Mathematics
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Mathematics
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Item A Sixth Order Implicit Hybrid Backward Differentiation Formulae (HBDF) for Block Solution of Ordinary Differential Equations(American Journal of Mathematics and Statistics, 2012) Muhammad R; Yahaya.Y.AThe Hybrid Backward Differentiation Formula (HBDF) for case K=5 was reformulated into continuous form using the idea of multistep collocation. Multistep Collocation is a continuous finite difference (CFD) approximation method by the idea of interpolation and collocation. The hybrid 5-step Backward Differentiation Formula (BDF) and additional methods of order (6,6,6,6,6,)𝑇𝑇 were obtained from the same continuous scheme and assembled into a block matrix equation which was applied to provide the solutions of IVPs over non-overlapping intervals.The continous form was im-mediately employed as block methods for direct solution of Ordinary Differential Equation (𝑦𝑦′=𝑓𝑓(𝑥𝑥,𝑦𝑦)). Some benefits of this study are, the proposed block methods will be self starting and does not call for special predictor to estimate 𝑦𝑦’ in the integrators and all the discrete methods obtained will be evaluated from a single continuous formula and its derivatives at various grids and off grid points. These study results help to speed up computation, also the requirement of a starting value and the overlap of solution model which are normally associated with conventional Linear Multistep Methods were elimi-nated by this approach. In conclusion, a convergence analysis of the derived hybrid schemes to establish their effectiveness and reliability was presented. Numerical example carried out on stiff problem further substantiates their performance.Item Approximate Solution of SIR Infectious Disease Model Using Homotopy Pertubation Method (HPM).(Pacific Journal of Science and Technology, 2013-11-20) Abubakar, Samuel; Akinwande, N. I.; Jimoh, O. R.; Oguntolu, F. A.; Ogwumu, O. D.In this paper we proposed a SIR model for general infectious disease dynamics. The analytical solution is obtained using the Homotopy Perturbation Method (HPM). We used the MATLAB computer software package to obtain the graphical profiles of the three compartments while varying some salient parameters. The analysis revealed that the efforts at eradication or reduction of disease prevalence must always match or even supersede the infection rate.Item DERIVATION AND ANALYSIS OF BLOCK IMPLICIT HYBRID BACKWARD DIFFERENTIATION FORMULAE FOR STIFF PROBLEMS(Nigerian Journal of Mathematics and Applications, 2014) MUHAMMAD, R.; YAHAYA, Y. A.; IDRIS L.The Hybrid Backward Di erentiation Formula (HBDF) for the case k = 3 was reformulated into continuous form using the idea of multistep collocation. The continuous form was evaluated at some grid and o grid points which gave rise to discrete schemes employed as block methods for direct solution of rst order Ordinary Di erential Equation y0 = f(x; y). The requirement of a starting value and the overlap of solution model which are associated with conventional Linear Multistep Methods were eliminated by this approach. A convergence analysis of the derived hybrid schemes to establish their e ec- tiveness and reliability is presented. Numerical example carried out on sti problem further substantiates their performance.Item Formulation Of A Standard Runge- Kutta Type Method For The Solution First And Second Order Initial Value Problems(Researchjournali’s Journal of Mathematics, 2015-03) Muhammad R.; Y. A Yahaya; A.S AbdulkareemIn this paper, we present a standard Runge-Kutta Type Method (RKTM) for . The process produces Backward Differentiation Formula (BDF) scheme and its hybrid form which combined together to form a block method. The method is reformulated into a Runge-Kutta Type of the same step number for the solution of first and second order (special or general) initial value problem of Ordinary Differential Equation (ODE).Item Stability Analysis of Disease Free Equilibrium (DFE) State of a Mathematical Model of Yellow Fever Incorporating Secondary Host(Pacific Journal of Science and Technology, 2017-11-20) Somma, Samuel Abu; Akinwande, N. I.; Jiya, M.; Abdulrahaman, S.In this paper we formulate a mathematical model of yellow fever incorporating secondary host. We obtained the Disease Free Equilibrium (DFE) Points and compute the basic reproduction number. The local and global stability of the DFE was analyzed using Jacobian Matrix stability techniques and Lyapunov function respectively. The local and global stability was asymptotically stable if 1 0 R and 1 0 