Mathematics

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Mathematics

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A 4-Step Order (K + 1) Block Hybrid Backward Differentiation Formulae (BHBDF) for the Solution of General Second Order Ordinary Differential Equations
(2023-12) Muhammad R; Hussaini A
In this paper, the block hybrid backward differentiation formulae (BHBDF) for the step number 𝑘 = 4 was 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.
A Comparative Study Of Two Iterative Techniques For Systems Of Linear Algebraic Equations
(Academic Staff Union of Universities, Nigeria, 2021-12-20) Khadeejah James Audu
This study compares numerically two iterative methods for solving systems of linear algebraic equations: the Symmetric Accelerated Overrelaxation technique and the Symmetric Successive Overrelaxation method. Four numerical problems are applied to analyze and compare the convergence speeds of the two approaches. On the basis of performance metrics including spectral radius, convergence time, accuracy, and number of iterations required to converge, the numerical results demonstrate that the Symmetric Accelerated Overrelaxation approach needed less computing time, a smaller spectral radius, and fewer iterations than the Symmetric Successive Overrelaxation approach. This demonstrates that the Symmetric Accelerated Overrelaxation is superior to the Symmetric Successive Overrelaxation. Researchers and numerical analysts can benefit from the findings of this study; it will help them comprehend iteration techniques and adopt an appropriate or more efficient iterative strategy for solving systems of linear algebraic equations.
A MATHEMATICAL MODEL OF SCABBY MOUTH DISEASE INCORPORATING THE QUARANTINE CLASS.
(39th Annual Conference of the Nigerian Mathematical Society, (NMS), 2021-04-23) Abdurrahman, Nurat Olamide; Somma S. A.; Aboyeji Folawe Ibironke; Akinwande Ninuola Ifeoluwa
We propose a mathematical model to study the transmission and control of scabby mouth disease in sheep, incorporating the vaccinated and quarantine classes. The Disease-free equilibrium was obtained, and the reproduction number was also computed. The local stability of DFE was analyzed for stability. Sensitivity analysis of the basic reproduction number with respect to some parameters of the model was carried out, and the sensitive parameters withR_0 are presented graphically. The local stability of DFE is stable if R_0<1. The sensitivity analysis shows that the contact rateα is the most sensitive parameter to increase the spread of the disease, and vaccination rate ω is the highest sensitive parameter to control the transmission of scabby.
A MATHEMATICAL MODEL OF YELLOW FEVER DISEASE DYNAMICS INCORPORATING SPECIAL SATURATION INTERACTIONS FUNCTIONS
(1st SPS Biennial International Conference Federal University of Technology, Minna, Nigeria, 2017-05-05) Akinwande, N. I.; Abdulrahman, S.; Ashezua, T. T.; Somma, Samuel Abu
We proposed an Mathematical Model of Yellow Fever Disease Dynamics Incorporating Special Saturation Process functions, obtained the equilibrium states of the model equations and analyzed same for stability. Conditions for the elimination of the disease in the population are obtained as constraint inequalities on the parameters using the basic reproduction number 0 R demographic and epidemiological data. . Graphical simulations are presented using some
A Mathematical Model of Yellow Fever Disease Dynamics Incorporating Special Saturation Interactions Functions
(1st SPS Biennial International Conference Federal University of Technology, Minna, Nigeria, 2017-05-05) Akinwande, N. I.; Abdulrahman, S.; Ashezua, T. T.; Somma, Samuel Abu
We proposed an Mathematical Model of Yellow Fever Disease Dynamics Incorporating Special Saturation Process functions, obtained the equilibrium states of the model equations and analyzed same for stability. Conditions for the elimination of the disease in the population are obtained as constraint inequalities on the parameters using the basic reproduction number 0R . Graphical simulations are presented using some demographic and epidemiological data.
A MULTIGRID METHOD FOR NUMERICAL SOLUTION OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
(IJSAR Journal of Mathematics and Applied Statistics (IJSAR-JMAS), 2022-12) I. O. Isah; A. Ndanusa; R. Muhammad; K. A. Al-Mustapha
Techniques and analyses of multigrid method for solving elliptic partial differential equations (PDEs) in two dimensions are presented. The focal point of this paper is the applicability of the parametric reaccelerated overrelaxation (PROR) iterative method as a smoother in multigrid solution of elliptic PDEs. The two-dimensional Poisson equation on a unit square domain with Dirichlet boundary conditions is adopted as the model PDE. We present some practical formulae and techniques for building the various multigrid components using Kronecker tensor product of matrices. In addition, we carryout smoothing analysis of the PROR method using Local Fourier Analysis (LFA) and show how optimal relaxation parameters and smoothing factors can be obtained from analytic formulae derived to ensure better convergence. This analysis combines full standard coarsening strategy (doubling) and second order finite difference scheme. The result of PROR smoothing factors in comparison with those of other widely used smoothers is also presented. Results obtained from numerical experiment are displayed and compared with theoretical results.
