Browsing by Author "R. Muhammad"

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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.
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.
Direct Solution of 𝒚′′(𝒙) = 𝒇(𝒙, 𝒚, 𝒚′) Using Four Points Block-Hybrid Linear Multistep Method of Order Seven with Applications
(WORLD SCIENTIFIC NEWS, 2022-11) A. A. Oyedeji; R. Muhammad
The study aims to construct an implicit block hybrid method with four points to tackle general second order initial value problems of ordinary differential equations (ODEs) directly. Power series is used as the basis function to obtain the proposed method which involved the first and second derivatives of 𝑓(𝑥, 𝑦, 𝑦′). From the investigation done, it was found that the proposed method is consistent and zerostable, hence it is convergent. The proposed method’s efficiency was obtained and a comparison was made in terms of accuracy to some existing methods with similar order and the ones higher than it. The new proposed method is able to solve linear, nonlinear and systems of equations of general second order Initial Value Problems and outperformed existing methods with impressive results. Applications of the proposed method to a real-life problem is discussed.
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.