http://dspace.dtu.ac.in:8080/jspui/handle/123456789/28 2026-06-11T00:53:08Z 2026-06-11T00:53:08Z BENIWAL, KIRTI Aggarwal, Vivek Kumar (SUPERVISOR) http://dspace.dtu.ac.in:8080/jspui/handle/repository/22768 2026-06-08T05:46:36Z 2026-02-01T00:00:00Z

Title: NUMERICAL AND MACHINE LEARNING APPROACH FOR SINGULARLY PERTURBED PROBLEMS WITH APPLICATION TO CYCLONE MODELING Authors: BENIWAL, KIRTI; Aggarwal, Vivek Kumar (SUPERVISOR) Abstract: This thesis contributes to numerical methods and Machine learning approach for singularly perturbed differential and its application in cyclone modeling. The pur- pose of this research is to approximate singularly perturbed differential equations using approximation methods. We investigate, develop, and analyze numerical methods, as well as their implementations, for such challenging problems. The chapter-by-chapter structure of the thesis is as follows. Chapter 1 is introductory in nature and gives a brief review of Singularly Per- turbed Problems (SPPs) and a survey on numerical analysis of Singularly Per- turbed Differential equations (SPDEs) and numerical techniques. Objectives, lit- erature survey, and a brief summary of the present work are also included in this chapter. Chapter 2 is about the numerical collocation methods based on Bernstein collo- cation method using equispaced nodes and Chebyshev-Gauss-Lobatto Nodes for solving linear and non linear singularly perturbed differntial equations, and results are then compared with traditional methods and error analysis is carried out. In chapter 3, is devoted to the Szasz Mirakyan operator . In this chapter we solved both linear and non linear singularly perturbed different equations and compared with traditional Methods Stability Analysis and error analysis is car- ried out. In chapter 4, This chapter presents a novel framework combining the Bernstein- Chlodowsky operator with neural networks termed the Bernstein-Chlodowsky Neu- ral Network (BNN) and the Bernstein-Chlodowsky collocation method (CCM) for solving singularly perturbed differential equations (SPDEs) in the context of at- mospheric science and cyclone modeling. We solve linear and nonlinear SPDEs using the Bernstein-Chlodowsky collocation method and BNN, emphasizing how well these methods capture the convection dominated processes observed in cy- clonic systems. The superior convergence of the operator and the resolution of the boundary layer make it an effective tool for cyclone modeling. Chapter 5, This chapter focuses on predicting the Accumulated Cyclone En- ergy (ACE) in the North Indian Ocean (NIO) during monsoon using an optimized Artificial Neural Network (ANN) model. The permutation feature is essential in finding the most influential features to improve model performance Chapter 6, This chapter aims to forecast, using yearly cyclone data, the Accu- mulated Cyclone Energy (ACE) values for the North Indian Ocean (NIO) area. We study a predictive framework for estimating Accumulated Cyclone Energy (ACE) in the North Indian Ocean (NIO) using historical cyclone data from 1982 to 2023, encompassing the Arabian Sea (AS) and Bay of Bengal (BOB) basins. Interpo- lation was performed using the Szász-Mirakyan operator, which preserves the statistical properties of the underlying distribution while reconstructing missing and zero values more effectively than conventional methods. XGBoost and neu- ral networks are two machine learning methods compared in the study. Chapter 7 Finally, this Chapter is devoted to conclusion of the study and discus- sion on some future directions of the current research work. Finally, the bibliography and list of author’s publications have been given at the end of the thesis.

