A Class of Collocation Hybrid Backward Differentiation Method for Initial Value Problems of Ordinary Differential Equations

Adewole Ajileye; Ojo Olamiposi Aduroja

doi:10.53704/fujnas.v13i2.515

Authors

Adewole Ajileye University of Ilesa, Ilesa, Osun State.
Ojo Olamiposi Aduroja University of Ilesa, Ilesa, Osun State, Nigeria

DOI:

https://doi.org/10.53704/fujnas.v13i2.515

Keywords:

Differential equations, collocation hybrid backward differential equations, initial value problems

Abstract

The development of a linear multistep method for initial value problems of Ordinary Differential Equations (O.D.E.s) has been the subject of investigation for some time. In this work, a class of hybrid backward differentiation methods with step number k=1 for initial value problems of first-order ordinary differential equations was presented. The method entails interpolation of approximate solutions and collocation of the differential system. The scheme has been tested and found to be consistent and zero-stable. Numerical examples are given to demonstrate the efficiency of the new method.

Keywords: Differential equations; collocation hybrid backward differential equation; initial value problems

References

Awoyemi, D. O. (1992) “Some continuous Linear Multistep Methods for Initial Value Problems”. Ph. D. Thesis (Unpublished), University of Ilorin, Ilorin

Ademiluyi, R. A (1987) “New Hybrid Methods for System of Stiff Ordinary Differential Equations”. Ph. D. dissertation; University of Benin, Benin City.

Ayinde, A. M., James A. A., Ishaq A. A. and Oyedepo T. A. (2022). A new numerical approach using Chebyshev third kind polynomial for solving integro-differential equations of higher order. Gazi University Journal of Science. Part A: Engineering and Innovation, 9(3), 259 - 266

Butcher, J. C. (1965) “A Modified Method for Numerical Integration of Ordinary Differential Equations”, Journal of Association of Computer, 124 – 135

Dahlquist, G. (1963) A special Stability Problem for Linear Multistep Methods, B17 327- 434

Fatula, S. O (1988) Numerical Methods for Initial value Problems for Ordinary Differential Equations. Acdemic Press Inc., New York.

Gear, C. W. “Algorithm 407 DIFUB for Solution of Ordinary Differential equations” Communication of ACM 14, 185 – 190

Ghimire, B. K., Li X, Chen C. S.and Lamichane A. R. (2020) Hybrid Chebychev polynomial scheme for solving elliptic partial differential equations. Journal of Computational and Applied Mathematics, 364, 112324

Gragg, W. B. and Stetter H. J.(1964) “Generalized Multistep Corrector – Predictor Methods. Journal of Association of Computer, 108 – 209

Kayode, S. J. (2004) “A class of Multi-derivative Method for Solving Second Order Initial Value Problem of Ordinary Differential Equation” Ph. D. Thesis (unpublished), Federal University of Technology, Akure.

Lambert, J. D. (1973) Computational Methods in Ordinary Differential Equation, New York, John Willey and Sons

Ma’al A. I., Mohammed U, Akitububo B. Jikantoro Y. D. (2022). Three-Step Numerical Approximation Based on Block Hybrid Backward Differentiation Formula for Stiff System of Ordinary Differential Equations.European Journal of Mathematics and Statistics, 3(3), 48 – 53

Oladele, J. O. (1991) Some New Collocation Formulae for Continuous Numerical Solutions of Initial Value Problems, M. Sc. Degree Dissertation (Unpublished), University of Ilorin, Ilorin.

Onumanyi P., Awoyemi D. O. Jater S. N. and Sirisena U. W. (1994) New Linear Multistep Methods with Continuous Coefficients for first order Difference Approximations for solving Differntial Equations, Comp, Maths 72, 15 - 27

Oyedepo T., Ayoade A. A., Ajileye G., and Ikechukwu N. P. (2023). Lagendre computational algorithm for linear integro-differential equations. Cumburriyet Science Journal. 44(3), 561 -566

Oyedepo T., Ayoade A. A., Oluwayomi M. O. and Pandurangan R. (2023). Solution of Volterra-Fredholm integro-differential equations using the Chebyshev computational approach. International Conference onScinece, Engineering and Business for Sustainable Development Goals (SEB-SDG), Omu-Aran, Nigeria 1, 1 - 6

Sumon, M. M. I. and Nurulhoque M (2023). A comparative Study of Numerical Methods for solving Initial Value Problem (IVP) of Ordinary Differential Equations (ODE). American Journal of Applied Mathematics, 11(6), 106 -118

Zwillinger D and Dobrushkin V. (2021). Handbook of differential equations. CRC Press