Department of Mathematics

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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.
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
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.
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.
Analytical solutions of some fractional diffusion boundary value problems
(Elsevier, 2022-11) Mathur, Trilok; Agarwal, Shivi
Fractional Order Systems and Applications in Engineering presents the use of fractional calculus (calculus of non-integer order) in the description and modelling of systems and in a range of control design and practical applications. The book covers the fundamentals of fractional calculus together with some analytical and numerical techniques, and provides MATLAB® codes for the simulation of fractional-order control (FOC) systems. The use of fractional calculus can improve and generalize well-established control methods and strategies. Many different FOC schemes are presented for control and dynamic systems problems. These extend to the challenging control engineering design problems of robust and nonlinear control. Practical material relating to a wide variety of applications including, among others, mechatronics, civil engineering, irrigation and water management, and biological systems is also provided. All the control schemes and applications are presented with either system simulation results or real experimental results, or both. Fractional Order Systems and Applications in Engineering introduces readers to the essentials of FOC and imbues them with a basic understanding of FOC concepts and methods. With this knowledge readers can extend their use of FOC in other industrial system applications, thereby expanding their range of disciplines by exploiting this versatile new set of control techniques.
Applications of fractional calculus in computer vision: A survey
(Elsevier, 2022-06) Agarwal, Shivi; Mathur, Trilok
Fractional calculus is an abstract idea exploring interpretations of differentiation having non-integer order. For a very long time, it was considered as a topic of mere theoretical interest. However, the introduction of several useful definitions of fractional derivatives has extended its domain to applications. Supported by computational power and algorithmic representations, fractional calculus has emerged as a multifarious domain. It has been found that the fractional derivatives are capable of incorporating memory into the system and thus suitable to improve the performance of locality-aware tasks such as image processing and computer vision in general. This article presents an extensive survey of fractional-order derivative-based techniques that are used in computer vision. It briefly introduces the basics and presents applications of the fractional calculus in six different domains viz. edge detection, optical flow, image segmentation, image de-noising, image recognition, and object detection. The fractional derivatives ensure noise resilience and can preserve both high and low-frequency components of an image. The relative similarity of neighboring pixels can get affected by an error, noise, or non–homogeneous illumination in an image. In that case, the fractional differentiation can model special similarities and help compensate for the issue suitably. The fractional derivatives can be evaluated for discontinuous functions, which help estimate discontinuous optical flow. The order of the differentiation also provides an additional degree of freedom in the optimization process. This study shows the successful implementations of fractional calculus in computer vision and contributes to bringing out challenges and future scopes.
Assessing the Radial Efficiency Performance of Bus Transport Sector Using Data Envelopment Analysis
(CRC Press, 2021) Mathur, Trilok; Agarwal, Shivi
This study proposed an application of data envelopment analysis (DEA) to estimate the overall technical efficiency (OTE) for 52 bus depots of Rajasthan State Road Transport Corporation (RSRTC) for the financial year 2016–2017. The proposed study measures the efficiency and thoroughly explores the scope for optimal utilization of the input resources owned by depots of the RSRTC. The new slack model (NSM) of DEA is used as it calculates slacks for input and output variables. Also, this model satisfies the radial properties, unit invariance, and translation invariance. An input-oriented NSM model explicitly reduces input quantities (X) proportionally and renders no change in output level (Y). Four inputs have been considered, namely the number of buses, number of employees, total fuel consumption, and number of routes, and a single output, passenger kilometers occupied. The empirical results indicated that only 4 out of 52 depots are on the efficient frontier. All other depots possess the OTE value in the range of 0.7 to 1 relative to efficient depots. This study enables policy-makers to evaluate inputs for consistent output up to the optimum level and improve the performance of the inefficient depots.
Assessment of offshore wind farm site suitability in India using GIS and MCDM methods
(Springer, 2025-03) Agarwal, Shivi; Mathur, Trilok
The growing demand for sustainable energy sources has prompted investigation of offshore wind farms (OfWF) as viable options. This study proposes an innovative framework combining GIS with MCDM methods to identify highly suitable sites and efficiently prioritize alternative sites for OfWF, specifically focusing on regions viable for fixed foundations within the Indian EEZ. It takes into account technical, economic, and socio-environmental criteria to identify the optimal sites for OfWF. Two types of criteria are involved: exclusion criteria, refining study area by excluding areas unsuitable for renewable energy projects, and evaluation criteria, influencing the viability of specific sites in line with defined objective. The methodology involves screening criteria from existing literature, preprocessing data through GIS, and calculating criteria weights with AHP and CRITIC methods. This is followed by ranking potential sites using TOPSIS and VIKOR, offering structured decision support. The final phase of the analysis involved evaluating the consistency between individual site rankings using the Spearman rank correlation, ensuring reliability in the ranking outcomes across selected methods. The findings suggest the top fifteen sites for installing wind farms, ranging in offshore regions from Pallam town in Kanyakumari to Kooduthalai in Tirunelveli district, Tamil Nadu. This study’s insights on offshore renewable energy site selection in India are invaluable for policymakers, investors and sustainability initiatives, offering a roadmap for a cleaner and more sustainable energy future worldwide.