2026-02-01T00:00:00Z KUMARI, REENU http://dspace.dtu.ac.in:8080/jspui/handle/repository/22669 2026-02-24T09:03:17Z 2026-01-01T00:00:00Z

Title: FUZZY PORTFOLIO SELECTION VIA RANKING MODELS IN DEA AND MULTI-CRITERIA DECISION MAKING Authors: KUMARI, REENU Abstract: A portfolio selection problem implemented through an optimization technique is called portfolio optimization. The mathematical model for portfolio optimization al- locates total capital among various assets following investors’ preferences about return/risk. Generally, investors seek to reduce risk while enhancing returns, yet attaining higher expected returns inevitably involves accepting greater levels of risk. Therefore, an investor faces a trade-off between risk and expected return. Hence, portfolio optimization is a technique used to construct an optimal basket of assets, where an optimal is understood in the context of an investor’s objec- tives and desires. In many real-world situations, the return from an asset cannot be anticipated accurately based on historical data. The presence of vagueness and fuzziness in the input and output data can not be resolved by using proba- bility theory. The unpredictable dynamic nature of the financial market motivates researchers to use the concept of fuzzy set theory in the field of portfolio selec- tion. The possibility theory is an uncertainty theory devoted to the handling of incomplete information. Besides an accurate determination of a risk measure of a return distribution, investors also wish to evaluate the performance of their portfolios concerning a benchmark index or to rank different portfolio strategies. Generally, the role of an asset’s performance in optimal portfolio construction has not been considered so far. When selecting assets for a portfolio, an investor considers several fac- tors. Data Envelopment Analysis (DEA) simultaneously accommodates multiple inputs and outputs, providing a composite efficiency score. As DEA measures the relative efficiency of several similar processing units, it also helps in asset selection before portfolio construction. However, the DEA allows each financial asset to evaluate its efficiency relative to other homogeneous financial assets by assigning favorable weights. This often results in unrealistic weight schemes. To address this issue, the DEA cross-efficiency framework is employed, which elim- v inates such unrealistic weight allocations. In financial markets, assets compete for higher efficiency scores, often leading to multiple optimal weights in standard cross-efficiency. DEA game cross-efficiency introduces a noncooperative frame- work where competing assets jointly determine balanced weights, reducing non- uniqueness and producing more stable and fair rankings for portfolio selection. In certain instances, DEA models may yield an efficiency score of one for sev- eral decision-making units (DMUs), making it challenging to rank these DMUs. Further, in DEA, every approach uses a distinct theory and framework to rank the DMUs, so each DMU has a different ranking order. The decision-maker’s reliability of the results is a critical consideration when choosing a ranking sys- tem. Multi-Criteria Decision Making (MCDM) approaches, which differ from DEA models, can be used to solve the problem of ranking efficient DMUs. The challenge of aggregating self- and peer-evaluated cross-efficiencies into a final score has been widely discussed. Notably, using a simple arithmetic average inherently assumes that all DMUs’ evaluations are equally valid or dependable, which is not always true. To address this issue, we propose a novel use of the Ordered Visibility Graph Averaging (OVGA) operator for more meaningful aggre- gation. Furthermore, in the same work, we introduce a portfolio selection model for constructing the most efficient optimal portfolio. In DEA game cross-efficiency, the final score is obtained through an iterative algorithm, where each iteration aggregates the game cross-efficiency scores of all DMUs using the arithmetic averaging method. This thesis presents the OVGA aggregated DEA game cross-efficiency method, which considers the competition among DMUs in portfolio selection. These Game cross-efficiency scores serve as a tool for efficient portfolio selection. Further, a multi-objective portfolio selection model is proposed, where the Maverick index and variance of cross-efficiency are treated as risk metrics, and the OVGA game cross-efficiency scores are used as return characteristics. The Semi-oriented radial measure (SORM) model of DEA effectively handles the negative input-output data. However, it has a limitation of producing negative cross-efficiencies. We propose a modified SORM model to deal with this issue. Also, a novel multi-objective portfolio selection model is introduced, using the maverick index to represent risk and the diversity index to represent return. The maverick index is calculated using the column average of the cross-efficiency vi matrix, while the diversity index is determined using the row average. In another research study in this thesis, an innovative approach is introduced to portfolio selection derived from the RDM cross-efficiency matrix. In practical appli- cations, the column average of the cross-efficiency matrix is commonly employed for decision-making, as it helps identify efficient and consistent performers. How- ever, the row average also provides valuable insight into how fairly or aggressively each DMU evaluates its peers. We provide a method for categorization of the as- sets, which utilizes both row and column averages of the RDM cross-efficiency matrix. An essential aspect of investment management is the unique rating of portfolios, which enables investors to identify and assess the most effective portfolios based on criteria such as risk, return, etc. We present a hybrid approach for ranking investment portfolios by combining the Modified Slack-Based Measure (MSBM) of DEA with a multi-criteria decision-making method. Techniques like the MSBM and TOPSIS incorporate traditional performance metrics while adding flexibility to address fuzzy environments and handle imprecise data. aims to evaluate fuzzy portfolios using the MSBM model, with trapezoidal fuzzy numbers for returns and possibilistic measures for risk and mean return. Efficient portfolios are further ranked using the TOPSIS technique. This thesis entitled “Fuzzy Portfolio Selection via Ranking Models in DEA and Multi-criteria Decision Making” aims to highlight the advantages of DEA as an innovative tool for portfolio optimization, contributing to the development of more robust and efficient investment strategies. The methodologies introduced and developed in this thesis are rigorously tested on real-world case studies, demon- strating their practical applicability and effectiveness in enhancing portfolio selec- tion processes.

