Journal of Engineering Mathematics & Statistics

Authors: Ravindra K. Joshi, Meena L. Patwardhan

Abstract: Partial Differential Equations (PDEs) play a central role in modeling a wide variety of physical, biological, and engineering phenomena such as heat conduction, fluid flow, wave propagation, electromagnetics, and population dynamics. Analytical solutions to PDEs are available only for a limited class of problems with simplified geometries and boundary conditions. As a result, numerical methods have become an essential tool for obtaining approximate solutions to PDEs arising in real-world applications. This paper presents a comprehensive review of numerical methods for solving PDEs, with emphasis on classical and modern approaches. Finite difference methods, finite element methods, and finite volume methods are discussed in detail, along with spectral methods and meshless techniques. Issues related to stability, convergence, accuracy, and computational efficiency are also examined. The paper further highlights recent developments in adaptive schemes and high-performance computing for PDE solvers. The aim of this review is to provide a structured understanding of numerical PDE methods for students, researchers, and practitioners in engineering mathematics and applied sciences.

Keywords: Partial differential equations; numerical methods; finite difference method; finite element method; finite volume method; stability and convergence

Authors: Surender Venkatesh, Ranjeet Prasad, Neha Kulkarni

Abstract: The rapid growth of data across engineering, science, economics, and social domains has increased the need for advanced statistical models capable of handling multiple interrelated variables simultaneously. Multivariate statistical models have long served as a core analytical framework for understanding complex dependency structures among variables. In recent years, learning-based statistical models, particularly those inspired by machine learning, have significantly expanded the scope and performance of traditional multivariate approaches. This paper presents a comprehensive review of multivariate and learning-based statistical models, highlighting their theoretical foundations, methodological developments, and practical applications. Classical techniques such as multivariate regression, principal component analysis, and factor analysis are discussed alongside modern learning-based models including neural networks, support vector machines, and ensemble methods. The paper also examines hybrid frameworks that integrate probabilistic modeling with data-driven learning strategies. Challenges such as high dimensionality, interpretability, and computational complexity are critically analyzed. The review concludes by outlining emerging trends and open research problems in multivariate and learning-based statistical modeling.

Keywords: Multivariate analysis, statistical learning, high-dimensional data, machine learning models, probabilistic modeling, data-driven methods

Author: Reshma Srivastaw

Abstract: Fluid dynamics is one of the most important and challenging branches of applied mathematics and engineering science. It deals with the study of motion of fluids such as liquids and gases and their interaction with solid boundaries. Mathematical modeling plays a crucial role in understanding, predicting, and controlling fluid flow phenomena that arise in nature and industrial applications. From atmospheric circulation and ocean currents to blood flow in arteries and fuel injection systems, fluid dynamics models provide a theoretical framework to analyze complex flow behavior. This paper presents a comprehensive review of mathematical modeling of fluid dynamics, emphasizing governing equations, modeling assumptions, analytical and numerical techniques, and selected applications. Classical models such as the Euler and Navier–Stokes equations are discussed along with simplified models like boundary layer equations and potential flow theory. The paper also highlights challenges related to nonlinearity, turbulence, and computational complexity. Although significant progress has been made, many open problems remain, particularly in turbulence modeling and multiphase flows. The study aims to provide a clear and structured overview for researchers and postgraduate students working in applied mathematics, physics, and engineering fields.

Keywords: Fluid dynamics, Mathematical modeling, Navier–Stokes equations, Turbulence, Computational fluid dynamics

Authors: R. K. Mishra, Devender Singh, S. D. Iyer

Abstract: High-dimensional data analysis has become a central topic in modern scientific research due to rapid advances in data acquisition technologies. In many contemporary applications, the number of variables or features is extremely large compared to the number of observations, leading to unique analytical and computational challenges. Traditional statistical and machine learning techniques often fail or perform poorly in such settings due to issues like the curse of dimensionality, overfitting, and interpretability problems. This paper presents a comprehensive review of high-dimensional data analysis, focusing on theoretical foundations, dimensionality reduction techniques, feature selection methods, and modern machine learning approaches. We also discuss practical challenges, including computational complexity and data sparsity, and highlight applications in fields such as bioinformatics, image processing, finance, and social network analysis. The paper aims to provide a balanced overview that is accessible to researchers from mathematics, statistics, and engineering backgrounds, while also identifying open research directions in this rapidly evolving field.

