Department of Mathematics

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An Adaptive Fractional Guard Channel Based CAC Scheme for Heterogeneous Traffic in Wireless Cellular Networks
(IEEE, 2019-03) Kulshrestha, Rakhee
In this paper, we propose an adaptive fractional guard channel based call admission scheme for heterogeneous traffic. We have developed a generalised model for investigating the quality of service (QoS) and the effect of mobility of subscribers through simulation and also analysed the results. We evaluate the proposed scheme by computing the new call blocking probability and handoff blocking probability. Moreover, our scheme being relevant to the choice of the wireless medium is extensible to LTE Networks that utilise MIMO wireless technology.
An adaptive mesh based computational approach to the option price and their greeks in time fractional black–scholes framework
(Springer, 2025-02) Santra, Sudarshan
This article deals with an efficient numerical method for solving the time fractional Black–Scholes equation governing the European option pricing model and their Greeks. The Caputo fractional derivative involved in time results a mild singularity and forms a layer near the initial time. For discretization, a graded mesh is introduced in the temporal direction, and in space, a uniform mesh is constructed. The L1 scheme is used to discretize the time fractional derivative, while the second-order finite difference approximations are used for the spatial derivatives. The proposed approach effectively resolves the initial layer with a graded mesh in time, achieving higher temporal accuracy of . It provides valuable insights into the error bounds through stability and convergence analysis and captures the behavior of option Greeks, highlighting the impact of fractional derivatives. Compared to uniform mesh-based methods and other existing approaches, it demonstrates superior accuracy and efficiency for time-fractional Black–Scholes equations, ensuring space-time higher-order accuracy. Some numerical results on the solution and their Greeks prove the theoretical analysis. The proposed scheme is applied to European option pricing models governed by the time fractional Black–Scholes equation to examine the impact of the fractional derivative on option pricing.
Admission Control Policy of Maintenance for Unreliable Server Machining System with Working Vacation
(Springer, 2017-03) Shekhar, Chandra
This investigation is concerned with the performance modeling of machining system operating under the admission control F-policy and server working vacation policy. The repair of failed machines is provided by an unreliable server, who also renders the service with the slower rate rather than completely terminating the service during the vacation period. The failed machines are allowed to enter the system till the system capacity (K) is full; then after failed machines are not allowed to join the system until the system size again decreases to the prespecified threshold level ‘F’. At that instant, the server takes start-up time in order to start allowing the failed machines to enter into the system for the repair job. Numerical method based on successive over-relaxation is applied to obtain the steady-state probabilities and various performance indices including the cost function. The numerical simulation is performed to explore the sensitivity of the system indices with respect to various parameters. Quasi-Newton method and direct search method are used to determine the optimal service rate and threshold parameter.
Adomian decomposition and homotopy perturbation method for the solution of time fractional partial integro-differential equations
(Springer, 2021-07) Santra, Sudarshan
This article deals with two different methods to solve a time fractional partial integro-differential equation. The fractional derivatives are defined here in Caputo sense. The model problem is solved using the Adomian decomposition method and homotopy perturbation method. Moreover, this paper proves the convergence analysis of the solution based on the present methods. Numerical evidences are illustrated in support of the theoretical analysis.
Advancements of solar energy research in the context of SDG-7 attainment: a bibliometric analysis using spar-4-slr protocol
(IEEE, 2025-05) Agarwal, Shivi; Mathur, Trilok
Renewable energy sources, free of environmental risks, are vital for achieving net-zero CO2 emissions and addressing climate change to meet Sustainable Development Goals. This study explores the evolution of solar energy research using bibliographic coupling and keyword co-occurrence analysis of 6,460 articles from 1988 to 2024. The findings reveal a significant increase in solar power-related publications, with China leading in research output, followed by the United States and India. Top journals include Renewable Energy and Energies, with a growing focus on Energy and Engineering. This analysis serves as a vital reference for solar energy researchers and professionals.
