Department of Mathematics

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http://nits.ndl.gov.in/handle/123456789/145

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Study On Singularly Perturbed Integro-Differential Parabolic Equation With Space Delay
(Department of Mathematics, NITS, 2025) Jitesh
This thesis presents a comprehensive numerical study of a class of singularly perturbed delay differential equations (SP-DDEs), which arise frequently in applications across science and engineering. The presence of both a small perturbation parameter and a delay term introduces significant challenges in analysis and computation. To address these, the solution is decomposed into regular and singular components, facilitating both theoretical and numerical treatment. Parameter-uniform bounds for these components and their derivatives are rigorously estab lished. A finite element framework is developed, beginning with the weak formulation of the problem and leading to a fully discretized numerical scheme. The associated bilinear form is shown to be coercive and continuous, thereby ensuring the stability and well-posedness of the method. A piecewise-uniform Shishkin mesh is employed to effectively resolve boundary and interior layers, enabling accurate approximations even in regions with steep gradients. The theoretical results are supported by numerical experiments, which confirm the parameter-uniform convergence and demonstrate the efficiency of the proposed method in capturing the complex layer behavior induced by singular perturbation and delay.
Enhancement Of Security Of Diffie-Hellman Key Exchange Protocol Using Dual Rsa Cryptography
(Department of Mathematics, NITS, 2025) Hazra, Dipanwita
The field of cryptography focuses on securing sensitive information against unauthorized access and cyber threats. One of the most widely used techniques in secure communication is the Diffie-Hellman (DH) key exchange, which enables two parties to establish a shared secret key over a public network. However, DH is susceptible to Man-in-the-Middle (MitM) attacks, where an attacker can intercept or alter the exchanged messages. To mitigate this critical vulnerability, this thesis introduces a novel cryptographic protocol: the Dual RSA-Integrated Diffie Hellman Key Exchange Protocol. The proposed method enhances the standard DH protocol by incorporating two layers of RSA encryption, providing mutual authentication and stronger defense against interception and tampering. The performance and security of the proposed protocol are compared with traditional DH and standalone RSA schemes. Results demonstrate that the integrated approach significantly improves resistance to MitM attacks while maintaining the confidentiality of key exchanges. This makes the Dual RSA-Integrated protocol a more secure and reliable solution for modern encrypted communication systems.
A mathematical model on EMHD peristaltic flow of UO2 nanofluid with microbes through a porous conduit: Application to flows in esophagus
(Department of Mathematics, NITS, 2025) Mannan, Abdul
An analytical study is presented on the bioconvective peristaltic flow of a Newtonian nanofluid within a porous, symmetric channel featuring flexible walls. The nanofluid behavior is examined under the influence of externally applied electric and magnetic fields. The model incorporates effects from motile microorganisms. The electroosmotic flow is described using the Nernst–Planck and Poisson equations. To simplify the mathematical formulation, assumptions of long wavelength and low Reynolds number are employed. The Homotopy Perturbation Method (HPM), along with the lubrication approximation, is utilized to derive analytical solutions for the reduced, non-dimensional governing equations. Graphical analysis is used to explore the impacts of various physical parameters on axial velocity, temperature, nanoparticle concentration, and microorganism density. It is observed that increasing buoyancy forces promote increased axial velocity. The Hartmann number and Joule heating parameter contribute to temperature elevation by inhibiting convective heat loss and producing internal heat. Furthermore, higher thermophoresis effects lead to greater nanoparticle accumulation, an effect magnified by Brownian motion. Increased Peclet numbers and bioconvection parameters are found to diminish microorganism density due to restricted diffusion and intensified convective disturbances. we believe that the outcomes of this study have potential applications in several areas, including bioengineering, bioreactors, enzyme processing, solar thermal systems, and energy storage technologies.
A study on q-series and partitions
(Department of Mathematics, NITS, 2025) Shit, Sabuj Baran
This Thesis explores the rich field of the q-series, highlighting how it interacts with three important mathematical theorems: Euler’s identity, the Jacobi triple product and the Pentagonal number Theorem, Ramnujan’s Partition Congruences. A foundation for comprehending the structure of q-series and their generating functions is provided by Euler’s identity. The depth of q-series is further demonstrated by the Pentagonal Number Theorem, which relates to combinatorics and integer partitions. Together, these results illustrate the profound coherence of the q− series within mathematics.
