Mathematics
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Mathematics
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Item Comparison of Refinement Accelerated Relaxation Iterative Techniques and Conjugate Gradient Technique for Linear Systems.(Mathematical Association of Nigeria (MAN), 2023-08-20) Khaddejah James AuduIterative methods use consecutive approximations to get more accurate results. A comparison of three iterative approaches to solving linear systems of this type 𝑀𝑦=𝐵 is provided in this paper. We surveyed the Refinement Accelerated Relaxation technique, Refinement Extended Accelerated Relaxation technique, and Conjugate Gradient technique, and demonstrated algorithms for each of these approaches in order to get to the solutions more quickly. The algorithms are then transformed into the Python language and used as iterative methods to solve these linear systems. Some numerical investigations were carried out to examine and compare their convergence speeds. Based on performance metrics such as convergence time, number of iterations required to converge, storage, and accuracy, the research demonstrates that the conjugate gradient method is superior to other approaches, and it is important to highlight that the conjugate gradient technique is not stationary. These methods can help in situations that are similar to finite differences, finite element methods for solving partial differential equations, circuit and structural analysis. Based on the results of this study, iteration techniques will be used to help analysts understand systems of linear algebraic equationsItem A groundwater-based irrigation modeling system that optimizing water use efficiency and ensuring long-term sustainability of groundwater resources.(Maths Model Research Group, FUT, Minna, Nigeria, 2025-03-20) Y. Y. Alheri; N. Nyor; Khadeejah James AuduItem Numerical Assessment of Some Almost Runge-Kutta and Runge-Kutta Methods for First- Order Differential Equation(Maths Model Research Group. FUT, Minna, Nigeria, 2025-02-20) Khadeejah James Audu; Muideen Taiwo Kharashi; Yusuph Amuda Yahaya; James Nkereuwem Essien; Abraham Ajeolu OluwasegunNumerical methods play a critical role in solving first-order Ordinary Differential Equations (ODEs), with their efficiency and accuracy being key considerations. This study conducts a detailed comparative analysis of four numerical schemes: the Almost Runge-Kutta fourth-order scheme (ARK4), the Almost Runge-Kutta third-order fourth-stage scheme (ARK34), the classical Runge- Kutta fourth-order scheme (RK4), and the Runge-Kutta fourth-order fifth-stage scheme (RK45). The methods are evaluated based on their computational accuracy, error behavior, and efficiency. Numerical experiments reveal that all methods deliver highly accurate solutions, with ARK4 emerging as the most effective due to its lower computational complexity. ARK4 demonstrates superior performance in achieving minimal absolute error with reduced computational effort, making it a suitable choice for solving first-order ODEs. This study highlights ARK4 as a viable alternative to conventional Runge-Kutta methods for practical applications.Item Stability and Bifurcation Analysis of a Mathematical Modeling of Measles Incorporating Vitamin a Supplement(Sule Lamido University Journal of Science and Technology (SLUJST), 2021-01-20) Somma, Samuel Abu; Akinwande, N. I.; Gana, P.; Ogwumu, O. D.; Ashezua, T. T.; Eguda, F. YMeasles is transmissible disease that is common among children. The death caused by measles among children of five years and below is alarming in spite of the safe and effective vaccine. It has been discovered that Vitamin A Deficiency (VAD) in children increases their chances of measles infection. In this paper, the mathematical model of measles incorporating Vitamin A supplement as treatment was formulated and analyzed. The equilibrium points are obtained and analyzed for stability. Bifurcation and sensitivity analyses were carried out to gain further insight into the spread and control of measles. The stability analysis revealed that Disease Free Equilibrium (DFE) is stable if R0 1. The bifurcation analysis revealed forward bifurcation while the sensitivity analysis shows the most sensitive parameters of the model that are responsible for the spread and control of the diseases. The effect of sensitive parameters on Basic Reproduction Number, 0 R were presented graphically. Vaccination, recovery and Vitamin A supplement rates have been shown from the graphical presentation as the important