2026-01-01T00:00:00Z SRIVASTAVA, ABHAY http://dspace.dtu.ac.in:8080/jspui/handle/repository/22651 2026-02-10T04:47:30Z 2025-10-01T00:00:00Z

Title: MODELING AND SIMULATION OF INFECTIOUS DISEASE USING FRACTIONAL CALCULUS Authors: SRIVASTAVA, ABHAY Abstract: In recent years, the world has faced a sharp rise in infectious diseases, which continue to be a serious threat to public health. Despite progress in medical science, surveil- lance systems, and control measures, outbreaks such as influenza, SARS, and most recently COVID-19 have shown that our societies remain highly vulnerable. These events have also revealed some of the limitations of the classical models used to study and predict the spread of infections. In particular, standard models often ignore mem- ory effects, individual behaviour, and environmental influences. To overcome these gaps, this thesis applies fractional calculus in the modeling and simulation of infec- tious diseases. Fractional-order models have the advantage of incorporating memory and history, which makes them more realistic for studying epidemics where past expo- sure, immunity, and behavioural changes play an important role. The work begins with a study of vaccination strategies followed in five countries that were badly affected during the first half of 2022: the USA, India, Brazil, France, and the UK. A detailed comparison shows that most countries gave priority first to frontline workers and health professionals, and then to elderly or immunocompromised people. The main difference was how countries divided the age groups for priority. By comparing these strategies with confirmed cases and deaths per population, as well as with population density and median age, the study highlights how vaccine distribution policies must be designed carefully to suit the demographics of each country. Motivated by these findings, different fractional-order models are developed in this thesis. The first is an SIS model with Beddington-De Angelis incidence, used to capture the effect of fear-driven behaviour. When people become afraid of infection, they may self-isolate or reduce contact with others. Such actions can strongly influence disease spread, and fractional calculus is especially suitable to model this because fear and behaviour are shaped by past experiences. A second contribution is an SVIR model that divide vaccinated people into two groups: partially vaccinated (those who did not complete the prescribed course of the doses) and fully vaccinated (those who completed the vaccination schedule and followed health guidelines). This distinction is important, as many people worldwide xiii xiv ACKNOWLEDGMENTS showed hesitancy in taking vaccines, often due to doubts about safety or mistrust of governments. The model allows us to study how partial vaccination affects recovery compared with full vaccination, giving a clearer picture of real vaccination outcomes. The thesis also extends the SEIQR model by including two realistic features: psy- chological effects during transmission (using Monod-Haldane incidence) and a limited quarantine capacity (Holling type-III function). These changes reflect how quarantine in practice cannot be increased indefinitely and is often constrained by resources. An associated fractional optimal control problem is studied using Pontryagin’s principle, showing how time-dependent controls can be used to reduce infections at minimum cost. Beyond vaccination and quarantine, the thesis considers environmental effects. A Susceptible-Pollution affected-Infected-Recovered (SPIR) model is proposed to study how exposure to pollutants weakens immunity and increases vulnerability to infec- tions. This model even accounts for prenatal exposure in newborns, reflecting the long-term consequences of pollution. A fractional optimal control problem with two controls is solved to examine how information campaigns and other interventions can help reduce infections in polluted environments. Another area studied is the role of bacteria. Due to rising household waste and urbanization, bacterial populations in the environment are growing, leading to more bacterial and vector-borne diseases. To address this, a fractional SIR model with bac- teria in the environment and in organisms is developed. An optimal control problem with three controls is analyzed to show how disease transmission can be reduced effi- ciently. Across all these models, the unifying theme is the use of fractional-order sys- tems. By including memory, they allow us to model more realistic epidemic be- haviours, whether due to human psychology, environmental stress, or bacterial growth. Numerical simulations are carried out using the Adams-Bashforth-Moulton predictor- corrector method, which validates the theoretical results and demonstrates how the models behave under different conditions. In summary, this thesis presents a set of new fractional-order models that bring together vaccination strategies, fear and behaviour, quarantine measures, environmen- tal pollution, and bacterial effects in infectious disease dynamics. The results show that fractional models are not only mathematically richer but also practically more meaningful, as they reflect the role of memory and history in epidemic processes. By combining theory, simulations, and control strategies, the thesis provides insights that can support better decision-making in managing infectious diseases and preparing for future outbreaks.