A study on the Monkeypox virus disease using the Mathematical Modelling
(Department of Mathematics, NITS, 2025) Gupta, Abhishekh
In this study, we proposed a novel mathematical model to analyze the transmission dynamics of the Monkeypox virus. The model is formulated as a system of non-linear ordinary differential equations, incorporating key epidemiological classes for both hu man and rodent populations: susceptible, exposed, infected, and recovered humans (SEIR), and susceptible and infected rodents (SI). We conducted a thorough anal ysis of the model’s equilibrium points, identifying the Monkeypox-Free Equilibrium point (MFE), the Rodent-Free Endemic Equilibrium point (RDEE), and the Endemic Equilibrium point (EE). We rigorously established the local asymptotic stability of the model at the MFE and RDEE. Furthermore, the global asymptotic stability of the endemic equilibrium was demonstrated through the construction and analysis of a suitable Lyapunov function. A sensitivity analysis was performed on the parameters comprising the basic reproduction number to determine their relative influence on dis ease transmission. Finally, the theoretical findings were complemented and illustrated through specific numerical simulations.
Numerical Analysis For Singularly Perturbed Delay Differential Equations
(Department of Mathematics, NITS, 2025) Hazarika, Pradyut
This thesis investigates a singularly perturbed delay differential equation modeled as a parabolic partial differential equation (PDE). Such problems are characterized by the presence of a small perturbation parameter and a delay term, which together induce boundary and interior layers, making both analysis and numerical approximation challenging. We begin by decomposing the solution and the differential operator into regular and singular components, enabling precise characterization of the solution structure. Rigorous bounds are derived for both components, shedding light on their respective behaviors and interactions. To approximate the solution numerically, the domain is discretized, and a weak formulation of the model problem is developed. This formulation is further employed to construct a finite element method (FEM) tailored to handle the singular perturbation and delay effects efficiently. Stability and accuracy of the FEM are established through analytical and numerical validation. The methodology ensures robustness in capturing sharp gradients and delayed responses in the solution. Overall, the thesis provides a comprehensive framework for understanding and approximating singularly perturbed delay PDEs. The results lay the groundwork for future research in higher dimensions, nonlinear models, and applications in science and engineering involving time delays and rapid transitions.
Mathematical Study on Wave Reflection in Graded Electro-elastic Waveguide
(Department of Mathematics, NITS, 2025) Parasar, Hemanga Jyoti
The incidence of quasi-P (qP) wave on a waveguide made of pre-stressed electro-elastic material with rigid boundary surface is studied in this work to determine how elastic waves are reflected. For reflected qP, quasi-SV (qSV) and electro-acoustic (EA) waves, closed form expressions for the reflection coefficients (RCs) are obtained by introducing suitable mechanical and electrical boundary conditions. Furthermore, functional grading and material rotation are taken into account as well. The dissertation consists of two chapters, outlined as follows: • In the introductory chapter, Chapter 1, important ideas related to wave reflection are defined and explained. It covers the fundamental ideas that are important to this in vestigation, such as the effects of pre-stress, electro-elastic materials, and elastic wave reflection. It also investigates how these waves behave at rigid boundary surfaces. Addi tionally, the chapter introduces the role of functionally graded materials and the influence of rotational motion on wave propagation. • Elastic wave reflection in a rotating graded electro-elastic waveguide is the primary focus of Chapter 2. It provides a thorough articulation of the problem, taking into account the complications brought out by rotational motion and functional grading. The chapter gives the analytical solution of the wave equations under the boundary conditions that correspond to a rigid surface. The RCs for the reflected waves, resulting from the inci dence of elastic waves on the rigid boundary, are derived and discussed. Furthermore, a number of particular instances are examined to establish the dependency of various physical parameters on RCs.
Unification, Dilation, and von Neumann’s Inequality
(Department of Mathematics, NITS, 2025) Roshan
In this project, we present elegant proofs of four fundamental theorems using Zabreiko’s Lemma. Subsequently, we delve into Dilation Theory and von Neumann’s Inequality, exploring their diverse applications. Inspired by [1, 2], our primary objective is to explicitly check the generalizations of Ando’s dilation for a Q-commutative setup, where Q is a unitary operator. Additionally, we would like to establish a concrete connection between von Neumann’s Inequality and distinguished varieties. We aim to investigate the Q-commutative extensions of both Dilation Theory and von Neumann’s Inequality.In a Q-commutative setup, we focus on an isometric dilation of a Q−commuting pairof contractions. The present attempt is inspired by the work done by Sibaprasad Barikand Bappa Bisai on the subject ”A Generalization of Ando’s Dilation, and Isometric Dilations for a Class of Tuples of q-Commuting Contractions” [3]. We are taking aQ−commuting pair of contractions to perform Ando’s dilation modulo some conditions.
