Browsing by Author "Muhammad R"

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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.
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    A Sixth Order Implicit Hybrid Backward Differentiation Formulae (HBDF) for Block Solution of Ordinary Differential Equations
    (American Journal of Mathematics and Statistics, 2012) Muhammad R; Yahaya.Y.A
    The Hybrid Backward Differentiation Formula (HBDF) for case K=5 was reformulated into continuous form using the idea of multistep collocation. Multistep Collocation is a continuous finite difference (CFD) approximation method by the idea of interpolation and collocation. The hybrid 5-step Backward Differentiation Formula (BDF) and additional methods of order (6,6,6,6,6,)𝑇𝑇 were obtained from the same continuous scheme and assembled into a block matrix equation which was applied to provide the solutions of IVPs over non-overlapping intervals.The continous form was im-mediately employed as block methods for direct solution of Ordinary Differential Equation (𝑦𝑦′=𝑓𝑓(𝑥𝑥,𝑦𝑦)). Some benefits of this study are, the proposed block methods will be self starting and does not call for special predictor to estimate 𝑦𝑦’ in the integrators and all the discrete methods obtained will be evaluated from a single continuous formula and its derivatives at various grids and off grid points. These study results help to speed up computation, also the requirement of a starting value and the overlap of solution model which are normally associated with conventional Linear Multistep Methods were elimi-nated by this approach. In conclusion, a convergence analysis of the derived hybrid schemes to establish their effectiveness and reliability was presented. Numerical example carried out on stiff problem further substantiates their performance.
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    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)
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    Error and Convergence Analysis of a Hybrid Runge- Kutta Type Method
    (International Journal of Science and Technology Publications UK, 2015-04) Muhammad R; Y. A Yahaya,; A.S Abdulkareem
    Implicit Runge- Kutta methods are used for solving stiff problems which mostly arise in real life problems. Convergence analysis helps us to determine an effective Runge- Kutta Method (RKM) to use, but due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation. In this paper, we derived a hybrid Runge -Kutta Type method (RKTM) for 𝑘=1, obtained the order and error constant and convergence analysis of the method.
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    Reformulation of Block Implicit Linear Multistep Method into Runge Kutta Type Method for Initial Value Problem
    (International Journal of Science and Technology Publications UK, 2015-04) Muhammad R; Y.A Yahaya; A.S. Abdulkareem
    In this research work, we reformulated the block hybrid Backward Differentiation Formula (BDF) for 𝑘=4 into Runge Kutta Type Method (RKTM) of the same step number for the solution of Initial value problem in Ordinary Differential Equation (ODE). The method can be use to solve both first and second order (special or general form). It can also be extended to solve higher order ODE. This method differs from conventional BDF as derivation is done only once
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    THE ORDER AND ERROR CONSTANT OF A RUNGE-KUTTA TYPE METHOD FOR THE NUMERICAL SOLUTION OF INITIAL VALUE PROBLEM
    (Federal University Dutsin MA Journal of Sciences (FJS), 2020-06) Muhammad R
    In this paper, we examine in details how to obtain the order, error constant, consistency and convergence of a Runge-Kutta Type method (RKTM) when the step number 𝑘 = 2. Analysis of the order, error constant, consistency and convergence will help in determining an effective Runge- Kutta Method (RKM) to use. Due to the loss of linearity in Runge –Kutta Methods and the fact that the general Runge –Kutta Method makes no mention of the differential equation makes it impossible to define the order of the method independently of the differential equation.