Keywords: High-dimensional data, curse of dimensionality, dimensionality reduction, feature selection, machine learning, big data analytics

Authors: Raj Mondal, Shivesh Mishra, Aman Chopra

Abstract: Extreme-value theory and reliability analysis are two closely related areas that play a crucial role in modern engineering, environmental science, and risk assessment. While reliability analysis focuses on the probability of survival or failure of systems over time, extreme-value analysis concentrates on modeling rare and extreme events such as maximum loads, peak stresses, floods, or extreme temperatures. These rare events, though infrequent, often dominate system design and safety decisions. This review paper presents a comprehensive overview of the fundamental concepts, statistical models, and applications of extreme-value theory and reliability analysis. Classical distributions used in extreme-value modeling, including Gumbel, Fréchet, and Weibull distributions, are discussed along with their practical relevance. Reliability measures such as reliability function, hazard rate, and mean time to failure are reviewed in detail. The paper also highlights methods for parameter estimation, system reliability modeling, and the integration of extreme-value models into reliability frameworks. Applications in civil engineering, mechanical systems, environmental risk, and industrial safety are presented. The discussion emphasizes both strengths and limitations of existing approaches and outlines future research directions. The presentation is intended to be accessible to researchers and graduate students, though some parts may appear slightly informal in expression.

Keywords: Extreme-value theory, reliability analysis, Weibull distribution, hazard function, risk assessment

Author: Arjun Patel

Abstract: Computational Linear Algebra (CLA) is a core area of applied mathematics and scientific computing that focuses on the numerical solution of linear algebraic problems arising in science, engineering, economics, and data analysis. Problems such as solving systems of linear equations, eigenvalue computation, matrix factorizations, and least squares approximation form the backbone of modern computational methods. With the growth of large-scale simulations, big data analytics, and machine learning, computational linear algebra has gained renewed importance. This paper presents a comprehensive review of fundamental concepts, classical and modern algorithms, numerical stability issues, and practical applications of computational linear algebra. Emphasis is placed on direct and iterative methods, matrix decompositions, sparse and structured matrices, and parallel computing aspects. The paper also highlights current challenges and future research directions in the field. The discussion is intended to serve as a reference for students, researchers, and practitioners working in applied mathematics and computational sciences.

Keywords: Computational Linear Algebra, Numerical Methods, Matrix Decomposition, Iterative Methods, Sparse Matrices

Authors: Justine Nkundwe Mbukwa, G.V.S. R. Anjaneyulu

Abstract: This paper has been motivated as a result of an existence of high dimensionality problem in maize yield. This means that an application of the Sparse Principal Component Analysis (SPCA) pattern recognition technique is unknown in selecting few consistent features and easier interpretation as opposed to classical PCA. This paper fulfills the existing knowledge gap in the context of Tanzania. A structure questionnaire was used to collect primary data from Mbozi and Mvomero Districts among small farming household in rural areas. The study was designed on the basis of hierarchical random sampling. The breakdown of facts was made by R-Statistical computing (version 3.3.2) whereas the findings were depicted using graphs and tables. The statistical estimates like percentage, mean and variance were also used. In line with SPCA, PCA and Robust PCA were also fitted for comparison purpose. Results showed 19 variables were condensed to six components explaining 63.7 per cent variations under PCA. Contrary to these findings, there were great improvements of the loadings, consistent and easier to interpret in each PC of the modified model (SPCA). However, the paper discovered that the Robust PCA condensed the p-variable to two PCs such that PC1 explained (81.0 per cent) variances. The study recommends the Sparse and Robustness as the best filtering techniques with reliable results as contrasted to the ordinary PCA.

Keywords: Classical Principal Components Analysis, Sparse Principal Component Analysis, Dimensionality Reduction, Robustness, Smallholder Farmers and Maize Yield

Authors: PriyankaMajumder, Apu Kumar Sahab, Mrinmoy Majumder

Abstract: The ever-growing demand for luxury has increased the stress on conventional energy sources and encourages scientists and engineers to look for alternatives. Hydropower is by far the most inexpensive but reliable source of energy which is deemed to have the capacity to substitute for conventional energy sources. The worldwide contribution of hydropower plants (HPP) in supplying the demand for electricity is 1106 TWh. The problem with hydropower lies with the fact that its efficiency depends on multiple factors which are a function of climatic, hydraulic and socio-economic parameters. All these parameters again depend upon hydraulic loss imposed due to time in use, change in energy requirements, locational interference and quality of the machine installed. As there are multiple parameters having different levels of influence on the performance efficiency of HPP. Thus some factors are overrated and some others remain under rated which results in erroneous decision-making. The present study proposes a new hybrid model based on Decision-Making Trial and Evaluation Laboratory (DEMATEL) with ORESTE. The priorities are determined by hybrid method namely DEMATEL-ORESTE. According to the results, efficiency of generator is most significant for efficiency of HPP.

Keywords: Hydropower Plants, Hybrid MCDM Method, Decision-Making Trial and Evaluation Laboratory (DEMATEL), ORESTE