R , respectively. The basic reproduction number and control parameters of the model were presented graphically.Item Semi-Analytical Solution for the Mathematical Modeling of Yellow Fever Dynamics Incorporating Secondary Host(Communication in Mathematical Modeling and Applications, 2019-04-15) Somma, Samuel Abu; Akinwande, N. I.; Abah, R. T.,; Oguntolu, F. A.; Ayegbusi, F. D.In this paper we use Differential Transformation Method (DTM) to solve the mathematical modeling of yellow fever dynamics incorporating secondary host. The DTM numerical solution was compared with the MAPLE RungeKutta 4-th order. The variable and parameter values used for analytical solution were estimated from the data obtained from World Health Organization (WHO) and UNICEF. The results obtained are in good agreement with Runge-Kutta. The solution was also presented graphically and gives better understanding of the model. The graphical solution showed that vaccination rate and recovery rate play a vital role in eradicating the yellow fever in a community.Item THE ORDER AND ERROR CONSTANT OF A RUNGE-KUTTA TYPE METHOD FOR THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM(Federal University Dutsin MA Journal of Sciences (FJS), 2020-06) Muhammad RIn this paper, we examine in details how to obtain the order, error constant, consistency and convergence of a Runge-Kutta Type method (RKTM) when the step number 𝑘 = 2. Analysis of the order, error constant, consistency and convergence will help in determining an effective Runge- Kutta Method (RKM) to use. Due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation.Item REFINEMENT OF PRECONDITIONED OVERRELAXATION ALGORITHM FOR SOLUTION OF THE LINEAR ALGEBRAIC SYSTEM 𝑨𝒙=𝒃(Faculty of Science, Kaduna State University, 2021) Ramatu Abdullahi; Raihanatu MuhammadIn this paper, a refinement of preconditioned successive overrelaxation method for solving the linear system 𝐵𝑥=𝑐 is considered. The coefficient matrix 𝐵∈𝑅𝑛,𝑛 is a nonsingular real matrix, 𝑐∈𝑅𝑛 and 𝑥 is the vector of unknowns. Based on the usual splitting of the coefficient matrix 𝐵 as 𝐵=𝐷−𝐿𝐵−𝑈𝐵, the linear system is expressed as 𝐴𝑥=𝑏 or (𝐼−𝐿−𝑈)𝑥=𝑏; where 𝐿=𝐷−1𝐿𝐵, 𝑈=𝐷−1𝑈𝐵 and 𝑏=𝐷−1𝑐. This system is further preconditioned with a preconditioner of the type 𝑃=𝐼+𝑆 as 𝐴̅𝑥=𝑏̅ or (𝐷̅−𝐿̅−𝑈̅)𝑥=𝑏̅. A refinement of the resulting preconditioned successive overrelaxation (SOR) method is performed. Convergence of the resulting refinement of preconditioned SOR iteration is established and numerical experiments undertaken to demonstrate the effectiveness and efficiency of the method. Results comparison revealed that the refinement of SOR method converges faster than the preconditioned as well as the classical SOR methodItem Application of Adomian Decomposition Method (ADM) for solving Mathematical Model of Measles(National Mathematical Center, 2021-03-22) Abdurrahman, Nurat Olamide; Somma S. A.; Ayegbusi F. D.; Gana P.; Adama P. W.; Yisa E. M.Item A 3-step block hybrid backward differentiation formulae (bhbdf) for the solution of general second order ordinary differential equation(New Trends in Mathematical Sciences, 2021-07-12) Hussaini Alhassan; Muhammad RaihanatuIn this paper, the block hybrid Backward Differentiation formulae (BHBDF) for the step number k=3 is developed using power series as basis function for the solution of general second order ordinary differential equation. The idea of interpolation and collocation of the power series at some selected grid and off- grid points gave rise to continuous schemes which were further evaluated at those points to produce discrete schemes combined together to form block methods. Numerical problems were solved with the proposed methods and were found to perform effectively.Item A Backward Diffrention Formula For Third-Order Inttial or Boundary Values Problems Using Collocation Method(Islamic Azad University,Rasht ', Iran, 2021-09-19) AbdGafar Tunde Tiamiyu; Abosede Temilade Cole; Khadeejah James AuduWe propose a new self-starting sixth-order hybrid block linear multistep method using backward differentiation formula for direct solution of third-order differential equations with either initial conditions or boundary conditions. The method used collocation and interpolation techniques with three off-step points and five-step points, choosing power series as the basis function. The convergence of the method is established, and three numerical experiments of initial and boundary value problems are used to demonstrate the efficiency of the proposed method. The numerical results in Tables and Figures show the efficiency of the method. Furthermore, the numerical