1,721,122 research outputs found
A new fractional-order model for defining the dynamics of ending student strikes at a university
No full textKumar, Pushpendra/0000-0002-7755-2837Nowadays, different real-life phenomena are being modelled using fractional-order operators. In this paper, a Caputo-type fractional-order mathematical model is proposed for defining the dynamics of ending student strikes at a university by taking per-year constant admissions. We analyse the possible strategies to control the strikes on the university campus. We prove the existence of a unique global solution for the given fractional-order model using a new characteristic of the well-known Mittag-Leffler function and fixed-point theory. We derive the numerical solution of the proposed model via the Haar wavelet method, which is one of the efficient numerical algorithms. A number of plots are performed, taking different cases, for a good understanding of the proposed problem. The aim of this study is to understand how fractional derivatives are useful to capture memory effects in such problems. All results are given with supporting arguments.Emerging Sources Citation Inde
A high-order space-time spectral method for the distributed-order time-fractional telegraph equation
No full textKumar, Pushpendra/0000-0002-7755-2837In this paper, a high-order and fast numerical method based on the space-time spectral scheme is obtained for solving the space-time fractional telegraph equation. In the proposed method, for discretization of temporal and spatial variables, we consider two cases. We use the Legendre functions for discretization in time. To obtain the full discrete numerical approach, we use a Fourier-like orthogonal function. The convergence and stability analysis for the presented numerical approach is studied and analyzed. Some numerical examples are given for the effectiveness of the numerical approach
A novel numerical method to solve fractional ordinary differential equations with proportional Caputo derivatives
No full textKumar, Pushpendra/0000-0002-7755-2837In this paper, we develop a novel numerical scheme, namely 'NPCM-PCDE,' to integrate fractional ordinary differential equations with proportional Caputo derivatives of the type (pc)D(alpha)u(t) = f(1)(t, u(t)), t >= 0, 0 < alpha < 1 involving a non-linear operator f(1). A new method is developed using a natural discretization of the proportional Caputo derivative and the decomposition method to decompose the non-linear operator f(1). The error and stability analyses for the proposed method are provided. Some illustrated examples are given to compare the solution curves graphically with the exact solution and to prove the utility and efficiency of the method. The proposed NPCM-PCDE is found to be efficient, easy to implement, convergent, and stable.Science Citation Index Expande
A Fractional-Order Improved Fitzhugh-Nagumo Neuron Model
No full textKumar, Pushpendra/0000-0002-7755-2837We propose a fractional-order improved FitzHugh-Nagumo (FHN) neuron model in terms of a generalized Caputo fractional derivative. Following the existence of a unique solution for the proposed model, we derive the numerical solution using a recently proposed L1 predictor-corrector method. The given method is based on the L1-type discretization algorithm and the spline interpolation scheme. We perform the error and stability analyses for the given method. We perform graphical simulations demonstrating that the proposed FHN neuron model generates rich electrical activities of periodic spiking patterns, chaotic patterns, and quasi-periodic patterns. The motivation behind proposing a fractional-order improved FHN neuron model is that such a system can provide a more nuanced description of the process with better understanding and simulation of the neuronal responses by incorporating memory effects and non-local dynamics, which are inherent to many biological systems.Science Citation Index Expande
A linear B-spline interpolation/Galerkin finite element method for the two-dimensional Riesz space distributed-order diffusion-wave equation with error analysis
No full textKumar, Pushpendra/0000-0002-7755-2837This paper focuses on the distributed-order time-fractional diffusion-wave equations with the Riesz space fractional derivatives. A combined method based on the midpoint quadrature rule, linear B-spline interpolation, and the Galerkin finite element method is proposed to obtain the approximate solution. Two steps are used to calculate the approximate solution to this type of equation. The first step approximates the temporal direction by combining a midpoint quadrature rule and linear B-spline interpolation. In the second step, a Galerkin finite element method in the space direction is applied to compute a full-discrete method. Furthermore, the error estimate has been displayed to demonstrate unconditional stability and convergence. Finally, two numerical examples are reported to show the simplicity and efficiency of the proposed method
A variable-order fractional mathematical model for the strategy to combat the atmospheric level of carbon dioxide
No full textKumar, Pushpendra/0000-0002-7755-2837In this article, we define a nonlinear model for exploring the strategy of combating the atmospheric level of carbon dioxide (CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}) considering development activities in terms of variable-order Liouville-Caputo fractional derivatives. There are two types of variable-order Liouville-Caputo fractional derivatives used to derive the proposed model. We prove the existence and uniqueness of the solution for the given model using fixed-point theory. The numerical solution is derived by using a recently proposed predictor-corrector scheme. We perform several graphical simulations to describe the outcomes of the given model. The outputs performed at various fractional-order values provide novel findings to understand how to combat atmospheric CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. A novel variable-order fractional model that captures memory effects in the proposed dynamics, along with a recent numerical methodology, are the key features of this study. The simulation analysis shows that the leafy tree plantation on the excess land will be efficient against atmospheric CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}