Almost Periodic Functions and Their Applications
(Department of Mathematics, NITS, 2025) Samal, Jajati Keshari
Periodic functions are among the most frequently occurring classes of functions in both applied and analytical mathematics. Due to this wide range of applicability, we aim to extend this class of functions to enable broader analysis. The extended class we discuss is known as almost periodic functions. In this thesis, we begin by exploring the motivation for and development of almost periodic functions, which arise from the theory of periodic functions in Chapter 1. Chapter 2 presents the history of the development of almost periodic functions. In Chapter 3, we revisit some fundamental results from analysis that are useful for our proofs. Chapter 4 reviews key properties and characterizations of almost periodic functions, including Bochner’s criterion, and provides relevant counterexamples. In Chapter 5, we discuss important applications of these functions in the context of ordinary differential equations. Finally, Chapter 6 concludes the thesis with the future scope of almost periodic functions.
Quadratic Approximation based GA for Function Optimization
(Department of Mathematics, NITS, 2025) Kumar, Nitish
This thesis presents an enhanced Genetic Algorithm (GA) framework utilizing Quadratic Approximation (QA) to solve both constrained and unconstrained optimization problems more efficiently. The primary objective is to obtain better fitness values by accelerating convergence and improving solution accuracy. In the proposed approach, a local quadratic approximation is integrated into the GA to refine solutions, thus balancing global exploration and local exploitation. To evaluate the effectiveness of the proposed QA-based GA (QA-GA), extensive experiments were conducted on standard benchmark functions, including Sphere, Rastrigin, and Ackley functions, among others. The performance of the QA-GA was systematically compared with the classical GA in terms of convergence speed and optimal solution quality. Experimental results indicate that QA-GA consistently outperforms the traditional GA by converging faster and locating better global optima across various test cases. The findings confirm that the integration of quadratic approximation significantly enhances the search capability of GA, making QA-GA a robust approach for complex optimization tasks.
Mathematical modeling on particulate suspension couple stress fluid flow in a stenosed artery under electro-magnetic effects
(Department of Mathematics, NITS, 2025) Verma, Santosh Kumar
An analytical study is presented on the fluid dynamics of blood flow in stenosed arteries, with a focus on pulsatile flow, electric and magnetic fields, and the non-Newtonian nature of blood. The narrowing of arteries caused by stenosis induces significant alterations in blood flow, making it essential to explore these changes for improved disease evaluation and therapeutic strategies. A mathematical model is proposed to describe pulsatile, laminar, incompressible, and electrically conductive flow, represented by the Jeffrey fluid model with couple stress effects. This model accounts for the effects of electromagnetic forces and periodic body forces, under the assumptions of low Reynolds number, mild stenosis, and low wall zeta potential. To analyze key parameters, including axial velocity, wall shear stress, and flow resistance, the governing equations are solved using Laplace and Hankel transforms. Graphical representations of the flow characteristics are generated using MATLAB, providing insights into the complex flow dynamics in stenosed arteries. This research aims to contribute to the development of medical devices and enhance treatment methodologies for cardiovascular diseases.
A mathematical study on the impact of media awareness during the COVID-19 pandemic
(Department of Mathematics, NITS, 2025) Kumar, Sarvesh
This study introduces a novel SEQIR mathematical model in response to the global challenges posed by COVID-19, which is recognized as one of the most contagious and potentially fatal diseases worldwide, with reinfection further exacerbating its severity. The model is designed to examine the transmission dynamics of COVID-19, evaluate behavioural changes influenced by media-induced public awareness through social media platforms, and formulate effective control strategies. We deduce the disease-free and conditional multiple endemic equilibria, and determine the basic reproduction number using the next-generation matrix approach. The local stability of these equilibria is investigated with the help of the Routh–Hurwitz criterion. Furthermore, we establish global asymptotic stability by constructing appropriate Lyapunov function. A sensitivity analysis is conducted to identify the key parameters that significantly influence the basic reproduction number and future dynamics. The model further demonstrates the presence of backward bifurcation under specific parametric conditions. Finally, numerical simulations are presented to validate the effectiveness of different intervention strategies aimed at reducing COVID-19 infection and enhancing public health outcomes.