parameter that will eradicate the measles from the population while contact and loss of immunity rates have shown that measles will persist in the population. People should be sensitized on the danger of living with infected persons. Government should do more in routine immunization and administration of Vitamin A Supplement.Item A 3-Person Non-Zero-Sum Game for Sachet Water Companies(Asian Research Journal of Mathematics, 2022-08-12) Nyor, N.; Muazu, M. I.; Somma, Samuel AbuThe business of Sachet water (popularly called pure water) in Nigeria is often competitive due to the high demand for Sachet water by the populace. This is so because sachet water is the most affordable form of pure drinking water in Nigeria. As such, Sachet Water Firms that want to succeed in an ever increasing competitive market need to have the knowledge of Game Theory to identify which strategy will yield better profit independent of the strategy adopted by other competitors. This paper is aimed to investigate and determine the equilibrium point for three Sachet Water Firms using the Nash Equilibrium Method as it provides a systematic approach for deciding the best strategy in competitive situation. The result showed two Nash Equilibriums (promo, promo) and (stay-put, stay-put) with their respective payoffs of (82; 82; 82) and (147; 147; 147).Item Modelling the Impacts of Media Campaign and Double Dose Vaccination in Controlling COVID-19 in Nigeria(Alexandria Engineering Journal, 2023-01-15) Akinwande, N. I.; Somma, Samuel Abu; Olayiwola, R. O.; Ashezua, T. T.; Gweryina, R. I.; Oguntolu, F. A.Corona virus disease (COVID-19) is a lethal disease that poses public health challenge in both developed and developing countries of the world. Owing to the recent ongoing clinical use of COVID-19 vaccines and noncompliance to COVID-19 health protocols, this study presents a deterministic model with an optimal control problem for assessing the community-level impact of media campaign and double-dose vaccination on the transmission and control of COVID-19. Detailed analysis of the model shows that, using the Lyapunov function theory and the theory of centre manifold, the dynamics of the model is determined essentially by the control reproduction number (𝑅𝑚𝑣). Consequently, the model undergoes the phenomenon of forward bifurcation in the absence of the double dose vaccination effects, where the global disease-free equilibrium is obtained whenever 𝑅𝑚𝑣 ≤ 1. Numerical simulations of the model using data relevant to the transmission dynamics of the disease in Nigeria, show that, certain values of the basic reproduction number ((𝑅0 ≥ 7)) may not prevent the spread of the pandemic even if 100% media compliance is achieved. Nevertheless, with assumed 75% (at 𝑅0 = 4)) media efficacy of double dose vaccination, the community herd immunity to the disease can be attained. Furthermore, Pontryagin’s maximum principle was used for the analysis of the optimized model by which necessary conditions for optimal controls were obtained. In addition, the optimal simulation results reveal that, for situations where the cost of implementing the controls (media campaign and double dose vaccination) considered in this study is low, allocating resources to media campaign-only strategy is more effective than allocating them to a firstdose vaccination strategy. More so, as expected, the combined media campaign-double dose vaccination strategy yields a higher population-level impact than the media campaign-only strategy, double-dose vaccination strategy or media campaign-first dose vaccination strategy.Item 4-Step Block Hybrid Backward Differentiation Formula For Solving Second Order (BHBDF II) Ordinary Differential Equations(2024) Hussaini Hajarat; Muhammad Raihanatu; Yusuf AbdulhakeemThis research work presents the derivation and implementation of a 4-step linear multistep method of block hybrid backward differentiation formula for solving nonlinear second-order initial value problems of ordinary differential equations. Collocation and interpolation methods are adopted in the derivation of the proposed numerical scheme where the legendary polynomial is adopted as a basic function. The 4-step BHBDF has higher order of accuracy p = 11 which implies that it is consistent. The proposed numerical block method is further applied to finding direct solution to nonlinear second order ordinary differentiation equations. This implementation strategy is more accurate than some existing methods considered in the literature.Item An Investigative Comparison