2025-10-01T00:00:00Z KUMAR, SANDEEP http://dspace.dtu.ac.in:8080/jspui/handle/repository/22548 2025-12-29T08:48:09Z 2025-11-01T00:00:00Z

Title: CONVERGENCE ANALYSIS OF SOME APPROXIMATION OPERATORS Authors: KUMAR, SANDEEP Abstract: The thesis is divided into seven chapters, the contents of which are organized as follows: Chapter1 of the thesis covers the literature and historical foundation of certain important approximation operators. We provide a brief overview of the chapters that constitute this thesis and discuss some of the preliminary tools we will use to delve into the subject’s depth. Chapter 2 introduces a new sequence of operators involving Apostol-Genocchi polynomials and Baskakov operators and their integral variants. We estimate some direct convergence re- sults using the second-order modulus of continuity, Voronovskaja type approximation theorem. Moreover, we find weighted approximation results of these operators. Next Chapter 3 is mainly focused on the difference operators of two positive linear operators (generalized pˇaltˇanea type operators Lλ n,c ( f ; x) and M. Heilmann type operators Mn,c( f ; x) ) with same basis functions. First, we estimate quantitative difference of these operators in terms of modulus of continuity and Peetre’s K−functional In Chapter 4, we present a recurrence relation for the semi-exponential Post-Widder operators and provide estimates for their moments. We then examine convergence results within Lipschitz- type spaces, analyzing the convergence rate using the Ditzian-Totik modulus of smoothness and the weighted modulus of continuity. Finally, we estimate the convergence rate for functions whose derivatives are of bounded variation. Chapter 5 introduces a novel Bézier variant within the family of Phillips-type generalized positive linear operators. The moments of these operators are derived to enhance understand- ing of their fundamental properties. The chapter further explores convergence properties in Lipschitz-type spaces, with particular focus on the Ditzian-Totik modulus of smoothness. Fi- nally, it provides a rigorous analysis of the convergence rate for functions whose derivatives are of bounded variation, contributing valuable insights to the field of approximation theory. The aim of Chapter 6 is to introduce the sequence of Baskakov-Durrmeyer type operators linked with the generating functions of Boas-Buck type polynomials. After calculating the mo- ments, including the limiting case of central moments of the constructed sequence of operators, in the subsequent sections, we estimate the convergence rate using the modulus of continuity and Ditzian-Totik modulus of smoothness and some convergence results in Lipchitz-type space and the end we estimates the convergence for the functions of bounded variations. The thesis is summarised in Chapter 7, before providing some insight into the author’s thoughts about the future research.

2025-11-01T00:00:00Z