Journal of Engineering Mathematics & Statistics

Authors:- Rahul Mondal

Abstract:- In this paper, an inventory model in a crisp environment is established. This is a model for nonmanufacturing inventories. Demand is assumed to be price sensitive. The rate of deterioration is assumed to remain constant. The model was created for the scenario of no shortage. For deterministic inventory parameters, profit and optimal order quantity are derived. The model is shown with numerical parameter values, and sensitivity analysis is offered.

 Keywords:- Deterioration Price dependent demand; Inventory model.

Authors:- Dr. B. R Narayan, Vishal Raghuvansi

Abstract:- The purpose of this essay is to apply statistical approaches to dependability difficulties using an experiment from the aerospace sector. To that purpose, failure data from this region is collected and categorised into two separate groups, with the category of interest being failures happening on components that will need to be changed in order to resolve the technical issues. In order to represent lifespan data from the category of interest, a Weibull distribution is presented, which is based on a non-parametric technique (the so-called Median Rank) for estimating cumulative distribution functions (c. d. f.), and some practical findings are provided using two basic instances.

Keywords:- line replaceable unit, Weibull distribution, Median Rank, time to first failure.

Authors:- Gurupadavva Ingalahalli , H.R. Mahadevaswamy, C.S. Bagewadi

Abstract:- The purpose of the current paper is to study W_2-flat, ξ-W_2-flat, W_2-Ricci pseudo symmetric, W_2-Ricci semisymmetric, ϕ-W_2-semisymmetric β-Kenmotsu manifold and Ricci almost solitons.

Keywords:- Kenmotsu manifold, connection, Symmetric. AMS (2010) Subject Classification: 53C05, 53D10, 53C20, 53C15, 53C25;

Authors: Ambika Bansal, Pooja Sharma, Pooja Choraria, Ritik Bansal

Abstract: An unnatural death results from an external cause, including accidental explosion, accidental fire, traffic accidents, drowning, electrocution, homicides, suicides, accidents, medical errors, drug overdoses, poisoning, etc. In case of women, causes may also include factors like pregnancy, abortion, reproductive years, etc. In this paper, we have studied the main factors of accidental deaths that increasing death rates in India. Research methodology is descriptive & exploratory (to explore the hidden problem more clearly).

For this study, secondary data is used and various statistical tools like chi square test of independence of attributes & Z–Test for equality of two proportions are applied to analyze the data.

Keywords: Accidental death, Disease, Unnatural death, Accidental Explosion, Accidental Fire, Traffic Accidents, Drowning, Association, NCRB, Chi Square Test, Z–Test, p–value, proportion, etc. 

Authors: T.Srinivasarao, Akula Boby

Abstract: Measure and probability distribution are closely related in the real time situations. For a measurable function, A system of measurable functions forming a hollow beam about the K – axis that is parallel to the X – axis formed by rotating through an angle   . if the given function f  is a function of bounded variation, then every function forming this beam is a generator of the beam and is a function of bounded variation. the projection of the beam upon the plane having K – axis as the axis of symmetry will be a symmetric function  and these two functions will enclose a symmetric region between the vertical limits  x = a and x = b.  in the previous paper titled “K – Horizontal Symmetric Measurable functions, the function  is noted to be the K – Horizontal Symmetric Measurable function of f. now, it is termed as the adjoint of the function f.  From the above discussion,    is also a generator of the hollow beam and is a function of bounded variation whenever f   BV. From the properties of the functions of bounded variation, we have   and consequently, about adjoint function, it follows 

Keywords: Complement Operator, Adjoint function, Measurable Function, X- axis 

Author: Dr. Prasana Kumar Mishra

Abstract: A Cauchy algebraic polynomial is a random algebraic polynomial  whose coefficients are independent real-valued random variables with a common Cauchy distribution then for every large enough integer n₀₁ where s is a finite number greater than 2+β; 0< β<1 and μ’s are positive constants. For this theorem we get, for s>3, a probability less than   1991 mathematics subject classification (Amer. math. soc.): 60 B 99.

Keywords: Independent Identically Distributed Random Variables, Random Algebraic Polynomial, Random Algebraic Equation

Author: Dr. Prasana Kumar Mishra

Abstract: Let us consider the random trigonometric polynomial  in the interval (Φ’, Φ’’). We have to prove that in the interval   all save a certain exceptional set of the functions (Tn (θw) have    zeros when n is large.  1991 Mathematics subject classification (Amer. Math. Soc.): 60 B 99.

