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Author: search.filters.author.Aliyu Rasheed Taiwo

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    Assessment of Numerical Performance of Some Runge-Kutta Methods and New Iteration Method on First Order Differential Problems
    (Federal University, Dutse, Nigeria, 2023-12-10) Khadeejah James Audu; Aliyu Rasheed Taiwo; Abdulganiyu Alabi Soliu
    This research focuses on the assessment of the numerical performance of some Runge-Kutta methods and New Iteration Method “NIM” for solving first-order differential problems. The assessment is conducted through extensive numerical experiments and comparative analyses. Accuracy, efficiency, and stability are among the key factors considered in evaluating the performance of the methods. A range of first-order differential problems with diverse characteristics and complexity levels is employed to thoroughly examine the methods' capabilities and limitations. The numerical investigation that is defined in the study as well as the results that are stated in the Tables, demonstrates that all the approaches produce extremely accurate results. However, the “NIM” was shown to be the most effective of the three methods used in this study. Conclusively, the “NIM” should be employed to solve first-order nonlinear and linear ordinary differential equations in place of Runge-Kutta Fourth order method (RK4M) and Butcher Runge-Kutta Fifth order method (BRK5M). In addition, BRK5M is more applicable and efficient than RK4M when solving first order ordinary differential problems.
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    A THIRD REFINEMENT OF JACOBI METHOD FOR SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS
    (Federal University, Dutsin Ma, Nigeria, 2023-10-15) Khadeejah James Audu; James Nkereuwem Essien; Abraham Baba Zhiri; Aliyu Rasheed Taiwo
    Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and 𝑀 -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.