BITS Faculty Publications

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Start date: 2024Subject: search.filters.subject.Mathematics

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    Solving extended assignment problem using stochastic DEA approach
    (IEEE, 2025-04) Agarwal, Shivi; Mathur, Trilok
    The assignment model is a particular application of linear programming problems where tasks are assigned to agents with the goal of either maximization of profit or minimization of cost (in terms of both money and time) with provided deterministic data. But in real-life cases, more than one attribute may occur. Also, all these attributes need not be deterministic; some attributes may be stochastic in nature. The existing assignment model cannot handle these types of issues. To overcome these drawbacks, the study proposes the integrated extended assignment model with stochastic theory and the data envelopment analysis (DEA) technique. To illustrate the suggested concept, a numerical example is provided.
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    Fuzzy DEA model with exogenously fixed variables for ranking of renewable energy sources
    (Springer, 2025-09) Agarwal, Shivi; Mathur, Trilok
    As the global population grows, so does the demand for energy. India, with its fast growth, industrialization, and urbanization, is struggling to meet energy needs using traditional sources. To tackle energy shortages, pollution, and climate change, it’s important to find cost-effective and environment friendly alternatives. Renewable energy sources (RESs) offer a promising solution, making it important to prioritize them. India has strong potential in technologies like solar, geothermal, hydro, biomass, wave energy, and onshore and offshore wind energy. However, prioritizing these energy options involves considering many factors, often with conflicting priorities. This study proposed a fuzzy Data Envelopment Analysis (DEA) method to prioritize renewable energy sources in India, considering exogenously fixed variables that can’t be controlled, and handling undesirable variables. The proposed model ranks RESs effectively. It is revealed from results that Offshore wind energy is found to be the most efficient, followed by onshore wind and hydro energy, while geothermal energy ranks the lowest. The proposed methodology and findings can help developing nations and policymakers make better decisions when adopting renewable energy sources.
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    An optimal criteria selection in efficiency assessment through integration of dea with rough set theory
    (Springer, 2025-09) Agarwal, Shivi; Mathur, Trilok
    Data Envelopment Analysis (DEA) is a prominent nonparametric technique used to assess the efficiency of decision-making units (DMUs) by using multi criteria. However, traditional DEA models can be significantly impacted by criteria that do not contribute significantly to the efficiency analysis, thereby reducing accuracy and discriminatory power. Additionally, for DEA models to produce reliable results, the number of DMUs should be greater than the number of criteria included. This paper introduces a Rough Data Envelopment Analysis (RDEA) approach, which integrates Rough Set Theory (RST) with DEA to effectively handle this problem. RST is used by the RDEA framework to find and remove less contributing criteria from the input and output data in efficiency analysis. RST generates lower and upper approximations which helps in identifying criteria that are not significantly contributing to the efficiency analysis. Once these criteria have eliminated from the data set, the DEA models may be utilized to provide a more accurate and reliable efficiency evaluation of DMUs. This theoretical framework leverages the capabilities of RST to streamline input and output data, enhancing the effectiveness of DEA in evaluating efficiency. Also, a numerical example is provided to show implementation of this method.
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    Analyzing unemployment dynamics: a fractional-order mathematical model
    (Wiley, 2025-03) Mathur, Trilok
    The persistent rise in unemployment rates poses a significant threat to global economic stability. Addressing this challenge effectively requires a deeper understanding of workforce dynamics, particularly through the integration of an individual's employment history into analytical models. This research introduces a fractional mathematical model, leveraging the Caputo fractional derivative and three key variables: the number of skilled unemployed individuals, the number of employed individuals, and the number of available job vacancies. The model's well-posedness and global stability are rigorously established using fixed-point theory. Additionally, the basic reproduction number is analyzed to identify critical factors that facilitate the creation of new job opportunities. Real-world data from India are employed for MATLAB simulations, offering predictions of unemployment trends in the coming years. A graphical analysis highlights the impact of the COVID-19 pandemic on unemployment rates. The model's predictive accuracy is demonstrated through error analysis, showing that fractional-order forecasts achieve less than 5% error, outperforming integer-order models in capturing the nuances of unemployment dynamics. Sensitivity analysis reveals that the employment rate is the most influential parameter; a 40% increase in this rate could lead to 192,200 additional employed individuals. The fractional-order model further exhibits superior performance metrics, including lower root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) values, alongside a higher correlation coefficient ( ). These findings underscore the model's potential to enhance the understanding and mitigation of unemployment challenges.
