Mathematics
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Mathematics
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Item Refinements of Some Iterative Methods for Solving Linear System of Equations(Nigerian Women in Mathematics (NWM), 2023-06-20) Khadeejah James Audu; James Nkereuwem EssienThe efficient and accurate solution of linear systems of equations is a fundamental problem in various scientific and engineering fields. In this study, we focus on the refinements of iterative methods for solving linear systems of equations (▁A k=▁b). The research proposes two methods namely, third refinement of Jacobi method (TRJ) and third refinement of Gauss-Seidel (TRGS) method, which minimizes the spectral radius of the iteration matrix significantly when compared to any of the initial refinements of Jacobi and Gauss-Seidel methods. The study explores ways to optimize their convergence behavior by incorporating refinement techniques and adaptive strategies. These refinements exploit the structural properties of the coefficient matrix to achieve faster convergence and improved solution accuracy. To evaluate the effectiveness of the proposed refinements, numerical examples were tested to see the efficiency of the proposed TRJ and TRGS on a diverse set of linear equations. We compare the convergence behavior, computational efficiency, and solution accuracy of the refined iterative methods against their traditional counterparts. The experimental results demonstrate significant improvements in terms of convergence rate and computational efficiency when compared to their initial refinements. The proposed refinements have the potential to contribute to the development of more efficient and reliable solvers for linear systems, benefiting various scientific and engineering applications.Item Numerical Solution of Parabolic Partial Differential Equations via Conjugate Gradient Technique(Nigeria Mathematical Science (NMS), 2023-06-19) Khadeejah James AuduParabolic partial differential equations (PPDEs) arise in many areas of science and engineering, including heat transfer, diffusion, and fluid dynamics. Analytical solutions to these PPDEs are often difficult or impossible to obtain, so numerical methods are needed to approximate the solution. In this research, we investigate the use of the conjugate gradient technique for numerically solving parabolic PDEs. The technique involves discretizing the PPDE with regard to both space and time. The parabolic partial differential equations are then transformed into systems of linear algebraic equations using the Crank-Nicholson centred difference approach. Then, these equations are solved to yield the unknown points in the grids, which are subsequently substituted into the assumed solution to obtain the required estimated solution, which is reported in tabular format. A comparison was made between the conjugate gradient solutions and those produced using the Jacobi preconditioned conjugate gradient technique in terms of the time required and rate of convergence at that point. Results indicate that conjugate gradient techniques are suitable for solving parabolic-type partial differential equations, with Jacobi-preconditioned conjugate gradient technique converging faster. This research has potential applications in various areas of science and engineering where parabolic PDEs ariseItem Numerical Solution for Magnetohydrodynamics Mixed Convection Flow Near a Vertical Porous Plate Under the Influence of Magnetic Effect and Velocity Ratio(Maths Model Research Group, FUT, Minna, Nigeria, 2024-02-18) Ibrahim Yusuf; Umaru Mohammed; Khadeejah James AuduThis paper investigates the effects of thermal radiation on MHD mixed convection flow, heat and mass transfer, Dufour and Soret effects over a porous plate having convective boundary condition under the influence of magnetic field. The governing boundary layer equations are formulated and transformed into nonlinear ordinary differential equations using similarity transformation and numerical solution is obtained by using Runge-Kutta fourth order scheme with shooting technique. The effects of various physical parameters such as velocity ratio parameter, mixed convection parameter, melting parameter, suction parameter, injection parameters, Biot number, magnetic parameter, Schmit and pranditl numbers on velocity and temperature distributions are presented through graphs and discussed.Item Linear Programming for Profit Maximization of Agricultural Stock(Maths Model Research Group, FUT, Minna, Nigeria, 2024-02-18) Jacob Rebeccal; Nyor Ngutor; Khadeejah James AuduThis paper discusses a few common issues that are specific to agricultural investing, such as the challenge of choosing which stocks to buy in order to maximize returns. The linear programming model was applied to ten (10) agricultural stocks, and the simplex approach was used as the numerical technique to calculate the best possible outcome. The TORA programmer was used to verify the best option, and the findings indicated that not every item should be invested order to maximize profit.Item Numerical Solutions of Higher Order Differential Equations via New Iterative Method(Maths Model Research Group, FUT, Minna, Nigeria, 2024-02-18) Khadeejah James AuduHigher order differential equations play a fundamental role in various scientific and engineering disciplines, but their numerical solutions often pose formidable challenges. The New Iterative Method (NIM) has emerged as a promising technique for addressing these challenges. This study is to explore and assess the efficiency and accuracy of New Iterative Method in solving higher-order differential equations. By applying NIM to a range of problems from diverse scientific disciplines, we aim to provide insights into the method's adaptability and its potential to revolutionize numerical analysis. The method is well-suited for numerically integrating both and nonlinear higher-order differential equations. To showcase the efficiency and accuracy of this approach, some numerical tests have been conducted, comparing it to existing methods. The numerical results obtained from these tests strongly suggest that the new iterative scheme outperforms the previously employed method in estimating higher-order problems, thus confirming its convergence.Item 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 EquationsItem Thermal explosion with convection in porous media: A Mathematical Approach(University of Lagos, Nigeria, 2022-11-04) R. O. Olayiwola; S. I. Yusuf; A. D. Abubakar; Khadeejah James Audu; I. B. S. Mohammed; E. O. Anyanwu; U. A. Abdullahi; J. P. Oyubu.This paper studies the interaction between natural convection and thermal explosion in porous media. The model consists of the heat equation with a nonlinear source term describing heat production due to an exothermic chemical reaction coupled with the Darcy law. The conditions for the existence of unique solutions of the energy equation are established by the Lipschitz continuity approach. The analytical solution is obtained via Olayiwola’s generalized polynomial approximation method (OGPAM), which shows the influence of the parameters involved on the system. The effect of changes in values of parameters such as the Frank-Kamenetskii number, Rayleigh number, and inverse of Vadasz number are presented graphically and discussed. The results revealed that convection can change the conditions of thermal explosion.Item Numerical Assessment of Some Semi-Analytical Techniques for Solving a Fractional-Order Leptospirosis Model.(University of Malaysia, 2024-09-30) Khadeejah James Audu; AbdGafar Tunde Tiamiyu; Jeremiah Nsikak Akpabio; Hijaz Ahmad; Majeed Adebayo OlabiyiThis research aims to apply and compare two semi-analytical techniques, the Variational Iterative Method (VIM) and the New Iterative Method (NIM), for solving a pre-formulated mathematical model of Fractional-order Leptospirosis. Leptospirosis is a significant bacterial infection affecting humans and animals. By implementing the VIM and NIM algorithms, numerical experiments are conducted to solve the leptospirosis model. Comparing the obtained findings demonstrates that VIM and NIM are effective semi-analytical methods for solving systems of fractional differential equations. Notably, our study unveils a crucial dynamic in the disease's spread. The application of VIM and NIM offers a refined depiction of the biological dynamics, highlighting that the susceptible human population gradually decreases, the infectious human population declines, the recovered human population increases, and a significant rise in the infected vector population is observed over time. This nuanced portrayal of the disease's dynamics is crucial for understanding the intricate interplay of Leptospirosis among human and vector populations. The study's outcomes contribute valuable insights into the applicability and performance of the methods in solving the Fractional Leptospirosis model. Results indicate rapid convergence and comparable outcomes for both methods.Item 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. TafidaIn 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.Item A Comparative Study Of Two Iterative Techniques For Systems Of Linear Algebraic Equations(Academic Staff Union of Universities, Nigeria, 2021-12-20) Khadeejah James AuduThis 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.