DEPARTMENT OF MATHEMATICS

Linearized water wave theory is a fundamental concept in fluid dynamics that has been extensively used to study wave propagation in various aquatic environments. Water waves play a crucial role in many engineering and scientific applications, including ocean and coastal engineering, ship hydrodynamics, and offshore engineering. However, the complexity of nonlinear wave dynamics has limited the accuracy of traditional numerical models, emphasizing the need for a simplified yet robust approach. Linearized water wave theory offers a promising solution by assuming small-amplitude waves, enabling the simplification of the governing equations and providing an efficient tool for wave analysis. This project explores the mathematical and physical principles underlying linearized water wave theory and its application in various fields such as oceanography, coastal engineering and naval architecture. The study begins with an overview of the basic equations governing water wave motion including the linearized Euler equation and boundary conditions. The dispersion equation which relates the wave frequency to its wavenumber is derived and analysed to properly understand wave propagation characteristics. In this study, we developed and applied linearized water wave theory to investigate wave propagation in a simplified fluid domain. We also discretized the linearized Navier-Stokes equations and then introduced a wave-like solution to represent the small-amplitude waves. By substituting this solution into the linearized equations, we obtained a set of ordinary differential equations that describe the wave propagation characteristics. Through mathematical analysis and numerical simulations, this study aims to provide a comprehensive understanding of linearized water wave theory and its applications in fluid dynamics. The applications of this study are diverse and far-reaching. Our results can be used to improve the design and optimization of various aquatic structures, such as seawalls, breakwaters, and offshore platforms, by providing a better understanding of wave-structure interactions. Additionally, our findings can be applied to enhance the accuracy of wave forecasting models, which are crucial for coastal erosion prediction, ship navigation, and offshore operations. Furthermore, the linearized water wave theory can be extended to study more complex wave phenomena, such as wave-current interactions and wave-induced sediment transport, offering a promising avenue for future research.

Effective management of financial flows is essential for sustaining liquidity stability and operational efficiency within multi-tier supply chain networks. Traditional optimisation models
tend to prioritise either cost reduction or risk mitigation, often neglecting the balance between
working capital efficiency and financial stability. This study proposes a Hybrid Mathematical Framework that integrates dynamic discounting mechanisms and risk mitigation strategies to optimise financial flows across supply chains. The framework addresses four primary objectives: developing a financial flow optimisation model, incorporating dynamic discounting into the
model, embedding stochastic variables representing demand volatility and credit risk, and
evaluating model performance through numerical simulations. A quantitative modelling approach is employed, formulating an objective function that minimises total financial costs while controlling for risk using Conditional Value at Risk (CVaR). The model integrates early payment incentives, late payment penalties, and financial risk thresholds to support strategic decision-making. Numerical simulations using synthetic financial data were conducted to assess the model’s performance. Results indicate that the Hybrid Model offers a superior trade-off between cost efficiency and financial stability. Dynamic discounting reduces total financial costs, while CVaR integration ensures liquidity remains risk-sensitive. The study recommends adopting dynamic discounting with risk-sensitive optimisation models and exploring technologies like real-time analytics and AI. Future research could refine the framework via industry-specific adaptations and block chain enabled contracts.

Brief history, origin and relevant roles of some numerical methods in the solution of the Hamiltonian Differential Equation with the help of some definitions and theorem. Some important results, are incorporated in this work, extending, specifically the Runge-Kutta Methods. We study the application of Runge-Kutta schemes to Hamiltonian systems. Basic principles are illustrated by means of examples. This work has been selected carefully so that the work is useful for study in this area of research. Particularly, a survey of the effectiveness of the Runge-kutta Methods. The numerical methods developed are primarily intended for use with Hamiltonian systems, but many find uses in solving other forms of ordinary differential equations. Almost all the real conservative physical processes can be cast in suitable Hamiltonian formulation in phase spaces with symplectic structure, which has the advantages to make the intrinsic properties and symmetries of the underlying processes more explicit than in other mathematically equivalent formulations, so I choose the Hamiltonian formalism as the basis, together with the mathematical and physical motivations of our symplectic approach for the purpose of numerical simulation of dynamical evolutions.

Legendre polynomial is a second order ordinary differential equation as well as a type of Fourier Series written in the system of orthogonal polynomials with a vast number of mathematical properties and numerous applications. It is obtained through linear differential equation methods based on Sturm-Liouville theory.

Machine learning has emerged as a transformative technology with a profound impact on various industries. In this course of study, this abstract provides an overview of ML, its significance, limitations, types of ML we have. Machine Learning: This refers to the exploration of computer programs that utilize algorithm and statistical model to acquire knowledge by identifying patterns and making inferences all without explicit programming. Some of the significance of Machine Learning includes its contribution to technological advancement in various industries, it equips individual with skills to automate processes and streamline operations and it also enables automation of repetitive tasks which enhance efficiency and productivity. There are challenges or limitations associated with ML, some of which includes: Data dependency, Interpretability and Transparency, Overfitting and Generalization, Domain-specific expertise and Lack of casual understanding. The types of Machine Learning are: Supervised Learning, Unsupervised Learning, Semi-supervised Learning and Reinforcement Learning.

Machine learning has emerged as a transformative technology with a profound impact on various industries. In this course of study, this abstract provides an overview of ML, its significance, limitations, types of ML we have. Machine Learning: This refers to the exploration of computer programs that utilize algorithm and statistical model to acquire knowledge by identifying patterns and making inferences all without explicit programming. Some of the significance of Machine Learning includes its contribution to technological advancement in various industries, it equips individual with skills to automate processes and streamline operations and it also enables automation of repetitive tasks which enhance efficiency and productivity. There are challenges or limitations associated with ML, some of which includes: Data dependency, Interpretability and Transparency, Overfitting and Generalization, Domain-specific expertise and Lack of casual understanding. The types of Machine Learning are: Supervised Learning, Unsupervised Learning, Semi-supervised Learning and Reinforcement Learning.

ABSTRACT
Numerical schemes for the integration of stiff initial value problems are required to
possess wide region of absolute stability which include the entire left of the complex
plane. Numerical schemes that are explicit usually do not attain the requirement for
integration of stiff initial value problems. In this study, implicit second derivative RungeKutta methods are constructed for the integration of stiff initial value problems.
A family of generalised second derivative mono-implicit Runge-Kutta (GSDMIRK)
method is derived using the method of Tailor series expansion.
The proposed GSDMIRK methods are 𝐴-stable for stage 𝑠=3 𝑎𝑛𝑑 4 and 𝐴(∝)-stable for
𝑠=5 𝑎𝑛𝑑 6. Numerical experiments show that the GSDMIRK methods perform better
when compared to some numerical algorithms in the literature.