Assessment of working capital management efficiency – a two-stage slack-based measure of data envelopment analysis
(Emerald, 2024-06) Chadha, Saurabh; Agarwal, Shivi
This study aims to assess the efficiency of managing working capital in 1,388 Indian manufacturing firms from 2008 to 2019 and investigate the effects of firm-specific and macro-level determinants on working capital management (WCM) efficiency.
A Benchmarking Approach with Missing Values Using Data Envelopment Analysis for Non-Symmetrical Fuzzy Data
(Solid State Technology, 2020) Agarwal, Shivi; Mathur, Trilok
This study proposes a benchmarking approach where the missing data point denoted by fuzzy numbers and aims to outstretch the crisp Data Envelopment Analysis (DEA) model in a fuzzy system. The fuzzy DEA integrated model is capable for benchmarking of homogeneous decision making units (DMUs) with the missing data as non-symmetrical fuzzy data in terms of fuzzy efficiency. A numerical illustration is presented as a case study to explain the integrated fuzzy DEA model.
Complex order fractional differential equation in complex domain with mixed boundary condition
(Elsevier, 2024-08) Agarwal, Shivi; Mathur, Trilok
Fractional calculus of complex orders in the complex domain is a rapidly growing field of interest among many mathematicians. While fractional differential equations in real variables have received much attention recently, attempts to solve such equations in complex variables have been rather scant. This research work deals with the complex order fractional differential equation with boundary conditions. The existence of solutions is established by using Dhage’s fixed point theorem with some conditions, whereas the application of the Banach contraction principle obtains the uniqueness of the solution. Moreover, Ulam–Hyers stability of the considered problem is also discussed in this work. Examples and application are presented to verify the obtained results.
Corrigendum to “Complex order fractional differential equation in complex domain with mixed boundary condition” [Chaos Solitons Fractals 185 (2024) 115090]
(Elsevier, 2024-11) Agarwal, Shivi; Mathur, Trilok
The authors would like to inform the readers that the conditions of Theorem 2.2 and Theorem 3.1 of Yadav et al. (2024) is not adequate. The statement and proof of Theorems 2.2 and 3.1 of Yadav et al. (2024) have been modified by Prof. Kiran Kumar Saha. We appreciate him for noticing this error. Throughout the article Yadav et al. (2024), Theorem 2.2 is replaced by Theorem 1.1, and Theorem 3.1 is replaced by Theorem 1.2. The statements of both Theorem 1.1 and 1.2 are given in this corrigendum. Due to this error found, Section 4 (Examples) of Yadav et al. (2024) needs to be updated. So, we have modified the examples.
A Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEA Model
(ARXIV, 2021-02) Agarwal, Shivi
Data Envelopment Analysis (DEA) is a multi-criteria technique based on linear programming to deal with many real-life problems, mostly in nonprofit organizations. The slacks-based measure (SBM) model is one of the DEA model used to assess the relative efficiencies of decision-making units (DMUs). The SBM DEA model directly used input slacks and output slacks to determine the relative efficiency of DMUs. In order to deal with qualitative or uncertain data, a fuzzy SBM DEA model is used to assess the performance of DMUs in this study. The credibility measure approach, transform the fuzzy SBM DEA model into a crisp linear programming model at different credibility levels is used. The results came from the fuzzy DEA model are more rational to the real-world situation than the conventional DEA model. In the end, the data of Indian oil refineries is collected, and the efficiency behavior of the companies obtained by applying the proposed model for its numerical illu
DEA based estimation of the technical efficiency of state transport undertakings in India
(Springer, 2011-02) Agarwal, Shivi
This paper measures the technical efficiency of public transport sector in India. The study makes an attempt to provide an overview of the general status of the State Transport Undertakings (STUs) in terms of their productive efficiency. Data have been collected for 35 STUs for the year 2004-2005. Technical efficiency of the STUs is measured by applying Data Envelopment Analysis (DEA) technique with the use of four input and three output variables. Fleet size, Total staff, Fuel consumption and Accident per lakh kilometer are considered as inputs and Bus utilization, Passenger kilometers and Load factor as outputs. On the basis of the status of technical efficiency, it is concluded that the performance of the STUs are good but still very far from the optimal level. The mean overall technical efficiency (OTE) is 83.26% which indicates that an average STU has the scope of producing the same output with the inputs 16.74% lesser than their existing level. Significant variation in OTE across STUs is also observed.