The study on S-wave propagation under non-homogeneous anisotropic incompressible initially stressed medium
(Department of Mathematics, NITS, 2025) Nigam, Akshya
In this paper, we investigate the propagation of S-waves in a non-homogeneous, anisotropic, incompressible medium subjected to initial stress. The study aims to understand how initial stresses and material anisotropy influence the behavior of shear waves under varying conditions of non-homogeneity. By developing a comprehensive mathematical model and conducting analytical and numerical analysis, we examine the intricate relationship between wave velocity, stress distribution, and material anisotropy. This research provides significant insights into the fundamental dynamics of wave propagation in complex media and has practical implications for geophysical explorations, seismic analysis, and materials science. The findings contribute to a deeper understanding of elastic wave behavior in stressed and anisotropic environments, enhancing predictive capabilities in related engineering and scientific applications. In chapters 2 and 3, we discuss about the Introduction, literature Review. In chapter 4, we describe the mathematical Formulation of the problem . In chapter 5, 6 and 7 Solution of the problem, praticular cases and numerical solution. In chapter 8, we concluded all the outcomes, and discussed the future work.
A Study on Edge Wave Propagation in an Incompressible Anisotropic Initially Stressed Plate of Finite Thickness
(Department of Mathematics, NITS, 2025) Prakash, Ved
This study investigates the propagation of edge waves in an incompressible, anisotropic, initially stressed elastic plate of finite thickness and infinite length. Using Biot’s theory of incremental deformations, a closed-form dispersion relation for edge waves is derived. The analysis accounts for the influence of material anisotropy and initial stress components. Special cases are examined to validate the model against known solutions in stress-free and isotropic conditions. Notably, it is observed that edge waves cannot propagate in a thin isotropic plate without initial stress, whereas in the limit of a thick plate, the solution approaches Rayleigh wave propagation in a half-space. Numerical simulations are performed to explore how the normalized wave velocity c β depends on various physical parameters. The key variables include the normalized initial stress components S11 2Q and S22 2Q , as well as the scaled thickness parameter H = kh, where k is the wave number and h is the plate thickness. Initially, an isotropic case is assumed with N = Q. Results show that increasing S22 2Q leads to a significant reduction in wave velocity, consistent with acoustoelastic effects observed in compressively stressed media. In contrast, variation in S11 2Q causes a more modest reduction in velocity, highlighting the directional sensitivity of the wave response. Further simulations introduce anisotropy by allowing N Q = R to vary, representing different levels of directional stiffness. For fixed H, increasing R influences the dispersion behavior, with distinct shifts in the velocity profile depending on whether the medium is stiffer or more compliant in the transverse direction. Additionally, increasing H generally leads to higher wave velocities in the isotropic case, while in anisotropic media, the trend may reverse, indicating a complex interplay between geometry, pre-stress, and material anisotropy. These findings provide new insights into edge wave dynamics under realistic mate rial and stress conditions and have implications for non-destructive evaluation, micro acoustic device design, and the analysis of wave propagation in pre-stressed elastic structures.
Study On The Fibonacci Numbers And The Golden Ratio For Function Optimization
(Department of Mathematics, NITS, 2025) Mourya, Ashish
The Fibonacci sequence and its associated Golden Ratio appear prominently in nature and various art forms. Numerous natural phenomena—such as the branching patterns of trees, the arrangement of leaves (phyllotaxis), the growth patterns of pineapples, the flower arrangements of artichokes, cross-sections of trees, and the configuration of pine cone bracts—display characteristics that correspond with Fibonacci numbers. This exploration includes various theorems and puzzles surrounding Fibonacci num bers, such as Binet’s formula, the Fibonacci Q-Matrix, Cassini’s identity, Fibonacci bamboozlement, the Golden rectangle, spiraling squares, the Golden spiral, and the Fibonacci spiral. Furthermore, Fibonacci numbers and the Golden ratio are critical in contemporary computational optimization, especially in developing uni-modal and multi-modal functions that implement the Golden Search (GS) method. Genetic Al gorithms (GA) are a popular method for function optimization, leveraging selection strategies like Tournament Selection (TS) or Roulette-Wheel Selection (RS) and em ploying crossover techniques such as Single-Point Crossover (SC) or Uniform Crossover (UC), alongside bit-wise mutation operators. This study presents an optimization anal ysis using seven benchmark functions—Sphere, Ackley, Rosenbrock, Rastrigin, Schaffer N.1, Cosine Mixture, and Schwefel—by applying both GA and GS hydridized GA. This hybrid method aims to enhance convergence accuracy and improve overall optimiza tion performance by skillfully marrying the global search abilities of GA with the local refinement offered by GS.