Keywords: Independent identically distributed random variables, random algebraic polynomial, random algebraic equation, real roots.

Authors: R. K. Deshmukh, Anita Verma, S. P. Kulkarni

Abstract: Stochastic differential equations (SDEs) have emerged as a powerful mathematical framework for modeling systems influenced by random fluctuations and inherent uncertainties. In parallel, uncertainty quantification (UQ) has become an essential component of numerical analysis, particularly in numerical linear problems arising from physical, engineering, and biological systems. This review paper presents a comprehensive discussion on the theoretical foundations of stochastic differential equations, their numerical discretization techniques, and the role of uncertainty quantification in numerical linear systems. Special emphasis is placed on the integration of SDEs with numerical linear algebra methods, such as stochastic linear solvers, random matrix theory, and probabilistic interpretations of numerical errors. Various uncertainty propagation techniques, including Monte Carlo simulation, polynomial chaos expansion, and stochastic Galerkin methods, are critically examined. Applications in engineering mechanics, fluid flow, financial modeling, and inverse problems are also highlighted. The paper aims to provide researchers and postgraduate students with a structured overview of current developments, challenges, and future research directions in stochastic modeling and uncertainty-aware numerical computation.

Keywords: Stochastic differential equations; uncertainty quantification; numerical linear algebra; Monte Carlo methods; polynomial chaos; stochastic modeling; numerical analysis

Author: Shobhit Nayar

Abstract: Real-time control systems are increasingly deployed in complex and dynamic environments such as autonomous vehicles, smart grids, robotics, aerospace systems, and biomedical devices. A defining challenge in such systems is the presence of uncertainty arising from modeling errors, external disturbances, sensor noise, time delays, and unpredictable environmental changes. This paper presents a detailed review of real-time control under uncertainty, focusing on theoretical foundations, mathematical modeling, uncertainty representation, and modern control strategies. Classical approaches such as robust control and adaptive control are discussed alongside stochastic control, model predictive control, and data-driven methods. Special emphasis is placed on uncertainty quantification, real-time computational constraints, and stability considerations. The paper also highlights recent trends and practical challenges in implementing uncertainty-aware controllers in real-time environments. Tables and conceptual figures are included to summarize methods and compare their strengths and limitations. The review aims to provide a structured understanding of real-time control with uncertainty for researchers and practitioners in engineering and applied mathematics.

Keywords: Real-time control, uncertainty modeling, robust control, stochastic systems, adaptive control, uncertainty quantification, feedback systems

Authors: Anil Verma, Sneha R. Tiwari, Mohammad Irfan Shaikh, Pooja Srivastav, Shyam prasas

Abstract: In recent years, data-driven modelling has become a central paradigm in engineering analysis and design due to the rapid growth of sensing technologies, computational power, and availability of large-scale datasets. Quantitative methods play a crucial role in transforming raw engineering data into meaningful models that can describe, predict, and optimize system behavior. This paper presents a comprehensive review of quantitative methods used in data-driven modelling for engineering applications. Classical statistical techniques, regression-based models, numerical optimization, and modern machine learning approaches are discussed in detail. The integration of datadriven models with physical knowledge and engineering constraints is emphasized, highlighting hybrid and physics-informed approaches. The paper also reviews common engineering applications such as structural health monitoring, manufacturing process optimization, energy systems, and control engineering. Challenges related to data quality, model interpretability, uncertainty, and scalability are critically analyzed. Finally, future research directions are outlined, focusing on robust, explainable, and computationally efficient quantitative modelling frameworks. The review aims to provide a structured understanding for researchers and practitioners working at the intersection of data science and engineering systems.

Keywords: Data-driven modelling; quantitative methods; engineering systems; regression analysis; machine learning; uncertainty quantification; optimization

Authors: Savrav Dubey, Deepali Mehta

Abstract: The rapid growth of scientific computing and data-driven engineering applications has led to the increasing demand for solving large-scale numerical problems efficiently. Traditional sequential numerical algorithms often fail to meet performance and memory requirements when applied to high-dimensional or large systems of equations. Parallel and distributed numerical algorithms address these challenges by exploiting multiple processors and distributed memory systems to reduce computation time and enhance scalability. This paper presents a comprehensive review of parallel and distributed numerical algorithms, focusing on their theoretical foundations, algorithmic structures, communication models, and practical implementation issues. Key numerical problems such as linear systems, eigenvalue computations, numerical integration, and partial differential equations are discussed in the context of parallelization strategies. The paper also highlights performance metrics, programming models, and current challenges associated with distributed computing environments. The discussion aims to provide a balanced understanding of both classical approaches and recent developments in parallel numerical computing.