A NOTE ON COMBUSTIBLE FOREST MATERIAL (CFM) OF WILDLAND FIRE SPREAD
(Proceedings of 3rd SPS Biennial International Conference Federal University of Technology, Minna, Nigeria, 2021-10-28) Zhiri, A. B.; Olayiwola, R. O.; Somma Samuel Abu
fire is presented. The equations describing the fractional components of forest fire were carefully studied. The reaction before a forest can burn or before fire can spread must involves fuel, heat and oxygen. The coupled dimensionless equations describing the phenomenon have been decoupled using perturbation method and solved analytically using eigen function expansion technique. The results obtained were graphically discussed and analysed. The study revealed that varying Radiation number and Peclet energy number enhances volume fractions of dry organic substance and moisture while they reduced volume fraction of coke.
A Note on Combustible Forest Material (CFM) of Wildland Fire Spread
(Proceedings of 3rd SPS Biennial International Conference Federal University of Technology, Minna, Nigeria, 2021-10-28) Zhiri, A. B.; Olayiwola, R. O.; Somma, Samuel Aabu
In this paper, a mathematical model for combustible forest material of a wildland fire is presented. The equations describing the fractional components of forest fire were carefully studied. The reaction before a forest can burn or before fire can spread must involves fuel, heat and oxygen. The coupled dimensionless equations describing the phenomenon have been decoupled using perturbation method and solved analytically using eigen function expansion technique. The results obtained were graphically discussed and analysed. The study revealed that varying Radiation number and Peclet energy number enhances volume fractions of dry organic substance and moisture while they reduced volume fraction of coke.
A TWO POINT BLOCK HYBRID METHOD FOR SOLVING STIFF INITIAL VALUE PROBLEMS
(JOURNAL OF MATHEMATICAL SCIENCES, 2011) Muhammad R
In this paper, a self starting hybrid method of order (3, 3,3) is proposed for the solution of stiff initial value problem of the form y' = f(x.y). The continous formation of the integrator enables us to differentiate and evaluate at grid and off grid points. The schemes compared favourably with exact results and results from Okunuga (2008)
Agreement between the Homotopy Perturbation Method and Variation Iterational Method on the Analysis of One-Dimensional Flow Incorporating First Order Decay
(SCHOOL OF PHYSICAL SCIENCES, FEDERAL UNIVERSITY OF TECHNOLOGY, MINNA, 2019-06-28) JIMOH, OMANANYI RAZAQ; Aiyesimi, Y. M.; Jiya, M.
In this paper, a comparative study of reactive contaminant flow for constant initial concentration in one dimension is presented. The adsorption term is modeled by Freudlich Isotherm. An approximation of the one-dimensional contaminant flow model was obtained using homotopy-perturbation transformation and the resulting linear equations were solved semi-analytically by homotopyperturbation method (HPM) and Variational Iteration Method (VIM). Graphs were plotted using the solution obtained from the methods and the results presented and discussed. The analysis of the results obtained show that the concentration of the contaminant decreases with time and distance as it moves away from the origin.
An Accelerated Iterative Technique: Third Refinement of Gauss-Seidel Algorithm for Linear Systems
(MDPI, 2023-05-01)
This study presents a novel accelerated iterative method referred to as the Third Refinement of the Gauss-Seidel Algorithm (TRGS) for solving large-scale linear systems of equations. By integrating a three-level refinement strategy into the classical Gauss-Seidel method, the proposed technique significantly improves the convergence rate and computational efficiency. The method is rigorously analyzed for consistency, stability, and convergence, and is evaluated through numerical experiments on various benchmark problems. Results demonstrate that the TRGS algorithm outperforms both the traditional Gauss-Seidel and other refinement-based methods in terms of iteration count and solution accuracy. This advancement offers a valuable contribution to numerical linear algebra, particularly in scientific computing where fast and accurate solutions to linear systems are critical.