Advancing organic photovoltaic cells for a sustainable future: The role of artificial intelligence (AI) and deep learning (DL) in enhancing performance and innovation
(Elsevier, 2025-05) Sharma, Bhupendra Kumar
The convergence of Organic Photovoltaic (OPV) technology and artificial intelligence (AI) is examined in this review as a promising approach to advancing sustainable energy solutions. Recognized for their lightweight, flexible, and cost-effective properties, OPVs are highlighted as viable alternatives within renewable energy applications, particularly suited for integration in building infrastructure and portable energy sources. A discussion of OPV mechanisms and structures, such as single-layer, bilayer, and bulk heterojunction cells, is provided to outline the unique efficiencies and challenges each architecture presents. AI, especially through machine learning (ML) and deep learning (DL) models, is shown to transform OPV research, enhancing both material discovery and device optimization. Through AI-driven processes, rapid predictions of power conversion efficiency (PCE), material selection automation, and high-throughput screening are achieved, effectively minimizing experimental time and cost. Recent developments in AI applications, including convolutional neural networks (CNNs) and Bayesian optimization, are reviewed for their contributions to improving OPV performance, stability, and scalability. Case studies are included to demonstrate AI’s impact in areas such as predictive modeling, real-time monitoring, and optimization of device architecture, all of which contribute to efficiency gains and improved material durability. Challenges, however, are noted, with data quality issues, the need for interdisciplinary collaboration, and gaps in AI-aided material innovation identified as key areas for ongoing development. This review highlights how the intersection of AI and OPV technology not only accelerates the path toward efficient, scalable renewable energy but also underscores the importance of interdisciplinary research in advancing sustainable, high-performance photovoltaic solutions.
Affine Near-Semirings Over Brandt Semigroups
(Taylor & Francis, 2014) Kumar, Jitender
In order to study the structure of A +(B n )—the affine near-semiring over a Brandt semigroup—this work completely characterizes the Green's classes of its semigroup reducts. In this connection, this work classifies the elements of A +(B n ) and reports the size of A +(B n ). Further, idempotents and regular elements of the semigroup reducts of A +(B n ) have also been characterized and studied some relevant semigroups in A +(B n ).
Air Quality Prediction Using the Fractional Gradient-Based Recurrent Neural Network
(Hindawi Publishing Corporation, 2022) Agarwal, Shivi; Mathur, Trilok
In this study, the air quality index (AQI) of Indian cities of different tiers is predicted by using the vanilla recurrent neural network (RNN). AQI is used to measure the air quality of any region which is calculated on the basis of the concentration of ground-level ozone, particle pollution, carbon monoxide, and sulphur dioxide in air. Thus, the present air quality of an area is dependent on current weather conditions, vehicle traffic in that area, or anything that increases air pollution. Also, the current air quality is dependent on the climate conditions and industrialization in that area. Thus, the AQI is history-dependent. To capture this dependency, the memory property of fractional derivatives is exploited in this algorithm and the fractional gradient descent algorithm involving Caputo’s derivative has been used in the backpropagation algorithm for training of the RNN. Due to the availability of a large amount of data and high computation support, deep neural networks are capable of giving state-of-the-art results in the time series prediction. But, in this study, the basic vanilla RNN has been chosen to check the effectiveness of fractional derivatives. The AQI and gases affecting AQI prediction results for different cities show that the proposed algorithm leads to higher accuracy. It has been observed that the results of the vanilla RNN with fractional derivatives are comparable to long short-term memory (LSTM). 1. Introduction
Algebraic methods in difference sets and bent functions
(JACODESMATH, 2021) Keskar, Pradipkumar H.
We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the $\mathcal{C}$-condition introduced by him doesn't hold.
Algorithm for constructing an optimally connected rectangular floor plan
(Elsevier, 2014-09) Shekhawat, Krishnendra
In most applications, such as urbanism and architecture, randomly utilizing given spaces is certainly not favorable. This study proposes an explicit algorithm for utilizing the given spaces inside a rectangle with satisfactory results. In the literature, connectivity is not considered as a criterion for floor plan design, but it is deemed essential in architecture. For example, dining rooms are preferably connected to kitchens, toilets should be connected to many rooms, and each bedroom should be separated from the other rooms. This paper describes adjacency among spaces and proves that the obtained rectangular floor plan is one of the best ones in terms of connectivity. An architectural and mathematical object called extra spaces is introduced by the proposed algorithm and is subsequently examined in this work.
An algorithm for customizing slicing floor plan design
(Springer, 2023) Shekhawat, Krishnendra
This paper proposes a linear time algorithm for the customization of a slicing floor plan design, which can be done by customizing its modules in the following two ways: —-by modifying the aspect ratio or by modifying either its width or height while retaining its area,-by modifying its area while keeping either of its aspect ratio or initial width or height. Both of the aforementioned approaches demonstrate that a slicing floor plan can be generated for any aspect ratio and area while preserving the module adjacencies of the original floor plan. A demonstration has been provided for a devised prototype to validate the viability of the aforementioned approaches
Almost ϕ-integrally closed rings
(Taylor & Francis, 2023-09) Kumar, Rahul
Let R be a commutative ring with unity. The notion of almost 𝜙-integrally closed ring is introduced which generalizes the concept of almost integrally closed domain. Let ℋ be the set of all rings such that Nil⁡(𝑅) is a divided prime ideal of R and 𝜙:𝑇⁡(𝑅)→𝑅Nil⁡(𝑅) is a ring homomorphism defined as 𝜙⁡(𝑥)=𝑥 for all 𝑥∈𝑇⁡(𝑅). A ring 𝑅∈ℋ is said to be an almost 𝜙-integrally closed ring if 𝜙⁡(𝑅) is integrally closed in 𝜙⁡(𝑅)𝜙⁡(𝔭) for each nonnil prime ideal 𝔭 of R. Using the idealization theory of Nagata, examples are also given to strengthen the concept.