Keywords: Parallel computing, distributed algorithms, numerical methods, high performance computing, scalability, scientific computing

Authors: Rakesh Kulkarni, Anjali Mehra, Awdhesh Pandey, Meraj Khan

Abstract: Optimization plays a central role in modern engineering design by enabling engineers to achieve the best possible performance under given constraints. With increasing complexity of engineering systems, traditional trial-and-error design approaches have become inefficient and costly. Optimization techniques provide systematic and mathematical frameworks to improve design quality, reduce material usage, enhance reliability, and minimize cost and energy consumption. This paper presents a comprehensive review of optimization in engineering design, covering fundamental concepts, formulation of design problems, classical and modern optimization techniques, and their applications across various engineering disciplines. Both deterministic and stochastic methods are discussed, with particular emphasis on gradient-based methods, evolutionary algorithms, and multi-objective optimization. Practical challenges such as computational cost, uncertainty, and real-world constraints are also highlighted. The review aims to provide researchers and practicing engineers with a clear understanding of optimization methodologies and their relevance in contemporary engineering design practice.

Keywords: Engineering design, optimization techniques, design variables, constraints, multi-objective optimization, evolutionary algorithms

Authors: R. K. Somashekar, P. Ananya Rao, M. Irfan Malik, Arjun Singh

Abstract: Complex engineering systems are characterized by strong nonlinearity, multiscale interactions, uncertainty, and coupling between different physical domains. Mathematical modeling plays a crucial role in understanding, analyzing, predicting, and optimizing the behavior of such systems. From power grids and transportation networks to aerospace structures and biologicalinspired engineering systems, mathematical models act as a bridge between physical reality and computational analysis. This paper presents a comprehensive review of mathematical modeling approaches for complex engineering systems. Classical modeling techniques such as differential equations and network models are discussed along with modern approaches including multi-physics modeling, data-driven models, and hybrid methods. The challenges associated with model formulation, parameter estimation, computational complexity, and validation are also examined. Representative applications from mechanical, electrical, civil, and interdisciplinary engineering domains are presented through tables and conceptual figures. The review highlights current trends and future directions, emphasizing the need for robust, scalable, and interpretable models to address real-world engineering problems.

Keywords: Complex systems, Mathematical modeling, Engineering systems, Nonlinear dynamics, multi-physics models, System simulation

Authors: Sailendra Tyagi, Chukesh Mishra

Abstract: The increasing complexity of scientific and engineering problems necessitates the development of efficient and accurate numerical solvers. Traditional numerical solvers, while reliable, often face challenges with high-dimensional systems, nonlinearity, and computational cost. Recently, the integration of machine learning (ML) techniques into numerical methods has emerged as a promising approach to enhance solver performance. This review presents a comprehensive analysis of machine learning-enhanced numerical solvers, focusing on their principles, methodologies, applications, and limitations. Techniques such as neural network-based solvers, physics-informed neural networks, and surrogate models are discussed in detail. Additionally, the paper explores comparative studies demonstrating the computational advantages and accuracy improvements achieved by ML-assisted solvers. The review concludes by highlighting current challenges, future research directions, and potential interdisciplinary applications.

Keywords: Machine learning, numerical solvers, physics-informed neural networks, surrogate models, computational efficiency, high-dimensional systems

Authors: Om Prakash Yadav, Arun Deshpande, Supriya Rajbhar, Aman Thakur

Abstract: Large systems of equations arise naturally in many branches of science and engineering, including fluid dynamics, structural analysis, climate modeling, power systems, machine learning, and biological simulations. As the size and complexity of these systems continue to grow, traditional sequential computational approaches become insufficient due to limitations in memory, time, and energy consumption. High Performance Computing (HPC) has emerged as a crucial enabler for solving such large-scale systems efficiently. This paper presents a comprehensive review of high-performance computing methods used for solving large systems of linear and nonlinear equations. The discussion includes parallel numerical algorithms, domain decomposition techniques, iterative solvers, sparse matrix methods, and the role of modern hardware architectures such as multicore processors, graphics processing units (GPUs), and distributed memory clusters. Performance metrics, scalability issues, and communication overheads are also examined. Through selected examples and comparative tables, the paper highlights the strengths and limitations of different HPC strategies. The review aims to provide researchers and practitioners with a structured understanding of current methodologies and practical considerations when applying HPC to large systems of equations.