Numerical Study On Polar Driven Cavity Flow
(Department of Mathematics, NITS, 2025) Kumar, Ashish
In this paper, we present the mathematical formulation and numerical approach for solving the two-dimensional steady incompressible Navier–Stokes equations in stream function–vorticity form for flow inside a polar cavity bounded by concentric circular walls. The study focuses on the annular region between the inner and outer walls, where the flow is driven by motion along the boundaries. The governing equations, expressed in cylindrical coordinates, are discretized using a finite difference scheme on a fine uniform grid. Particular attention is given to the treat ment of boundary conditions and the iterative solution strategy to ensure numerical stability and accuracy at high Reynolds numbers.” In chapters 1 and 2, we discuss about the basic terminology introduction, literature survey. In chapter 3 we discuss the mathematical formulation and boundary condition. In Chapter 4 we presents a finite difference approach for solving 2D steady incompressible flow in a polar cavity using the stream function–vorticity formulation. The stream function is modeled as a Poisson equation, while vorticity is treated as a convection–diffusion equation. Numerical schemes ensure stability and accuracy across different Reynolds numbers. Chapter 5 presents the results and discussion, including numerical validation and comparison between numerical and experimental solutions. Vorticity plots are analyzed for various Reynolds numbers, along with stream function and vorticity contours to visualize flow behavior in the polar cavity.
Optimizing the Transportation Problem Using a Spherical Fuzzy programing Apporach
(Department of Mathematics, NITS, 2025) Kumar, Ravi
In a real-life problem, decision-making plays a vital role. It is always observable that uncertainty, hesitation, vagueness, etc., are involved in actual circumstances. Such factors increase the difficulty for decision makers to decide the accurate, precise, and crisp values of parameters involved in the specific problem. Unlike conventional fuzzy or intuitionistic fuzzy sets, spherical fuzzy sets provide greater flexibility by incorporating “membership, non-membership, and hesitancy degrees” simultaneously while maintaining a normalized structure. The bulk of real-world optimization issues are ambiguous, although they are frequently considered to be crisp for the sake of solving them. Many of them are multi-objective optimization problems in which multiple conflicting objectives must be satisfied while adhering to the same set of constraints. The proposed model formulates the transportation problem within the spherical fuzzy environment and employs a modified algorithm to derive optimal solutions.
Study On Electro-Osmotic Flow In Rectangular Microchannel
(Department of Mathematics, NITS, 2025) Mazumdar, Kamal
This article investigates the combined impacts of electro-osmosis as well as pressure-driven flow under an applied electric field in a rectangular nano channel which is filled an electrolyte solution. The Poisson-Boltzmann equation provides the distribution of the induced potential using the Debye-H¨uckel approximation, while the Navier-Stokes equation provides the distribution of fluid velocity in one dimension. In this study, analytical solutions of the velocity and induced potential of fluid in rectangular microchannel. The fluid flow relies on many variables such as channel height, direction of pressure gradient, electrolyte strength, zeta potential at the channel edge, etc. The axial velocity produces a parabolic shape profile while the electrolyte concentration is low, but a plug-like form profile as the electrolyte concentration increases. The axial velocity rises with electrolyte concentration, reaching constant values at high electrolyte concentrations.
Study On Electrokinetic Flow Through Polyelectrolyte Layer Coated Cylindrical Nanochannel
(Department of Mathematics, NITS, 2025) Kalal, Kaushal Kumar
This article presents a theoretical investigation into the combined effects of electroosmotic and pressure-driven flow of fractional Maxwell fluids within a cylindrical nanochannel filled with an electrolyte solution. The channel is coated with a polyelectrolyte layer (PEL), and the system is subjected to a low-frequency AC electric field. The model incorporates surface charge, fixed charges within the PEL, dielectric permittivity contrasts, and ion partitioning due to born energy differences at the PEL–electrolyte interface. The induced electrostatic potential distribution is obtained by solving the linearized Poisson–Boltzmann equation using the De bye–H¨uckel approximation, yielding analytical solutions in terms of modified Bessel functions. The fluid transport behavior is influenced by parameters such as channel geometry, pressure gradient, electrolyte concentration, and zeta potential, offering valuable insights for nanofluidic device and biosensor design.
A Study On p-adic Analysis
(Department of Mathematics, NITS, 2025) Devjibhai, Rathod Gautam
This dissertation explores foundational and advanced topics in p-adic analysis, a field that extends classical analysis into the realm of p-adic numbers, which are constructed through the p-adic valuation and associated metric. The non-Archimedean nature of p-adic fields leads to unique analytical and topological behaviours that differ funda mentally from real and complex analysis. In this report, we have studied the p-adic systems and their relevance to broader mathematical structures. This report focuses on the p-adic numbers and some of the properties of them. We start by defining a norm and the p-adic absolute value and studied that it is a norm on the rational numbers. We also studied that the p-adic norm is a non-Archimedean norm which has strange properties compared to the ordinary (Archimedean) norm. Then we define the p-adic numbers and discuss that it is a complete field with respect to the p-adic norm.

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