An Appraisal on the Application of Reproduction Number for the Stability Analysis of Disease - Free Equilibrium State for S-I-R Type Models
(Proceedings of International Conference on Mathematical Modelling Optimization and Analysis of Disease Dynamics (ICMMOADD) 2024, 2024-02-28) Abdurrahman, Nurat Olamide; Somma S. A.; Akinwande, N. I.; Ashezua, T. T.; Gweryina, R.
One of the key ideas in mathematical biology is the basic reproduction number, which can be utilized to comprehend how a disease epidemic profile might evolve in the future. The basic reproduction number, represented by R0 , is the anticipated number of secondary cases that a typical infectious individual would cause in a population that is fully susceptible. This threshold parameter is highly valuable in characterizing mathematical problems related to infectious diseases. If R0 < 1, this suggests that, on average, during the infectious period, an infected individual produces less than one new infected individual, suggesting that the infection may eventually be eradicated from the population. On the other hand, if R0 < 1, every infected person develops an average of multiple new infections, it suggests that the disease may continue to spread throughout the population. We discuss the Reproduction number in this work and provide some examples, both for straightforward and complicated situations.
An Implicit Runge-Kutta Type Method for the Solution of Initial Value Problems
(KASU JOURNAL OF MATHEMATICAL SCIENCES, 2020-06) R. Muhammad; Y. A. Yahaya; A. S. Abdulkareem
In this research paper, an implicit block hybrid Backward Differentiation Formula (BDF) for 𝑘=2 is reformulated into a Runge-Kutta Type Method (RKTM) of the same step number. The method can be used to solve both first and second order (special or general form) initial value problem in Ordinary Differential Equation (ODE). This method differs from conventional BDF as derivation is done only once. It can also be extended to solve higher order ODE.
Behaviour of Contaminant in a Flow due to Variations in the Cross-Flow dispersion under a Dirichlet Boundary Conditions.
(SCHOOL OF PHYSICAL SCIENCES, FEDERAL UNIVERSITY OF TECHNOLOGY, MINNA, 2024-04-18) JIMOH, OMANANYI RAZAQ; Adebayo A.; Salihu, N. O.; Bako, D.
The advection-dispersion equation (ADE) is mostly adopted in evaluating solute migration in a flow. This study presents the behavior of contaminant in a flow due to variations in the cross-flow dispersion under a Dirichlet boundary conditions. The analytical solution of a two-dimensional advection-dispersion equation for evaluating groundwater contamination in a homogeneous finite medium which is initially assumed not contaminant free was obtained. In deriving the model equation, it was assumed that there was a constant point-source concentration at the origin and a Dirichlet type boundary condition at the exit boundary. The cross-flow dispersion coefficients, velocities and decay terms are time-dependent. The modeled equation was transformed using some space and time variables and solved by parameter expanding and Eigen-functions expansion method. Graphs were plotted to study the behavior of the contaminant in the flow. The results showed that increase in the cross-flow coefficient decline the concentration of the contaminant with respect to increase in time, vertical distance and horizontal distance in different patterns.
CONTINUOUS FORMULATION OF HYBRID BLOCK MILNE TECHNIQUE FOR SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS
(Mathematical Association of Nigeria (MAN), 2022-12-13) Khadeejah James Audu; Y. A. Yahaya,; J. Garba; A. T. Cole; F. U. Tafida
In most scientific and engineering problems, ordinary differential equations cannot be solved by analytic methods. Consequently, numerical approaches are frequently required. A block hybrid Milne technique was formulated in this paper in order to develop a suitable algorithm for the numerical solution of ordinary differential equations. Utilizing power series as the basis function, the proposed method is developed. The developed algorithm is used to solve systems of linear and nonlinear differential equations, and it has proven to be an efficient numerical method for avoiding timeconsuming computation and simplifying differential equations. The fundamental numerical properties are examined, and the results demonstrate that it is zero-stable and consistent, which ensures convergence. In addition, by comparing the approximate solutions to the exact solutions, we demonstrate that the approximate solutions converge to the exact solutions. The results demonstrate that the developed algorithm for solving systems of ordinary differential equations is straightforward, efficient, and faster than the analytical method.
