical interpretation of unstable nonlinear models. The various constructed solutions demonstrate the effectiveness of the suggested approach, which proves that the current technique may be applied to other nonlinear physical problems encountered in mathematical physics.Öğe EFFECTS OF CLUSTER THINNING TREATMENTS ON CHEMICAL COMPOSITION AND PHENOLIC COMPOUNDS OF GRAPE JUICE AND WINES OF NARINCE (VITIS VINIFERA) GRAPE CULTIVAR(Parlar Scientific Publications (P S P), 2019) Uzun, Tuba; Cangi, Rustem; Bayram, MustafaThe present study was conducted to determine the effects of 4 different cluster thinning treatments (CTT) [control (C), 15% (12000 kg/ha), 30% (9000 kg/ha), 60% (6000 kg/ha)] on grape juice and wine quality of Narince grape cultivar grown in Tokat province (Turkey) in 2014-2015 growing seasons. The grapes harvested at technological maturity stage were processed into wines. The pH, total soluble solid contents, titratable acidity, specific gravity, ethyl alcohol, volatile acid, reducing sugar, total sulphur dioxide, total phenolics, total flavonoids and some phenolic compounds of the grape juice and wines were determined. Sensory evaluations revealed that cluster thinning applications increased wine quality. The greatest sensory evaluation score was obtained from 30% cluster thinning treatment and these wines were classified as excellent quality wines. As a result, according to analyzes performed in grape juice and wines, 30% (9000 kg/ha) cluster thinning treatment was found to be the most suitable thinning practice in Narince.Öğe Further study of eccentricity based indices for benzenoid hourglass network(Cell Press, 2023) Iqbal, Hifza; Aftab, Muhammad Haroon; Akgul, Ali; Mufti, Zeeshan Saleem; Yaqoob, Iram; Bayram, Mustafa; Riaz, Muhammad BilalTopological Indices are the mathematical estimate related to atomic graph that corresponds biological structure with several real properties and chemical activities. These indices are invariant of graph under graph isomorphism. If top(h1) and top(h2) denotes topological index h1 and h2 respectively then h1 approximately equal h2 which implies that top(h1) = top(h2). In biochemistry, chemical science, nano-medicine, biotechnology and many other science's distance based and eccentricity-connectivity(EC) based topological invariants of a network are beneficial in the study of structure-property relationships and structure-activity relationships. These indices help the chemist and pharmacist to overcome the shortage of laboratory and equipment. In this paper we calculate the formulas of eccentricity-connectivity descriptor(ECD) and their related polynomials, total eccentricity-connectivity(TEC) polynomial, augmented eccentricityconnectivity(AEC) descriptor and further the modified eccentricity-connectivity(MEC) descriptor with their related polynomials for hourglass benzenoid network.Öğe 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, ManuelThis 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.Öğe Mathematical modelling of COVID-19 outbreak using caputo fractional derivative: stability analysis(Taylor & Francis Ltd, 2024) Ul Haq, Ihtisham; Ali, Nigar; Bariq, Abdul; Akgul, Ali; Baleanu, Dumitru; Bayram, MustafaThe novel coronavirus SARS-Cov-2 is a pandemic condition and poses a massive menace to health. The governments of different countries and their various prohibitory steps to restrict the virus's expanse have changed individuals' communication processes. Due to physical and financial factors, the population's density is more likely to interact and spread the virus. We establish a mathematical model to present the spread of the COVID-19 in worldwide. In this article, we propose a novel mathematical model (' $ \mathbb {S}\mathbb {L}\mathbb {I}\mathbb {I}_{q}\mathbb {I}_{h}\mathbb {R}\mathbb {P} $ SLIIqIhRP') to assess the impact of using hospitalization, quarantine measures, and pathogen quantity in controlling the COVID-19 pandemic. We analyse the boundedness of the model's solution by employing the Laplace transform approach to solve the fractional Gronwall's inequality. To ensure the uniqueness and existence of the solution, we rely on the Picard-Lindelof theorem. The model's basic reproduction number, a crucial indicator of epidemic potential, is determined based on the greatest eigenvalue of the next-generation matrix. We then employ stability theory of fractional differential equations to qualitatively examine the model. Our findings reveal that both locally and globally, the endemic equilibrium and disease-free solutions demonstrate symptomatic stability. These results shed light on the effectiveness of the proposed interventions in managing and containing the COVID-19 outbreak.Öğe Series and closed form solution of Caputo time-fractional wave and heat problems with the variable coefficients by a novel approach(Springer, 2024) Liaqat, Muhammad Imran; Akguel, Ali; Bayram, MustafaThe mathematical efficiency of fractional-order differential equations in modeling real systems has been established. The first-order and second-order time derivatives are substituted in integer-order problems by a fractional derivative of order 0 < omega <= 1, resulting in time-fractional heat and wave problems with variable coefficients. In this research, we analyze fractional-order wave and heat problems with variable coefficients within the framework of a Caputo derivative (CD) using the Elzaki residual power series method (ERPSM), which is a coupling of the residual power series method (RPSM) and the Elzaki transform (E-T). It relies on a novel form of fractional power series (FPS), which provides a convergent series as a solution. The accuracy and convergence rates have been proven by the relative, absolute, and recurrence error analyses, demonstrating the validity of the recommended approach. By employing the simple limit principle at zero, the ERPSM excels at calculating the coefficients of terms in a FPS, but other well-known approaches such as Adomian decomposition, variational iteration, and homotopy perturbation need integration, while the RPSM needs the derivative, both of which are challenging in fractional contexts. ERPSM is also more effective than various series solution methods due to the avoidance of Adomian's and He's polynomials to solve nonlinear problems. The results obtained using the ERPSM show excellent agreement with the natural transform decomposition method and homotopy analysis transform method, demonstrating that the ERPSM is an effective approach for obtaining the approximate and closed-form solutions of fractional models. We established that our approach for fractional models is accurate and straightforward and researcher can use this approach to solve various problems.