Keywords: High performance computing, large systems of equations, parallel algorithms, iterative solvers, sparse matrices, scalability

Authors: Depesh Chaudhary, Aaub Ansari

Abstract: Large-scale engineering problems arising in areas such as structural analysis, power systems, fluid dynamics, machine learning, and network modeling often lead to mathematical formulations involving very large systems of algebraic equations. Algebra, particularly linear and multilinear algebra, plays a central role in the modeling, analysis, and numerical solution of these problems. As engineering systems grow in size and complexity, traditional algebraic methods become computationally expensive or even infeasible. This review paper presents a comprehensive discussion on algebraic techniques used for largescale engineering problems, focusing on matrix theory, sparse algebra, iterative solvers, eigenvalue problems, and decomposition methods. Emphasis is given to the practical challenges encountered in real engineering applications, such as memory limitations, numerical stability, and scalability. Selected case studies from engineering disciplines are discussed to highlight the relevance of algebraic approaches. The paper also outlines recent trends and open research issues in large-scale algebraic computations. Some grammatical imperfections are intentionally retained to maintain a natural academic writing style.

Keywords: Large-scale systems, linear algebra, sparse matrices, iterative methods, engineering computation, numerical algebra

Authors: Akhilesh Srivastav, Raghav Prajapati

Abstract: Nonlinear partial differential equations (PDEs) appear in numerous scientific and engineering applications, including fluid dynamics, plasma physics, biological systems, and financial modeling. These equations, unlike linear PDEs, often lack closed-form analytical solutions due to the complexity introduced by nonlinear terms. Consequently, advanced numerical methods have become critical for approximating solutions with sufficient accuracy and stability. This paper provides a comprehensive review of contemporary numerical approaches for solving nonlinear PDEs, including finite difference, finite element, spectral, and mesh-free methods. Additionally, it discusses stability and convergence considerations, high-performance computing strategies, and hybrid techniques that combine multiple approaches. Illustrative examples, comparative tables, and figures highlight the efficiency and accuracy of various methods. Future research directions include adaptive algorithms, machine-learning-assisted PDE solvers, and parallel computing optimizations to tackle large-scale nonlinear problems.

Keywords: Nonlinear PDEs, Finite Difference Method, Finite Element Method, Spectral Methods, Mesh-free Methods, Stability, Convergence, Computational Fluid Dynamics

Author: Dr. Priyaranjan Dash

Abstract: We have suggested four different classes of estimators in cluster sampling using transformations on both study and auxiliary variables under as well as under SRSWOR scheme. The proposed classes of estimators can be reduced to an infinite number of estimators namely HT-estimator, usual ratio and product estimators under different conditions. Again, the suggested classes of estimators provide a better estimate of population mean as it possesses the minimum mean square error.

Authors:-Raykundaliya, D.P, Patel, M.N

Abstract:-COVID-19 cases prediction is very important for betterment of planning and appropriate strategy developed by government for health care, mental peace and economy of country. In this paper, we try to predict daily new cases and total finished cases including daily recovered cases and daily death cases using Error, Trend and Season (ETS) model for India data taken from Worldometer website. We fit ETS (M,A,N) model for both daily new cases and daily finished cases. We give next 10 days prediction as well as suggest that under similar circumstances COVID – 19 cases leads to zero in India by August 22, 2020.

Authors:-D. T. Patel, M. N. Patel

Abstract:-In this paper we have used a Bayesian approach to determine the optimal warranty lengths. The power function distribution is employed to describe the product life time under multiply type-II censoring scheme. The optimal warranty is that which maximize the expected utility of the product. We have considered a combination of free replacement policy and pro-rata policy for warranty. A numerical data is used to exemplify the theory. A sensitivity analysis is carried out to check the effect of hyper parameters on the optimal warranty length and the optimal value of expected utility.

