Yazar "Liaqat, Muhammad Imran" seçeneğine göre listele

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    A novel approach for solving linear and nonlinear time-fractional Schrodinger equations
    (Pergamon-Elsevier Science Ltd, 2022) Liaqat, Muhammad Imran; Akgul, Ali
    There is significant literature on Schrodinger differential equation (SDE) solutions, where the fractional derivatives are stated in terms of Caputo derivative (CD). There is hardly any work on analytical and numerical SDE solutions involving conformable fractional derivative (CFD). For the reasons stated above, we are required to solve the SDE in the form of CFD. The main goal of this research is to offer a novel combined computational approach by using conformable natural transform (CNT) and the homotopy perturbation method (HPM) for extracting analytical and numerical solutions of the time-fractional conformable Schrodinger equation (TFCSE) with zero and nonzero trapping potential. We call it the conformable natural transform homotopy perturbation method (CNTHPM). The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the CNTHPM. The error analysis has confirmed the higher degree of accuracy and convergence rates, which indicates the effectiveness and reliability of the suggested method. Furthermore, 2D and 3D graphs compare the exact and approximate solutions. The procedure is quick, precise, and easy to implement, and it yields outstanding results. In addition, numerical results are also compared with other methods such as the differential transform method (DTM), split-step finite difference method (SSFDM), homotopy analysis method (HAM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), and two-dimensional differential transform method (TDDTM). The comparison shows excellent agreement with these methods, which means that CNTHPM is a suitable alternative tool to the methods based on CD for the solutions of the time-fractional SDE. Moreover, we can conclude that the CFD is a suitable alternative to the CD in the modeling of time-fractional SDE. The Banach fixed point theory was also used to test the uniqueness of the solution, convergence, and error analysis.
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    A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay
    (Hindawi Ltd, 2022) Liaqat, Muhammad Imran; Khan, Adnan; Akgul, Ali; Ali, Md. Shajib
    Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor's series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He's and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.
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    Öğe
    A novel technique using integral transforms and residual functions for nonlinear partial fractional differential equations involving Caputo derivatives.
    (Public Library of Science, 2024) Khan, Zareen A; Riaz, Muhammad Bilal; Liaqat, Muhammad Imran; Akgül, Ali
    Fractional nonlinear partial differential equations are used in many scientific fields to model various processes, although most of these equations lack closed-form solutions. For this reason, methods for approximating solutions that occasionally yield closed-form solutions are crucial for solving these equations. This study introduces a novel technique that combines the residual function and a modified fractional power series with the Elzaki transform to solve various nonlinear problems within the Caputo derivative framework. The accuracy and effectiveness of our approach are validated through analyses of absolute, relative, and residual errors. We utilize the limit principle at zero to identify the coefficients of the series solution terms, while other methods, including variational iteration, homotopy perturbation, and Adomian, depend on integration. In contrast, the residual power series method uses differentiation, and both approaches encounter difficulties in fractional contexts. Furthermore, the effectiveness of our approach in addressing nonlinear problems without relying on Adomian and He polynomials enhances its superiority over various approximate series solution techniques.
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    Adaptation on power series method with conformable operator for solving fractional order systems of nonlinear partial differential equations
    (Pergamon-Elsevier Science Ltd, 2022) Liaqat, Muhammad Imran; Khan, Adnan; Akgul, Ali
    The aim of this research work is to modify the power series solution method to fractional order in the sense of conformable derivative to solve a coupled system of nonlinear fractional partial differential equations. We called it the conformable fractional power series method. To evaluate its efficiency and consistency, absolute errors of three problems are considered numerically. Consequences established that our recommended method is unpretentious, accurate, valid, and capable. When solving the nonlinear complications, it has a powerful superiority over the homotopy analysis and Adomian decomposition methods. Additional as in the residual power series method through generating the coefficients for a series, it is compulsory to calculate the fractional derivatives on every occasion, whereas this method only needs the idea of equating coefficients. The convergence and error analyses of the series solutions are also presented.(c) 2022 Elsevier Ltd. All rights reserved.
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    Analytical Investigation of Some Time-Fractional Black-Scholes Models by the Aboodh Residual Power Series Method
    (Mdpi, 2023) Liaqat, Muhammad Imran; Akgul, Ali; Abu-Zinadah, Hanaa
    In this study, we use a new approach, known as the Aboodh residual power series method (ARPSM), in order to obtain the analytical results of the Black-Scholes differential equations (BSDEs), which are prime for judgment of European call and put options on a non-dividend-paying stock, especially when they consist of time-fractional derivatives. The fractional derivative is considered in the Caputo sense. This approach is a combination of the Aboodh transform and the residual power series method (RPSM). The suggested approach is based on a new version of Taylor's series that generates a convergent series as a solution. The advantage of our strategy is that we can use the Aboodh transform operator to transform the fractional differential equation into an algebraic equation, which decreases the amount of computation required to obtain the solution in a subsequent algebraic step. The primary aspect of the proposed approach is how easily it computes the coefficients of terms in a series solution using the simple limit at infinity concept. In the RPSM, unknown coefficients in series solutions must be determined using the fractional derivative, and other well-known approximate analytical approaches like variational iteration, Adomian decomposition, and homotopy perturbation require the integration operators, which is challenging in the fractional case. Moreover, this approach solves problems without the need for He's polynomials and Adomian polynomials, so the small size of computation is the strength of this approach, which is an advantage over various series solution methods. The efficiency of the suggested approach is verified by results in graphs and numerical data. The recurrence errors at various levels of the fractional derivative are utilized to demonstrate the convergence evidence for the approximative solution to the exact solution. The comparison study is established in terms of the absolute errors of the approximate and exact solutions. We come to the conclusion that our approach is simple to apply and accurate based on the findings.