method outperformed the results from existing literature in terms of accuracy as evident in the results of absolute errors producedItem Convergence of Triple Accelerated Over-Relaxation (TAOR) Method for M-Matrix Linear Systems(Islamic Azad University, Rasht, Iran, 2021-09-19) Khadeejah James Audu; Yusuph Amuda Yahaya; Rufus Kayode Adeboye; Usman Yusuf AbubakarIn this paper, we propose some necessary conditions for convergence of Triple Accelerated Over-Relaxation (TAOR) method with respect to 𝑀 − coefficient matrices. The theoretical approach for the proofs is analyzed through standard procedures in the literature. Some numerical experiments are performed to show the efficiency of our approach, and the results obtained compared favourably with those obtained through the existing methods in terms of spectral radius of their iteration matricesItem Application of Grey-Markov Model for Forecasting Nigeria Annual Rice Production(African Journal Online (AJOL), South Africa, 2021-11-21) Lawal Adamu; Didigwu, N. E.; Saidu, D. Y; Sadiq, S. L.; Khadeejah James AuduIn this paper, Grey system model (GM(1,1)) and Grey-Markov model that forecast Nigeria annual Rice production have been presented. The data used in the research were collected from the archive of Central Bank of Nigeria for a period of Six years (2010-2015). The fitted models showed high level of accuracy. Hence, the models can be used for food security plans of the nation.Item Application of Hidden Markov Model in Yam Yield Forecasting.(African Journal Online (AJOL), Soutrh Africa, 2022-06-06) 11. Lawal Adamu; Saidu Daudu Yakubu; Didigwu Ndidiamaka Edith; Abdullahi Abubakar; Khadeejah James Audu; Isaac Adaji.Providing the government and farmers with reliable and dependable information about crop yields before each growing season begins is the thrust of this research. A four-state stochastic model was formulated using the principle of Markov, each state of the model has three possible observations. The model is designed to make a forecast of yam yield in the next and subsequent growing seasons given the yam yield in the present growing season. The parameters of the model were estimated from the yam yield data of Niger state, Nigeria for the period of sixteen years(2001-2016). After which, the model was trained using Baum-Welch algorithm to attend maximum likelihood. A short time validity test conduct on the model showed good performance. Both the validity test and the future forecast shows prevalence of High yam yield, this attest to the reality on the ground, that Niger State is one of the largest producers of yam in Nigeria. The general performance of the model, showed that it is reliable therefore, the results from the model could serve as a guide to the yam farmers and the government to plan strategies for high yam production in the region.Item A 3-Person Non-Zero-Sum Game for Sachet Water Companies(Asian Research Journal of Mathematics, 2022-06-24) 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 Comparison of Refinement Accelerated Relaxation Iterative Techniques and Conjugate Gradient Technique for Linear Systems(Mathematical Association of Nigeria (MAN), 2022-10-05) 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 equations.Item Mathematical model of COVID-19 transmission dynamics incorporating booster vaccine program and environmental contamination(2022-11-12) Akinwande, N. I.; Ashezua, T. T.; Gweryina, R. I.; Somma, Samuel Abu; Oguntolu, F. A.; Usman, A.COVID-19 is one of the greatest human global health challenges that causes economic meltdown of many nations. In this study, we develop an SIR-type model which captures both human-to-human and environment-to-human-to-environment transmissions that allows the recruitment of corona viruses in the environment in the midst of booster vaccine program. Theoretically, we prove some basic properties of the full model as well as investigate the existence of SARS-CoV-2-free and endemic equilibria. The SARS-CoV-2-free equilibrium for the special case, where the constant inflow of corona virus into the environment by any other means, Ωis suspended (Ω =0)is globally asymptotically stable when the effective reproduction number 𝑅0𝑐<1and unstable if otherwise. Whereas in the presence of free-living Corona viruses in the environment (Ω >0), the endemic equilibrium using the centremanifold theory is shown to be stable globally whenever 𝑅0𝑐>1. The model is extended into optimal control system and analyzed analytically using Pontryagin’s Maximum Principle. Results from the optimal control simulations show that strategy E for implementing the public health