Author:-D. P. Patil

Abstract:-The diffusion phenomena such as conduction of heat in solids and diffusion of vorticity in the case of viscous fluid flow past a body are governed by a parabolic type partial differential equation. In recent years, many scholars are engaged in finding the solution of advance problems of engineering and sciences by using integral transforms method. To solve the parabolic boundary value problem, especially heat equation, which is also known as diffusion equation, we apply double Mahgoub transform method.

Authors: Samsun Nahar, Marin Akter, Md. Abdul Alim

Abstract: In this paper, a new statistical averaging technique is suggested to solve MOLFPP and MOLPP by using new arithmetic averaging method and new geometric averaging method. It is significantly noticeable same characteristics among all the solution techniques while taking maximum or minimum among all optimized values for multi-objective functions using simplex algorithm. The characteristics provided from the problems are verified by the examples.

Authors: Sudama Iyer, Shambhu Chandra, Vivek Tripathi, Jagnarayana Thakur, Ajaharu Din

Abstract: Time series analysis and stochastic process theory form the backbone of modern statistical modeling for data observed over time and under uncertainty. These methods are widely applied in engineering, economics, finance, environmental science, signal processing, and biological systems. The main objective of this paper is to present a comprehensive review of time series and stochastic process methods, highlighting their theoretical foundations, commonly used models, estimation techniques, and recent methodological developments. Classical approaches such as autoregressive moving average (ARMA) models, spectral analysis, and Markov processes are discussed alongside advanced techniques including state space models, stochastic differential equations, and non-linear time series models. The paper also emphasizes practical applications and challenges in real-world data analysis, such as non-stationarity, noise, and missing observations. Although the paper is primarily a review, it aims to provide intuitive explanations and comparative insights that may be useful for researchers and practitioners working in applied domains.

Keywords: Time series analysis, stochastic processes, ARIMA models, Markov processes, state space models, random processes, forecasting

Authors: Ananya Singh, Ramesh Mishra, Jeetesh Shah, Ravindera Ojha

Abstract: Stochastic Differential Equations (SDEs) play a central role in the mathematical modeling of systems influenced by randomness. Unlike ordinary differential equations, SDEs incorporate stochastic processes, typically in the form of Brownian motion, to represent uncertain or noisy dynamics. Such equations arise naturally in physics, finance, biology, engineering, and climate science, where real-world phenomena are rarely deterministic. This paper presents a comprehensive review of the theoretical foundations of SDEs, including stochastic calculus, Itô and Stratonovich formulations, and existence and uniqueness results. Numerical methods for solving SDEs, such as the Euler–Maruyama and Milstein schemes, are discussed with emphasis on stability and convergence. Applications of SDEs across different disciplines are also reviewed to highlight their practical importance. Although the mathematical framework of SDEs is well established, many challenges remain in efficient computation and interpretation of stochastic models. The paper aims to provide a clear and structured overview suitable for postgraduate students and early-stage researchers.

Keywords: Stochastic differential equations; Brownian motion; Itô calculus; Numerical simulation; Random processes

Authors: Anshika Tkaur, Rajesh Sinha, Anmol Singh

Abstract: Statistical learning and machine learning have evolved as two closely related yet historically distinct paradigms for data-driven modeling and decision making. Statistical learning theory emphasizes probabilistic modeling, inference, interpretability, and uncertainty quantification, whereas machine learning traditionally focuses on predictive performance, scalability, and algorithmic efficiency. In recent years, the boundaries between these two fields have become increasingly blurred due to the rapid growth of data availability, computational power, and complex real-world applications. This paper presents a comprehensive review of the integration of statistical learning and machine learning, highlighting their theoretical foundations, methodological overlaps, and complementary strengths. Key concepts such as bias–variance trade-off, regularization, probabilistic modeling, and generalization are discussed from both perspectives. The paper also examines hybrid frameworks that combine statistical rigor with machine learning flexibility, including Bayesian machine learning, regularized empirical risk minimization, and interpretable learning models. Practical applications in engineering, healthcare, finance, and social sciences are reviewed to illustrate the benefits of integration. Despite notable advances, challenges remain in terms of model interpretability, computational complexity, and reliable uncertainty estimation. The study concludes that a unified approach to statistical and machine learning methods is essential for developing robust, transparent, and efficient intelligent systems.

Keywords: Statistical learning, Machine learning, Bayesian methods, Regularization, Predictive modeling, Data-driven methods