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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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    Approximate and Exact Solutions of Some Nonlinear Differential Equa-tions Using the Novel Coupling Approach in the Sense of Conformable Fractional Derivative
    (Universal Wiser Publisher, 2024) Liaqat, Muhammad Imran; Akgul, Ali
    Several scientific fields utilize fractional nonlinear partial differential equations to model various phenomena. However, most of these equations lack exact solutions. Consequently, techniques for obtaining approximate solutions, which sometimes yield exact solutions, are essential. In this research, we develop a new approach by combining the homotopy perturbation method (HPM) and the conformable natural transform to solve the gas-dynamic equation (GDE), the Fokker-Planck equation (FPE), and the Swift-Hohenberg equation (SHE) in the context of conformable derivatives. The proposed approach is called the conformable natural homotopy perturbation method (CNHPM). This approach has the advantage of not requiring assumptions about significant or minor physical factors. Consequently, it eliminates some of the constraints associated with conventional perturbation methods and can solve both weak and highly nonlinear problems. We consider the absolute, relative, and residual errors numerically and graphically to assess the correctness of our approach. The results show that our approach serves as a suitable alternative to the approximate methods in the literature for solving fractional differential equations.
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    Qualitative Analysis of Stochastic Caputo-Katugampola Fractional Differential Equations
    (Mdpi, 2024) Khan, Zareen A.; Liaqat, Muhammad Imran; Akgul, Ali; Conejero, J. Alberto
    Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we achieve significant results for these equations within the context of Caputo-Katugampola derivatives. First, we establish the existence and uniqueness of solutions by employing the contraction mapping principle with a suitably weighted norm and demonstrate that the solutions continuously depend on both the initial values and the fractional exponent. The second part examines the regularity concerning time. Third, we illustrate the results of the averaging principle using techniques involving inequalities and interval translations. We generalize these results in two ways: first, by establishing them in the sense of the Caputo-Katugampola derivative. Applying condition beta=1, we derive the results within the framework of the Caputo derivative, while condition beta -> 0+ yields them in the context of the Caputo-Hadamard derivative. Second, we establish them in Lp space, thereby generalizing the case for p=2.
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    Revised and Generalized Results of Averaging Principles for the Fractional Case
    (Mdpi, 2024) Liaqat, Muhammad Imran; Khan, Zareen A.; Conejero, J. Alberto; Akgul, Ali
    The averaging principle involves approximating the original system with a simpler system whose behavior can be analyzed more easily. Recently, numerous scholars have begun exploring averaging principles for fractional stochastic differential equations. However, many previous studies incorrectly defined the standard form of these equations by placing epsilon in front of the drift term and epsilon in front of the diffusion term. This mistake results in incorrect estimates of the convergence rate. In this research work, we explain the correct process for determining the standard form for the fractional case, and we also generalize the result of the averaging principle and the existence and uniqueness of solutions to fractional stochastic delay differential equations in two significant ways. First, we establish the result in Lp space, generalizing the case of p=2. Second, we establish the result using the Caputo-Katugampola operator, which generalizes the results of the Caputo and Caputo-Hadamard derivatives.
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    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, Mustafa
    The 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.
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    Some important results for the conformable fractional stochastic pantograph differential equations in the Lp space
    (Int Scientific Research Publications, 2025) Liaqat, Muhammad Imran; Din, Fahim Ud; Akgul, Ali; Riaz, Muhammad Bilal
    Important mathematical topics include existence, uniqueness, continuous dependency, regularity, and the averaging principle. In this research work, we establish these results for the conformable fractional stochastic pantograph differential equations (CFSPDEs) in L-p space. The situation of p = 2 is generalized by the obtained findings. First, we establish the existence and uniqueness results by applying the contraction mapping principle under a suitably weighted norm and demonstrating the continuous dependency of solutions on both the initial values and fractional exponent 4). . The second section is devoted to examining the regularity of time. As a result, we find that, for each Phi is an element of ( 0, Phi - 1/2 ), the solution to the considered problem has Phi-Holder continuous version. Next, we study the averaging principle by using Jensen's, Gronwall-Bellman's, Holder's, and BurkholderDavis-Gundy's inequalities. To help with the understanding of the theoretical results, we provide three applied examples at the end.