Browsing by Author "Jahan, S"

Now showing 1 - 12 of 12
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    An efficient method for the fractional electric circuits based on Fibonacci wavelet
    (2023-07) Ahmed, H; Shah, K; Jahan, S
    In this article, we provide effective computational algorithms based on Fibonacci wavelet (FW) to approximate the solution of fractional order electrical circuits (ECs). The proposed computational algorithm is novel and has not been previously utilized for solving ECs problems. Firstly, we have constructed the operational matrices of fractional integration (OMFI). Secondly, we transform the given initial value problems into algebraic equations, we used the Riemann–Liouville (R–L) fractional integral operator. The proposed approach is capable of handling a wide range of fractional order dynamics in ECs. To validate the effectiveness of the method, four models of electrical circuits with fractional order parameter are considered. The numerical results are compared with exact solutions and absolute errors are calculated to demonstrate the accuracy and efficiency of the approach. The proposed method provides a valuable tool for analyzing and designing fractional order systems in electrical engineering, offering improved accuracy and capturing the intricate behavior of complex systems.
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    Fibonacci wavelet method for the numerical solution of a fractional relaxation–oscillation model
    (2023-10) Jahan, S; Ahmed, S; Yadav, P; Nisar, K
    In this paper, we have discussed the Fibonacci wavelet (FW) framework for numerical simulations of the fractional relaxation–oscillation model (FROM). Firstly, the fractional order operational matrices of integration associated with the FW are constructed via the block pulse functions. The operational matrices merged with the collocation method are used to convert the given problem into a system of algebraic equations that is solved by the Newton method. We conduct error analysis, perform numerical simulations, and present the corresponding results to establish the credibility and practical applicability of the proposed technique. Numerical examples are provided to show the efficiency of our approach. To show the accuracy of the FW-based numerical technique, the approximate solutions of FROM are compared with the exact solution and other existing methods. This research opens up new possibilities for using FW as a powerful tool for addressing complex mathematical problems in real-world systems.
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    Fibonacci wavelet method for the numerical solution of a fractional relaxation–oscillation model
    (2023-10) Jahan, S; Ahmed, S; Yadav, K; Nisar, K
    In this paper, we have discussed the Fibonacci wavelet (FW) framework for numerical simulations of the fractional relaxation–oscillation model (FROM). Firstly, the fractional order operational matrices of integration associated with the FW are constructed via the block pulse functions. The operational matrices merged with the collocation method are used to convert the given problem into a system of algebraic equations that is solved by the Newton method. We conduct error analysis, perform numerical simulations, and present the corresponding results to establish the credibility and practical applicability of the proposed technique. Numerical examples are provided to show the efficiency of our approach. To show the accuracy of the FW-based numerical technique, the approximate solutions of FROM are compared with the exact solution and other existing methods. This research opens up new possibilities for using FW as a powerful tool for addressing complex mathematical problems in real-world systems.
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    Fractional order mathematical model of Ebola virus under Atangana–Baleanu–Caputo operator
    (2023-11) Yadav, P; Jahan, S; Nisar, K
    The aim of this paper is to analyze a fractional model of the Ebola virus. This study is important because it contributes to our understanding of the Ebola virus transmission dynamics using the notion of non-local differential operators. We aim to apply the recently implemented Atangana– Baleanu–Caputo (ABC) fractional derivative with the Mittag-Leffler kernel to study the Ebola virus model closely. The Picard–Lindelof approach is used to do a comprehensive study of the existence and uniqueness of the model’s solutions. The approximate solutions of the fractional order Ebola virus model were obtained using a numerical technique with the ABC operator, a combination of the fundamental theorem of fractional calculus and the two-step Lagrange polynomial interpolation. This innovative approach may offer new insights into the Ebola virus model that were not previously explored. Finally, the numerical simulations illustrate how the control parameters impact specific compartments within the model. The geometrical representation gives significant information about the model’s complexity and reliable infor mation about the model. We simulate each model compartment at various fractional orders and compare them with integer-order simulations, highlighting the effectiveness of modern derivatives. The fractional analysis underscores the enhanced accuracy of non-integer order derivatives in capturing the Ebola virus model’s dynamics.
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    Fractional-order modelling and analysis of diabetes mellitus: Utilizing the Atangana-Baleanu Caputo (ABC) operator
    (2023-09) Yadav, P; Jahan, S; Shah, K
    This article aims to introduce and analyze a diabetes mellitus model of fractional order, utilizing the ABC derivative. Diabetes mellitus is a prevalent and significant disease worldwide, ranking among the top causes of mortality. It is characterized by chronic metabolic dysfunction, leading to elevated blood glucose levels and subsequent damage to vital organs including the nerves, kidneys, eyes, blood vessels, and heart. The fractional ABC derivative is used in this study to describe and analyzediabetes mellitus mathematically while removing hereditary influences. The investigation begins by exploring the initial points of the diabetes mellitus model. Under the fractional ABC operator, Picard’s theorem is used to prove the existence and uniqueness of solutions. For the numerical approximation of solutions in the fractional-order diabetes mellitus model, this study used a specialized technique that combines the principles of fractional calculus and a two-step Lagrange polynomial interpolation. Finally, the obtained results are visually presented through graphical representations, serving as empirical evidence to support our theoretical findings. The numerical experiments showed that the proportion of patients with diabetes mellitus increased as the fractional dimension (𝜃) reduced. The combination of mathematical modelling, analysis, and numerical simulations provides insights into the dynamics of diabetes mellitus, offering valuable contributions to the understanding and management of this prevalent disease. Additionally, the proposed scheme can be enhanced by incorporating the ABC operator, allowing for the simulation of real-world dynamics and behavior in the coexistence of diabetes mellitus and tuberculosis.