Differential Transformation Method (DTM) for Solving Mathematical Modelling of Monkey Pox Virus Incorporating Quarantine
(Proceedings of 2nd SPS Biennial International Conference Federal University of Technology, Minna, Nigeria, 2019-06-26) Somma, Samuel Abu; Akinwande, N. I.; Abdurrahman, N. O.; Zhiri, A. B.
In this paper the Mathematical Modelling of Monkey Pox Virus Incorporating Quarantine was solved semi-analytically using Differential Transformation Method (DTM). The solutions of difference cases were presented graphically. The graphical solutions gave better understanding of the dynamics of Monkey pox virus, it was shown that effective Public Enlightenment Campaign and Progression Rate of Quarantine are important parameters that will prevent and control the spread of Monkey Pox in the population.
Extended Block Hybrid Backward Differentiation Formula for Second Order Fuzzy Differential Equations Using Legendre Polynomial as Basis Function.
(Federal University of Technology, Minna, Nigeria, 2020-03-12) Ma’ali, A. I.,; Mohammed, U.; Khadeejah James Audu; Yusuf, A.; Abubakar, A. D.
In this paper, we developed an implicit continuous four-step Extended Block Hybrid Backward Differentiation Formulae (EBHBDF) for the direct solution of Fuzzy Differential Equations (FDEs). For this purpose, the Legendre polynomial was employed as the basis function for the development of schemes in a collocation and interpolation techniques. in this regard and the results are satisfied the convex triangular fuzzy number. We also compare the numerical results with the exact solution, and it shows that the proposed method is good approximation for the analytic solution of the given second order Fuzzy Differential Equations
Homotopy Perturbation Analysis of the Spread and Control of Lassa Fever
(Proceedings of International Conference on Mathematical Modelling Optimization and Analysis of Disease Dynamics (ICMMOADD), 2024-02-22) Tsado, D.; Oguntolu, F. A.; Somma, Samuel Abu
Lassa fever, a viral infection transmitted by rodents, has emerged as a significant global health concern in recent times. It continues to garner significant attention daily basis owing to its rapid transmission and deadly nature. In this study, the Homotopy Perturbation Analysis was conducted to examine the spread and control of Lassa fever. The human population was categorized into susceptible, exposed, infected, and recovered compartments, while the rodent population was divided into susceptible and infected recovered compartments. By applying the Homotopy Perturbation Analysis to the nonlinear differential equations associated with these compartments, we were able to obtain the analytical solution for the spread and control of Lassa fever. The nonlinear differential equations were integrated into the Homotopy Perturbation framework and solved to form a power series solution. Finally, the final approximate solutions were obtained and simulation results were generated from the general solution graphically.
Improving Accuracy Through the Three Steps Block Methods For Direct Solution of Second Order Initial Value Problem Using Interpolation and Collocation Approach
(KASU JOURNAL OF MATHEMATICAL SCIENCES (KJMS), 2020-06) R. Muhammad; I. D. Zakariyau
This paper presents three-step block method for direct solution of second order initial value problems of ordinary differential equations. The collocation and interpolation approach was adopted to generate a continuous block method using power series as basis function. The properties of the proposed approach such as order, error constant, zero-stability, consistency and convergence were also investigated. The proposed method competes favorably with exact solution and the existing methods.
INFLUENCE OF ZERO-ORDER SOURCE AND DECAY COEFFICIENTS ON THE CONCENTRATION OF CONTAMINANTS IN TWO-DIMENSIONAL CONTAMINANT FLOW
(SCHOOL OF PHYSICAL SCIENCES, FEDERAL UNIVERSITY OF TECHNOLOGY, MINNA, 2017-04-23) JIMOH, OMANANYI RAZAQ; Aiyesimi, Y. M.; Jiya, M.; Bolarin, G. A.
In this article, an eigenfunctions expansion method is used in studying the behavior of two-dimensional contaminant flow problem with non-zero initial concentration. The mathematical model describing the contaminant flow is described by advection, dispersion, adsorption, first order decay and zero-order source. It is assumed that the adsorption term is modeled by Freudlich isotherm. Before the application of the eigenfunctions method, the parameter expanding method is applied on the model and the boundary conditions are transformed to the homogeneous type. Thereafter, the approximate solution of the resulting initial value problem was obtained successively. The results obtained are expressed graphically to show the effect of change in the zero-order source and decay coefficients on the concentration of the contaminants. From the analysis of the results, it was discovered that the contaminant concentration decreases with increase in the distance from the origin as the zero-order source and decay coefficient increases.