DEVELOPMENT OF A CONTINUOUS MULTI-STEP ONE-FOURTH STEP HYBRID SCHEME FOR THIRD-ORDER IVP IN ORDINARY DIFFERENTIAL EQUATIONS

OLUWASAYO ESTHER  TAIWO

OLUWASAYO ESTHER TAIWO DEPARTMENT OF MATHEMATICAL SCIENCES, AJAYI CROWTHER UNIVERSITY, OYO, NIGE- RIA

Keywords: Block Method, Hybrid method, Initial Value Problems, One - fourth Step, Order seven, Third Order

Abstract

This study introduces a hybrid block approach for the approximate solution of initial value problems involving third-order ordinary differential equations. The formulation of the method employs orthogonal and Chebyshev polynomials as basis functions, with its performance enhanced through the incorporation of off-step points. This modification is aimed at achieving zero-stability while maintaining high computational accuracy. Several numerical experiments are carried out to demonstrate the methods effectiveness, and the results confirm its reliability and efficiency in solving such problems.

References

[1] Adeniyi, R.B. and Alabi, M.O. (2007). Continuous Formulation of a class Accurate Implicit Linear Multi-step Methods with Chebyshev Basis Function in a Collocation Technique. Journal of Mathematical Association of Nigeria (ABACUS), 34(2A):58-77.
[2] Awoyemi, D.O. (1992). On some continuous linear multistep methods for initial value problems, Unpublished doctoral thesis, University of Ilorin, Ilorin, Nigeria.
[3] Fotta, A.U. (2015) Block Method with one hybrid point for the solution of first order intial value problems of ordinary differential equations. 103(3):511-52.
[4] Lambert, J.D. (1973).Computational Methods in ODEs. New York: John Wiley.
[5] Odejide, S.A. and Adeniran, A.O. (2012). A hybrid linear collocation multistep scheme for solving first order initial value problems. Journal of Nigerian Mathematical Society. 31:229-241.
[6] Onumanyi, P., Awoyemi, D.O., Jator, S.N. & Sirisena, U.W. (1994). ”New linear multi- step methods with continuous coefficient for first order initial value problems”. Journal of Nigerian Mathematics Society., 13:37-51.
[7] Taiwo, O.E, Yakusak N.S. and Ogunniran, M.O. (2025). Hybrid continuous multi-step method for second order problems in ordinary differential equations.Istanbul Journal of Mathematics . 3(1):1-11.