Browsing by Author "Raihanatu Muhammad"

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An Order (K+5) Block Hybrid Backward Differentiation Formula for Solution of Fourth Order Ordinary Differential Equations
(Çankaya University Journal of Science and Engineering, 2024) Raihanatu Muhammad; Hajara Hussaini; Abdulmalik Oyedeji
This paper covers the derivation and implementation of the 4-step linear Multistep method of Block Hybrid Backward Differentiation Formula (BHBDF) for solving fourth-order initial value problems in ordinary differential equations. In the derivation of the proposed numerical method, the utilization of collocation and interpolation points was adopted with Legendre polynomials serving as the fundamental basis function. The 4-step BHBDF developed to solve fourth-order IVPs has a higher order of accuracy (p=9). Furthermore, the proposed numerical block methods are employed directly to solve fourth-order ODEs. In comparison to some existing methods examined in the prior studies, the proposed method has a robust implementation strategy and demonstrate a higher level of accuracy.
METHOD INTO RUNGE KUTTA TYPE METHOD FOR FIRST ORDER INITIAL VALUE PROBLEM (IVP)
(2025-03) Abubakar Aliyu; Raihanatu Muhammad; Abdulhakeem Yusuf
Problems arises from science and technology are expressed in differential equations. These differential equation are sometimes in ordinary differential equations. Reliability with high accuracy and stability are necessary for a numerical method for the solution of differential equations. This research paper presents the analysis of a reformulated block hybrid linear multistep method into Runge-Kutta type method (RKTM) for first order initial value problems (IVPs). In view of this, the block hybrid method derived is of uniform order 6 with error constants of , , , and while the Runge-Kutta type method reformulated maintain the order of the derived block hybrid linear multistep method which are of uniform order 6 but with error constants of . Testing for convergence of both the derived block hybrid linear multistep method and the Runge-Kutta type method shows that the two methods are consistent and are also zero stable
REFINEMENT OF PRECONDITIONED OVERRELAXATION ALGORITHM FOR SOLUTION OF THE LINEAR ALGEBRAIC SYSTEM 𝑨𝒙=𝒃
(Faculty of Science, Kaduna State University, 2021) Ramatu Abdullahi; Raihanatu Muhammad
In 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 method
The Algebraic Structure of an Implicit Runge- Kutta Type Method
(International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2024-11) Raihanatu Muhammad; Abdulmalik Oyedeji
In this paper, the theory of linear transformation (Homomorphism) and monomorphism is applied to a first-order Runge-Kutta Type Method illustrated in a Butcher Table and the extended second order Runge- Runge-Kutta type Method to substantiate their uniform order and error constants obtained. A homomorphism is a mapping from one group to another group which preserves the group operations. It’s sometimes called the operation preserving function. The methods which initially are Linear Multistep were reformulated into Runge-Kutta (R-K) Type to establish the advantages the R-K has over Linear Multistep. The first-order Linear multistep was reformulated into first-order R-K type which was further extended to second order. This extension can be made to higher order. For this study, the extension was limited to the second order.