Department of Mathematics

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Author: search.filters.author.Agarwal, ShiviSubject: search.filters.subject.Fractional Differential Equation

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Now showing 1 - 6 of 6
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    Fractional boundary value problem in complex domain
    (Elsevier, 2023-10) Mathur, Trilok; Agarwal, Shivi
    Fractional calculus of complex order in complex domain has emerged as a brand-new area of study. Over the past few years, fractional boundary value problems (FBVP) in real variables have been extensively studied but there are few attempts on these types of problems in complex variables. In this study, the existence and uniqueness for the solutions of fractional differential equation (FDE) in complex domain with boundary conditions is examined. We established the existence of solutions using the Krasnoselskii fixed point theorem; however, the uniqueness result is proved by applying the Banach contraction principle. To explain our findings, an illustrative example is presented. The special cases of the derived findings are equivalent to the theorems that already exist.
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    Impact of social media on academics: a fractional order mathematical model
    (Taylor & Francis, 2023-12) Mathur, Trilok; Agarwal, Shivi
    Due to the COVID pandemic and lockdown, usage of social platforms increased for academic and non-academic purposes. As a result, students are at significant risk of developing social media addiction, so techniques to control social media addiction throughout society are required. There are several positive and negative ways in which social media affects the academic performance of a student. Most of the mathematical models exclude the past of an individual, which is critical for controlling social media consumption. Hence, this study offers a fractional-order mathematical model to analyze the impact of social media on academics. There are two equilibrium points for the proposed model: social web-free and endemic equilibrium. Based on an evaluation of the threshold value, the social web-free equilibrium point is globally asymptotically stable whenever the threshold value is less than one. Endemic equilibrium points exist when the threshold value is greater than 1. Additionally, numerical simulations have been performed to examine changes in population dynamics and validate analytical outcomes. In summary, the findings of this research reveal that social media addiction decreases as the order of the derivative decreases, demonstrating the high efficiency of a fractional-order model over an integer-order model.
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    Dynamics of Crime Transmission Using Fractional-Order Differential Equations
    (World Scientific, 2022) Mathur, Trilok; Agarwal, Shivi
    Due to the alarming rise in types of crime committed and the number of criminal activities across the world, there is a great need to amend the existing policies and models adopted by jurisdictional institutes. The majority of the mathematical models have not included the history of the crime committed by the individual, which is vital to control crime transmission in stipulated time. Further, due to various external factors and policies, a considerable number of criminals have not been imprisoned. To address the aforementioned issues prevailing in society, this research proposes a fractional-order crime transmission model by categorizing the existing population into four clusters. These clusters include law-abiding citizens, criminally active individuals who have not been imprisoned, prisoners, and prisoners who completed the prison tenure. The well-posedness and stability of the proposed fractional model are discussed in this work. Furthermore, the proposed model is extended to the delayed model by introducing the time-delay coefficient as time lag occurs between the individual’s offense and the judgment. The endemic equilibrium of the delayed model is locally asymptotically stable up to a certain extent, after which bifurcation occurs.
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    Analysis of illegal drug transmission model using fractional delay differential equations
    (AIMS Press, 2022) Agarwal, Shivi; Mathur, Trilok
    The global burden of illegal drug-related death and disability continues to be a public health threat in developed and developing countries. Hence, a fractional-order mathematical modeling approach is presented in this study to examine the consequences of illegal drug usage in the community. Based on epidemiological principles, the transmission mechanism is the social interaction between susceptible and illegal drug users. A pandemic threshold value () is provided for the illegal drug-using profession, which determines the stability of the model. The Lyapunov function is employed to determine the stability conditions of illegal drug addiction equilibrium point of society. Finally, the proposed model has been extended to include time lag in the delayed illegal drug transmission model. The characteristic equation of the endemic equilibrium establishes a set of appropriate conditions for ensuring local stability and the development of a Hopf bifurcation of the model. Finally, numerical simulations are performed to support the analytical results.
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    Fractional boundary value problem in complex domain
    (Elsevier, 2023-10) Agarwal, Shivi; Mathur, Trilok
    Fractional calculus of complex order in complex domain has emerged as a brand-new area of study. Over the past few years, fractional boundary value problems (FBVP) in real variables have been extensively studied but there are few attempts on these types of problems in complex variables. In this study, the existence and uniqueness for the solutions of fractional differential equation (FDE) in complex domain with boundary conditions is examined. We established the existence of solutions using the Krasnoselskii fixed point theorem; however, the uniqueness result is proved by applying the Banach contraction principle. To explain our findings, an illustrative example is presented. The special cases of the derived findings are equivalent to the theorems that already exist.
  • No Thumbnail Available
    Item
    Dynamics of Crime Transmission Using Fractional Order Differential Equations
    (World Scientific, 2022) Agarwal, Shivi; Mathur, Trilok
    Due to the alarming rise in types of crime committed and the number of criminal activities across the world, there is a great need to amend the existing policies and models adopted by jurisdictional institutes. The majority of the mathematical models have not included the history of the crime committed by the individual, which is vital to control crime transmission in stipulated time. Further, due to various external factors and policies, a considerable number of criminals have not been imprisoned. To address the aforementioned issues prevailing in society, this research proposes a fractional-order crime transmission model by categorizing the existing population into four clusters. These clusters include law-abiding citizens, criminally active individuals who have not been imprisoned, prisoners, and prisoners who completed the prison tenure. The well-posedness and stability of the proposed fractional model are discussed in this work. Furthermore, the proposed model is extended to the delayed model by introducing the time-delay coefficient as time lag occurs between the individual’s offense and the judgment. The endemic equilibrium of the delayed model is locally asymptotically stable up to a certain extent, after which bifurcation occurs.