School of Physical Sciences (SPS)

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School of Physical Sciences (SPS)

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Start date: 2024Subject: search.filters.subject.Implicit

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    REFORMULATION OF TWO STEP IMPLICIT LINEAR MULTI-STEP BLOCK HYBRID METHOD INTO RUNGE KUTTA TYPE METHOD FOR THE SOLUTION OF SECOND ORDER INITIAL VALUE PROBLEM (IVP)
    (2025) ALIYU Abubakar; MUHAMMAD Raihanatu; ABDULHAKEEM Yusuf
    Second-order ordinary differential equations (ODEs) is unavoidable in scientific and engineering fields. This research focuses on the reformulation of two-step implicit linear multistep block hybrid method into a seven-stage Runge-Kutta type method for the solution of second-order initial value problems (IVPs). A two-step, four-off-grid-point implicit block hybrid collocation method for first-order initial value problems was derived. Its order and error constants were determined, which shows that the schemes were of order 8, 8, 8, 8, 8 and 9 with respective error constants of , , , , . The derived block method was reformulated into a seven-stage Runge-Kutta type method (RKTM) for the solution of first-order ordinary differential equations; this reformulation was extended to handle the required second-order ordinary differential equations. The second-order Runge- Kutta-type method derived was implemented on numerical experiments. The method was found to be better than existing methods in the literature.
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    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.