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    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.
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    Short-term wind speed prediction with adaptive signal processing based hybrid statistical models
    (Springer, 2025-03) Pasari, Sumanta
    The inherent nonlinearity, intermittency, and chaotic nature of wind speed make accurate forecasting challenging. Traditional approaches like standalone time series models and frequency domain analysis struggle to capture these complex characteristics effectively. In light of this, the present study utilizes three self-adaptive signal processing methods, namely empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD), and variational mode decomposition (VMD) and combines with ARIMA or window-sliding ARIMA (WSARIMA) to develop six hybrid models, namely EMD–ARIMA, EEMD–ARIMA, VMD–ARIMA, EMD–WSARIMA, EEMD–WSARIMA, and VMD–WSARIMA. To illustrate the efficacy of the proposed hybrid models in daily wind speed prediction, four study sites from India with different climates are considered. Based on the analysis of 7 years (08-2015–03-2023) of wind speed data, it is found that: (i) the extracted components of VMD overcome the limitations of EMD and EEMD methods; (ii) the combination of VMD and WSARIMA outperforms any other comparative model, such as ARIMA, WSARIMA, EMD–ARIMA, EEMD–ARIMA, VMD–ARIMA, EMD–WSARIMA, or EEMD–WSARIMA; the VMD–WSARIMA model reduces RMSE by 70–80% compared to the conventional ARIMA model; (iii) finally, as a part of post-processing, the residual analysis of the best fit VMD–WSARIMA model shows desirable characteristics. Therefore, the present study strongly recommends to consider adaptive decomposition based hybrid models in wind speed forecasting at shorter time horizon.
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    Earthquake cycle progression in major city regions of Taiwan through nowcasting technique
    (Springer, 2025-05) Pasari, Sumanta
    The complex tectonic framework of Taiwan makes it susceptible to devastating earthquakes that originate on both mapped faults, and at times, on unmapped faults. The unmapped faults especially highlight the limitation of conventional fault–based hazard assessment methods, emphasizing the need for alternative approaches. In this context, we implement a surrogate area–based earthquake nowcasting technique to assess the seismic cycle progression in 10 densely populated cities across Taiwan. We utilize the notion of natural times, the inter–event counts of small earthquakes between successive large events, to calculate the Earthquake Potential Score (EPS) for each city region. To derive natural time statistics, we analyze eight reference probability models, including exponential distribution and its variants, exponentiated group of distributions, and heavy–tailed distributions. Statistical inference of 114 observed natural times shows that the exponentiated exponential distribution provides the best fit. As of April 24, 2025, the EPS values (%) for M 6.0 earthquakes in the 10 cities range from 53% to 69%, with the following values: Taipei (69%), Hsinchu (68%), Keelung (67%), Hualien (59%), Nantou (58%), Taitung (57%), Chiayi (56%), Pingtung (55%), Tainan (54%), and Kaohsiung (53%). These EPS values indicate the progression in current earthquake cycle toward a M 6.0 earthquake in the corresponding city region. Moreover, there is a consistency in the nowcast scores despite some variations in threshold magnitudes and city regions. The studied approach and results therein offer valuable insights to decision makers to enhance earthquake preparedness and risk management across Taiwan.
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    Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations
    (Elsevier, 2025-01) Santra, Sudarshan
    An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at . In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.
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    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.
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    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.