A comparative study for mathematical modelling of the contest of CAR-T and tumour cells in solid cancers using fractional- and integer-order derivatives
No full textKumar, Pushpendra/0000-0002-7755-2837Cancer is a disease resulting from the fractious growth and division of abnormal cells and has gotten consistent and dedicated attention from scientists across multiple disciplines. To date, several mathematical studies have been done to study its dynamics. In this paper, we study two fractional-order mathematical models that describe the competition between CAR-T and tumour cells in terms of their immune-suppressive efficiency. We explore whether the use of a large number of CAR-T cells encountering the antigens of solid tumours could beat the immune-suppressive force of cancer. Our results are obtained through the implementation of the well-known Caputo fractional derivative as well as the Adams-Bashforth-Moulton scheme. The main aim of this study is to compare the results we obtained through the use of fractional derivatives with previously published integer-order simulations. Of interest are the instances when the results obtained via the fractional-order derivative contradict the solutions provided by the integer-order models.Emerging Sources Citation Inde
A novel L1-Predictor-Corrector method for the numerical solution of the generalized-Caputo type fractional differential equations
No full textS M, Sivalingam/0000-0003-0818-9007; Kumar, Pushpendra/0000-0002-7755-2837This paper proposes a novel L1 -based predictor-corrector method for the fractional differential equations involving generalized-Caputo type derivative. A decomposition scheme is used to obtain the three-point predictor-corrector formula. The error and stability of the proposed method are given in detail. A computer virus and a five-dimensional Hopfield neural network models are solved using the proposed approach.UGC NFOBC Ph.D. Fellowship, India [[202122]-TN13000109]; National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India [02011/18/2023 NBHM (R.P)/ RD II/5952]The first author received the financial support of UGC NFOBC Ph.D. Fellowship, India (Ref. [202122]-TN13000109). V. Govindaraj would like to thank the National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India, for funding the research project (File No. 02011/18/2023 NBHM (R.P)/ R & D II/5952) All authors approved the version of the manuscript to be published
A Study on the Transmission Dynamics of the Omicron Variant of COVID-19 Using Nonlinear Mathematical Models
No full textDickson, S/0000-0002-4805-3502; S, Padmasekaran/0000-0002-8873-7853; S, Dickson/0000-0002-4805-3502; Kumar, Pushpendra/0000-0002-7755-2837This research examines the transmission dynamics of the Omicron variant of COVID-19 using SEIQIcRVW and SQIRV models, considering the delay in converting susceptible individuals into infected ones. The significant delays eventually resulted in the pandemic's containment. To ensure the safety of the host population, this concept integrates quarantine and the COVID-19 vaccine. We investigate the stability of the proposed models. The fundamental reproduction number influences stability conditions. According to our findings, asymptomatic cases considerably impact the prevalence of Omicron infection in the community. The real data of the Omicron variant from Chennai, Tamil Nadu, India, is used to validate the outputs.Prince Sattam bin Abdulaziz University [PSAU/2023/R/1444]; Periyar University, Salem [PU/AD-3/URF/21F37237/2021]; DST [SR/FST/MSI-115/2016]This study is supported via funding from Prince Sattam bin Abdulaziz University Project Number (PSAU/2023/R/1444) . The first author is partially supported by the University Research Fellowship (PU/AD-3/URF/21F37237/2021 dated 09.11.2021) of Periyar University, Salem. The second author is supported by the fund for improvement of Science and Technology Infrastructure (FIST) of DST (SR/FST/MSI-115/2016)
A Chebyshev neural network-based numerical scheme to solve distributed-order fractional differential equations
No full textS M, SIVALINGAM/0000-0003-0818-9007; Kumar, Pushpendra/0000-0002-7755-2837This study aims to develop a first-order Chebyshev neural network-based technique for solving ordinary and partial distributed-order fractional differential equations. The neural network is used as a trial solution to construct the loss function. The loss function is utilized to train the neural network via an extreme learning machine and obtain the solution. The novelty of this work is developing and implementing a neural network-based framework for distributed-order fractional differential equations via an extreme learning machine. The proposed method is validated on several test problems. The error metrics utilized in the study include the absolute error and the L-2 error. A comparison with other previously available approaches is presented. Also, we provide the computation time of the method.National Board for Higher Mathematics, NBHM; University Grants Commission, UGC, (Ref.202122-TN13000109); Department of Atomic Energy, Government of India, DAE, (02011/18/2023 NBHM (R.P)/ R&D II/5952)UGCNFOBC Ph.D. Fellowship [202122-TN13000109]; National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India [02011/18/2023NBHM (R.P)/RDII/5952]S.M. Sivalingam received the financial support of UGCNFOBC Ph.D. Fellowship (Ref. 202122-TN13000109). V. Govindaraj would like to thank the National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India,for funding the research project(FileNo. 02011/18/2023NBHM (R.P)/R&DII/5952).Science Citation Index Expande
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