advocacy, booster vaccine program, treatment of isolated people and disinfecting or fumigating of surfaces and dead bodies before burial is the most effective control intervention for mitigating the spread of Corona virus. Importantly, based on the available data used, the study also revealed that if at least 70%of the constituents followed the aforementioned public health policies, then herd immunity could be achieved for COVID-19 pandemic in the community.Item BLOCK METHOD APPROACH FOR COMPUTATION OF ERRORS OF SOME ADAMS CLASS OF METHODS(Association of Nigerian Journal of Physics, 2022-12-12) Yahaya, Y. A.; Odeyemi, A. O.; Khadeejah James AuduTraditionally, the error and order constant of block linear multistep methods were analyzed by examining each block members separately. This paper proposes a block-by-block analysis of the schemes as they appear for implementation. Specifically, cases when k= 2, 3, 4, and 5 for Adams Moulton (implicit) are reformulated as continuous schemes in order to generate a sufficient number of schemes required for the methods to be self-starting. The derivation was accomplished through the continuous collocation technique utilizing power series as the basis function, and the property of order and error constants is examined across the entire block for each case of the considered step number. The findings of the study generated error constants in block form for Adams Bashforth and Adams Moulton procedures at steps 2, 3, 4, 5 k . Furthermore, the relevance of the study demonstrates that calculating all members' error constants at once, reduces the amount of time necessary to run the analysis. The new approach, for examining the order and error constants of a block linear multistep method, is highly recommended for application in solving real-world problems, modelled as ordinary and partial differential equationsItem An Accelerated Iterative Technique: Third Refinement of Gauss–Seidel Algorithm for Linear Systems(Multidisciplinary Digital Publishing Institute, Switszerland, 2023-04-28) Khaddejah James Audu; James Nkereuwem EssienObtaining an approximation for the majority of sparse linear systems found in engineering and applied sciences requires efficient iteration approaches. Solving such linear systems using iterative techniques is possible, but the number of iterations is high. To acquire approximate solutions with rapid convergence, the need arises to redesign or make changes to the current approaches. In this study, a modified approach, termed the “third refinement” of the Gauss-Seidel algorithm, for solving linear systems is proposed. The primary objective of this research is to optimize for convergence speed by reducing the number of iterations and the spectral radius. Decomposing the coefficient matrix using a standard splitting strategy and performing an interpolation operation on the resulting simpler matrices led to the development of the proposed method. We investigated and established the convergence of the proposed accelerated technique for some classes of matrices. The efficiency of the proposed technique was examined numerically, and the findings revealed a substantial enhancement over its previous modificationsItem Computational Algorithm for Volterra Integral Solutions via Variational. Iterative Method(Paper Presentation at University of Lagos, Nigeria, 2023-08-28) Khadeejah James AuduThe Volterra Integral Equations (VIE) are a class of mathematical equations that find applications in various fields, including physics, engineering, and biology. Solving VIEs analytically is often challenging, and researchers have turned to numerical methods for obtaining approximate solutions. In this research, we propose a computational algorithm based on the Variational Iterative Method (VIM) to efficiently and accurately solve VIEs. By incorporating this method into the computational algorithm, we aim to improve the accuracy and convergence rate of the solutions. The performance of our algorithm was evaluated through extensive numerical experiments on various types of VIEs. The results demonstrate the effectiveness of the VIM approach in terms of accuracy, convergence rate, and computational efficiency. In conclusion, the proposed computational algorithm based on VIM presents a valuable contribution to the field of solving VIEs. It offers an efficient and accurate approach for obtaining approximate solutions, enabling researchers and practitioners to tackle complex problems that rely on VIEs. The algorithm's versatility and robustness make it a promising tool for a wide range of applications, including physics, engineering, and biology.