Browsing by Author "Shah, K"

Now showing 1 - 5 of 5
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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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    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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    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.