An alternative finite difference weno-like scheme with physical constraint preservation for divergence-preserving hyperbolic systems
(2025-06) Bhoriya, Deepak
Alternative finite difference Weighted Essentially Non-Oscillatory (AFD-WENO) schemes allow us to very efficiently update hyperbolic systems even in complex geometries. Recent innovations in AFD-WENO methods allow us to treat hyperbolic system with non-conservative products almost as efficiently as conservation laws. However, some PDE systems,like computational electrodynamics (CED) and magnetohydrodynamics (MHD) and relativistic magnetohydrodynamics (RMHD), have involution constraints that require divergence-free or divergence-preserving evolution of vector fields. In such situations, a Yee-style collocation of variables proves indispensable; and that collocation is retained in this work. In previous works, only higher order finite volume discretization of such involution constrained systems was possible. In this work, we show that substantially more efficient AFD-WENO methods have been extended to encompass divergence-preserving hyperbolic PDEs.
Analysis of a chronological age-structured epidemic model with a pair of optimal treatment controls
(IOP, 2024-11) Das, Dhiraj Kumar
Emerging infectious diseases are one of the core concerns in epidemiology, and they often lead to a global public health emergency and consequently affect socio-economic sectors. The risk of contracting an infection varies with different age groups in a population. In this investigation, we present a continuous age-structured epidemic model that incorporates a chronological age-dependent bilinear disease incidence rate. We establish the model's feasibility from the population perspective and derive the basic reproduction number, . The analysis reveals that the model consistently exhibits a disease-free equilibrium, and a unique endemic equilibrium emerges whenever . Furthermore, the value of determines the stability of the age-structured model. We further formulate an optimal control problem by introducing a pair of age-dependent control variables, namely, (i) pre-cautions or medical care to the latently infected individuals and (ii) treatments or hospitalization of the infected individuals. We aim to minimize the cost of implementing these two controls so that the severity of an epidemic can be mitigated. We derived the adjoint equations to the age-dependent optimal control problem using Gateaux derivatives, then proved the existence and uniqueness of optimal control solution pair using Ekeland's principle. Finally, numerical simulations are conducted to verify the analysis and visualize the solution profiles of the model. Our observations suggest that while both control measures are effective in reducing the impact of the disease, taking precautions proves to be significantly more effective in mitigating the spread of the epidemic.
Analysis of a finite difference method based on L1 discretization for solving multi-term fractional differential equation involving weak singularity
(Wiley, 2022-03) Santra, Sudarshan
In this article, we consider a multi-term fractional initial value problem which has a weak singularity at the initial time . The fractional derivatives are defined in Caputo sense. Due to such singular behavior, an initial layer occurs near which is sharper for small values of γ1 where γ1 is the highest order among all fractional differential operators. In addition, the analytical properties of the solution are provided. The classical L1 scheme is introduced on a uniform mesh to approximate the fractional derivatives. The error analysis is carried out, and it is shown that the numerical solution converges to the exact solution. Further analysis proves that the scheme is of order over the entire region, but it is of order O(τ) on any subdomain away from the origin. τ denotes the mesh parameter. To show the efficiency of the proposed scheme, this method is tested on several model problems, and the results are in agreement with the theoretical findings.
Analysis of a higher-order scheme for multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels
(Springer, 2024-09) Santra, Sudarshan
This work is focused on developing a hybrid numerical method that combines a higher-order finite difference method and multi-dimensional Hermite wavelets to address two-dimensional multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels having bounded and unbounded time derivatives at the initial time . Specifically, the multi-term fractional operators are discretized using a higher-order approximation designed by employing different interpolation schemes based on linear, quadratic, and cubic interpolation leading to accuracy on a suitably chosen nonuniform mesh and accuracy on a uniformly distributed mesh. The weakly singular integral operators are approximated by a modified numerical quadrature, which is a combination of the composite trapezoidal approximation and the midpoint rule. The effects of the exponents of the weakly singular kernels over fractional orders are analyzed in terms of accuracy over uniform and nonuniform meshes for the solution having both bounded and unbounded time derivatives. The stability of the proposed semi-discrete scheme is derived based on -norm for uniformly distributed temporal mesh. Further, we employ the uniformly distributed collocation points in spatial directions to estimate the tensor-based wavelet coefficients. Moreover, the convergence analysis of the fully discrete scheme is carried out based on -norm leading to accuracy on a uniform mesh. It also highlights the higher-order accuracy over nonuniform mesh. Additionally, we discuss the convergence analysis of the proposed scheme in the context of the multi-term time-fractional diffusion equations involving time singularity demonstrating a accuracy on a nonuniform mesh with suitably chosen grading parameter. Note that the scheme reduces to accuracy on a uniform mesh. Several tests are performed on numerous examples in - and -norm to show the efficiency of the proposed method. Further, the solutions’ nature and accuracy in terms of absolute point-wise error are illustrated through several isosurface plots for different regularities of the exact solution. These experiments confirm the theoretical accuracy and guarantee the convergence of approximations to the functions having time singularity, and the higher-order accuracy for a suitably chosen nonuniform mesh.