of Higher-Order Runge-Kutta Techniques for Resolving First-Order Differential Equations(Fırat University, Turkey, 2025-07-14) Khadeejah James Audu; Victor James Udoh; Jamiu GarbaIn the context of solving first-order ordinary differential equations (ODEs), this paper thoroughly compares various higher-order Runge-Kutta methods. Reviewing the effectiveness, precision, and practicality of several Runge-Kutta schemes and highlighting their usage in numerical approximation is the main goal of the research. The study explores traditional approaches, including the fifth-order, six-stage Runge-Kutta (RK56), the sixth-order, seven-stage Runge-Kutta (RK67), and the seventh-order, nine-stage Runge-Kutta (RK79), with the goal of offering a comprehensive comprehension of their individual advantages and disadvantages. In order to help academics and practitioners choose the best approach based on the features of the problem, comparative benchmarks are constructed, utilizing both theoretical underpinnings and real-world implementations. Robustness evaluations and sensitivity analysis complement the comparison research by illuminating how flexible these techniques are in various context. The results of this study provide important new understandings of how higherorder Runge-Kutta methods function and provide a thorough manual for applying them to solve first-order differential problems in a variety of scientific and engineering fields. The study’s examination of three higher order Runge-Kutta algorithms reveals that the RK56 is more effective at solving first order ODEsItem Application of Backward Differentiation Formula on Fourth-Order Differential Equations.(Universiti Tun Hussein Onn Malaysia Publisher’s Office, Malaysia, 2022-12-30) Khadeejah James Audu; Jamiu Garba; Abdgafar Tunde Tiamiyu; Blessing Ashiodime ThomasHigher order ordinary differential equations are typically encountered in engineering, physical science, biological sciences, and numerous other fields. The analytical solution of the majority of engineering problems involving higher-order ordinary differential equations is not a simple task. Various numerical techniques have been proposed for higher-order initial value problems (IVP), but a higher degree of precision is still required. In this paper, we propose a novel two-step backward differentiation formula in the class of linear multistep schemes with a higher order of accuracy for solving ordinary differential equations of the fourth order. The proposed method was created by combining interpolation and collocation techniques with the use of power series as the basis function at some grid and off-grid locations to generate a hybrid continuous two-step technique. The method's fundamental properties, such as order, zero stability, error constant, consistency, and convergence, were explored, and the analysis showed that it is zero stable, consistent and convergent. The developed method is suitable for numerically integrating linear and nonlinear differential equations of the fourth order. Four Numerical tests are presented to demonstrate the efficiency and accuracy of the proposed scheme in comparison to some existing block methods. Based on what has been observed, the numerical results indicate that the proposed scheme is a superior method for estimating fourth-order problems than the method previously employed, confirming its convergence.Item Extended accelerated overrelaxation iteration techniques in solving heat distribution problems(Price of Songkla University, Thailand, 2023-10-05) Khadeejah James AuduSeveral stationary iteration techniques for numerical solutions to special systems of linear equation systems of the form 𝒜𝑢 = 𝑏 have been studied in an attempt to improve their convergence, suitability, and strength. Among such techniques is the Extended Accelerated Overrelaxation (EAOR) iterative scheme. In this paper, we studied the basics of the EAOR methods and applied them to compute the solution of a real-life problem, resolving the heat equation when a steady temperature is applied to a metal plate. We show how the real-life problem can be modeled into a partial differential equation, followed by discretization through the use of finite differences, and finally generating a largely sparse system of algebraic linear equations, from which the unknowns are to be solved. The techniques were compared with the Refinement of Accelerated Overrelaxation (RAOR) iterative scheme. The outcome of the numerical tests proves the effectiveness of the EAOR schemes for such problems.