School of Physical Sciences (SPS)

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School of Physical Sciences (SPS)

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    A 3-step block hybrid backward differentiation formulae (bhbdf) for the solution of general second order ordinary differential equation
    (New Trends in Mathematical Sciences, 2021-07-12) Hussaini Alhassan; Muhammad Raihanatu
    In this paper, the block hybrid Backward Differentiation formulae (BHBDF) for the step number k=3 is 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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    Formulation Of A Standard Runge- Kutta Type Method For The Solution First And Second Order Initial Value Problems
    (Researchjournali’s Journal of Mathematics, 2015-03) Muhammad R.; Y. A Yahaya; A.S Abdulkareem
    In this paper, we present a standard Runge-Kutta Type Method (RKTM) for . The process produces Backward Differentiation Formula (BDF) scheme and its hybrid form which combined together to form a block method. The method is reformulated into a Runge-Kutta Type of the same step number for the solution of first and second order (special or general) initial value problem of Ordinary Differential Equation (ODE).
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    DERIVATION AND ANALYSIS OF BLOCK IMPLICIT HYBRID BACKWARD DIFFERENTIATION FORMULAE FOR STIFF PROBLEMS
    (Nigerian Journal of Mathematics and Applications, 2014) MUHAMMAD, R.; YAHAYA, Y. A.; IDRIS L.
    The Hybrid Backward Di erentiation Formula (HBDF) for the case k = 3 was reformulated into continuous form using the idea of multistep collocation. The continuous form was evaluated at some grid and o grid points which gave rise to discrete schemes employed as block methods for direct solution of rst order Ordinary Di erential Equation y0 = f(x; y). The requirement of a starting value and the overlap of solution model which are associated with conventional Linear Multistep Methods were eliminated by this approach. A convergence analysis of the derived hybrid schemes to establish their e ec- tiveness and reliability is presented. Numerical example carried out on sti problem further substantiates their performance.