Analysis of a prion proliferation model with polymer coagulation in the presence of chaperone
(Wiley, 2023-03) Kumar, Rajesh
In the present work, a mathematical model which consists of a nonlinear partial integro-differential equation coupled with two ordinary differential equations (ODEs) is analyzed. This model describes the relation between infectious, noninfectious prion proteins, and chaperone. The well-posedness of the system is proved for bounded kernels by using evolution operator theory in the state space . The existence of a global weak solution for unbounded kernels is also discussed by a weak compactness argument. In addition, we investigated the stability analysis results theoretically and effect of chaperone on prion proliferation numerically.
Analysis of Arrhenius activation energy on magnetohydrodynamic gyrotactic microorganism flow through porous medium over an inclined stretching sheet with thermophoresis and Brownian motion
(Sage, 2022-10) Sharma, Bhupendra Kumar
This paper aims to examine the combined effects of Arrhenius activation and microorganisms on unsteady flow through a porous medium with thermophoresis and Brownian motion over an inclined stretching sheet. The governing partial differential equations are transformed into a set of non-linear ordinary differential equations using similarity analysis. The resultant non-linear coupled ordinary differential equations are solved numerically using the boundary value problem solver in MATLAB. The effects of the physical parameter such as magnetic field parameter (M), thermal radiation parameter (R), permeability parameter (K), Eckert number (Ec), thermophoresis parameter (Nt) and Brownian motion parameter (Nb) on the velocity, temperature, concentration profiles, skin friction coefficient, Nusselt number, and the local Sherwood number are presented and analysed graphically. The comparison has been made with previously published work, and there is a good agreement. These results may be helpful in geothermal engineering, energy conversation and disposal of nuclear waste material. Furthermore, scientists can employ this technique in medical fields such as gene therapy and the synthesis of drug delivery systems.
Analysis of G–queue with unreliable server
(Springer, 2012-12) Kulshrestha, Rakhee
This paper presents a model for a discrete time single server G-queue with two types of independent arrivals, namely positive and negative. The arrival of negative customer to queueing system removes one customer from the head of the queue (RCH), which causes server breakdown. The repair of the server is non-instantaneous and after repair server is assumed as good as new. The interarrival times of both positive and negative customers and the repair times of the server are geometrically distributed. We analyse the queueing system by using Matrix Geometric method. The expressions for various performance measures such as mean queue length, throughput, delay, etc. are derived and calculated numerically.
Analysis of Hybrid MCDM Methods for the Performance Assessment and Ranking Public Transport Sector: A Case Study
(MDPI, 2022-09) Agarwal, Shivi; Mathur, Trilok
The quality of the public transport sector affects the economy and the daily livelihoods of passengers. One of the most important objectives of policymakers is to choose the influencing criteria for performance evaluations. A variety of factors are crucial for raising the standards of public transportation services. In this investigation, we used a decision-based model with uncertainty in order to identify significant criteria in the public transport sector. We also performed a comparative analysis to rank the Rajasthan State Road Transport Corporation (RSRTC) bus depots based on their performance using hybrid multi-criteria decision-making (MCDM) techniques such as TOPSIS, VIKOR, and ELECTRE. To handle judgement ambiguities, in this work we incorporated the Delphi method (DM) and the analytic hierarchy process (AHP), along with fuzzy set theory. The fuzzy Delphi method was used to filter the important criteria. Using a fuzzy AHP approach, the screening criterion weights and rankings were determined. Furthermore, the bus depots were ranked using TOPSIS, VIKOR, and ELECTRE. Our findings can be applied in assisting policy-managers in formulating appropriate policies targeted at improving the overall health and competitiveness of bus depots using significant criteria and associated key indicators. In this study we investigated performance measures and proposed recommendations for the sustainable development of transportation in India.