DEA Model for Exogenously Fixed Variables Under Fuzzy Environment
(Springer, 2019-08) Mathur, Trilok; Agarwal, Shivi
Data Envelopment Analysis (DEA) is developed to estimate the performance of Decision-Making Units (DMUs) and it is a nonparametric approach based on linear programming. In the real world, some inputs and outputs are not precisely known which are dealt by fuzzy DEA, where these variables could be expressed as fuzzy numbers. In addition, exogenously fixed inputs and outputs are also involved in efficiency measurement, which are not under the control of management. DEA can deal these situations with crisp variables. This paper proposed a DEA model with exogenously fixed fuzzy inputs and outputs to estimate the performance of DMUs having exogenously fixed variables in a fuzzy environment. A numerical example is also presented.
A DEA Model for Systematic Screening of Sustainable Suppliers with Uncertainty
(Solid State Technology, 2020) Agarwal, Shivi; Mathur, Trilok; Routroy, Srikanta
Suppliers play a vital role to create a sustainable and environment friendly supply chain of manufacturers. The manufacturer must have a systematic screening of available sustainable suppliers along various dimensions including their environmental sustainability dimension so that they can make the right outsourcing decision. It was observed that there are quantitative, qualitative factors and vagueness of experts' opinions involved in the screening, so a modified integrated model of Data Envelopment Analysis (DEA) integrated with the concepts of fuzzy logic is proposed which can handle the above factors for studying, analyzing, comparing and screening the sustainable suppliers. To describe the salient features of the proposed model a numerical illustration as case study is presented.
A DEA-based approach for optimizing the extended transshipment problem
(Elsevier, 2025-09) Agarwal, Shivi; Mathur, Trilok
Transshipment issues represent a distinct category of transportation challenges that commonly arise in practical contexts. The conventional transshipment model typically focuses on a single variable (predominantly cost), with decision-makers endeavoring to enhance this variable. Nevertheless, in practical contexts, numerous cost-related and revenue-related variables significantly influence the transshipment process, necessitating their optimization to augment the efficacy of the transshipment strategy. To our knowledge, a deficiency exists in addressing such a transshipment problem characterized by multiple variables. To address this deficiency, the present study advances the conventional transshipment model to accommodate scenarios wherein various variables are present within the transshipment framework. To illustrate the validity of the proposed methodology, a numerical example is provided, utilizing empirical data concerning wheat production in Iran.
DEA-neural networks approach to assess the performance of public transport sector of India
(Springer, 2015-10) Agarwal, Shivi
This paper proposes the integrated Data Envelopment Analysis-Neural Networks approach to measures the efficiency of public transport sector of India. Data have been collected for 30 State Road Transport Undertakings (STUs) for the year 2011–2012. Efficiency of the STUs is measured with the use of three inputs and single output. Fleet Size, Total Staff and Fuel Consumption are considered as inputs and Passenger Kilometers as output. On the basis of the status of efficiency, it is concluded that efficiency of the STUs are not good and very far from the optimal level. In order to check the robustness of the results, regression and correlation analysis are also conducted which reveal that the efficiency scores measured by all the models having the common trends. The most efficient and the lowest efficient STUs are found same by all the models. The results also demonstrate that the proposed models are highly flexible and don’t require any prior assumptions about the functional form between inputs and outputs. The models also handle the problem of the presence of the outliers and statistical noise in the data points.
Deformable derivative in complex domain
(Springer, 2024) Mathur, Trilok; Agarwal, Shivi
In this article, we are proposing a limit-based definition, which we have termed as complex deformable derivative. The complex deformable derivative and complex deformable integral with non-integer order , coincides with the classical complex derivative for . The complex deformable derivative is defined by using a limit approach similar to the classical definition of derivative involving a parameter. This derivative is represented as the linear combination of the function and its integer-order derivative. We investigate the elementary properties of deformable derivative and integral. The results have been demonstrated for this derivative using illustrative examples.
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.