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    Fractional-order modelling and analysis of diabetes mellitus: Utilizing the Atangana-Baleanu Caputo (ABC) operator
    (2023-09) Yadav, P; Jahan, S; Shah, K
    This article aims to introduce and analyze a diabetes mellitus model of fractional order, utilizing the ABC derivative. Diabetes mellitus is a prevalent and significant disease worldwide, ranking among the top causes of mortality. It is characterized by chronic metabolic dysfunction, leading to elevated blood glucose levels and subsequent damage to vital organs including the nerves, kidneys, eyes, blood vessels, and heart. The fractional ABC derivative is used in this study to describe and analyze diabetes mellitus mathematically while removing hereditary influences. The investigation begins by exploring the initial points of the diabetes mellitus model. Under the fractional ABC operator, Picard’s theorem is used to prove the existence and uniqueness of solutions. For the numerical approximation of solutions in the fractional-order diabetes mellitus model, this study used a specialized technique that combines the principles of fractional calculus and a two-step Lagrange polynomial interpolation. Finally, the obtained results are visually presented through graphical representations, serving as empirical evidence to support our theoretical findings. The numerical experiments showed that the proportion of patients with diabetes mellitus increased as the fractional dimension (𝜃) reduced. The combination of mathematical modelling, analysis, and numerical simulations provides insights into the dynamics of diabetes mellitus, offering valuable contributions to the understanding and management of this prevalent disease. Additionally, the proposed scheme can be enhanced by incorporating the ABC operator, allowing for the simulation of real-world dynamics and behavior in the coexistence of diabetes mellitus and tuberculosis.
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    On mathematical modelling of measles disease via collocation approach
    (2024-05) Ahmed, S; Jahan, S; Shah, K; Abdeljawad, T
    Measles, a highly contagious viral disease, spreads primarily through respiratory droplets and can result in severe complications, often proving fatal, especially in children. In this article, we propose an algorithm to solve a system of fractional nonlinear equations that model the measles dis ease. We employ a fractional approach by using the Caputo operator and validate the model’s by applying the Schauder and Banach fixed-point theory. The fractional derivatives, which constitute an essential part of the model can be treated precisely by using the Broyden and Haar wavelet collocation methods (HWCM). Furthermore, we evaluate the system’s stability by implementing the Ulam-Hyers approach. The model takes into account multiple factors that influence virus transmission, and the HWCM offers an effective and precise solution for understanding insights into transmission dynamics through the use of fractional derivatives. We present the graphical results, which offer a comprehensive and invaluable perspective on how various parameters and fractional orders influence the behaviours of these compartments within the model. The study emphasizes the importance of modern techniques in understanding measles outbreaks, suggesting the methodology’s applicability to various mathematical models. Simulations conducted by using MATLAB R2022a software demonstrate practical implemen tation, with the potential for extension to higher degrees with minor modifications. The simulation’s findings clearly show the efficiency of the proposed approach and its application to further extend the field of mathematical modelling for infectious illnesses.
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    Shifted fractional order Gegenbauer wavelets method for solving electrical circuits model of fractional order
    (2023-10) Yadav, P; Jahan, S; Nisar, K
    This study introduces a novel numerical approach based on shifted fractional-order Gegenbauer wavelets. The technique aims to provide approximate solutions for a certain class of extremely significant fractional models of electrical LC , RC , RL, and RLC circuits. We use a collocated approach to generate numerical solutions for these circuits model by making use of the beneficial characteristics of shifted fractional-order Gegenbauer polynomials (SFGBP). We include a parameter that characterizes the presence of fractional structures inside the models in order to retain the dimensional properties of the physical parameters in the electrical circuits. Several particular instances of the various source terms have also been examined. The main findings include the close resemblance of current fractional-order models to classical cases, variations in current with resistance values, and error reduction through additional series terms. Numerical simulations are shown through geometrical interpretation to illustrate the exactness and reliability of our technique.
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    Solving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics
    (2023-05) Yadav, P; Jahan, S; Nisar, K
    This study introduces a new fractional order Fibonacci wavelet technique proposed for solving the frac tional Bagley-Torvik equation (BTE), along with the block pulse functions. To convert the specified initial and boundary value problems into algebraic equations, the Riemann–Liouville (R-L) fractional integral operator is defined, and the operational matrices of fractional integrals (OMFI) are built. This numerical scheme’s performance is evaluated and examined on particular problems to show its proficiency and effectiveness, and other methods that are accessible in the current literature are compared. The numer ical results demonstrate that the approach produces extremely precise results and is computationally more decisive than previous methods.
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    Wavelets collocation method for singularly perturbed differential–difference equations arising in control system
    (2023-12) Ahmed, S; Jahan, S; Ansari, K; Shah, K
    In this paper, we present a wavelet collocation method for efficiently solving singularly perturbed differential–difference equations (SPDDEs) and one-parameter singularly perturbed differential equations (SPDEs) taking into account the singular perturbations inherent in control systems. These equations represent a class of mathematical models that exhibit a combination of differential and difference equations, making their analysis and solution challenging. The terms that include negative and positive shifts were approximated using Taylor series expansion. The main aim of this technique is to convert the problems by using operational matrices of integration of Haar wavelets into a system of algebraic equations that can be solved using Newton’s method. The adaptability and multi-resolution properties of wavelet functions offer the ability to capture system behavior across various scales, effectively handling singular perturbations present in the equations. Numerical experiments were conducted to showcase the effectiveness and accuracy of the wavelet collocation method, demonstrating its potential as a reliable tool for analyzing and solving SPDDEs in control system.