Coherent hybrid block method for approximating fourth-order ordinary differential equations

Year 2023, Volume: 4 Issue: 1, 52 - 71, 02.07.2023

Faruk Muritala , Abdulazeez Kayode Jimoh , Muideen Ogunniran , Abdulmalik Oyedeji , Jafaar Lawal

https://doi.org/10.54559/jauist.1262994

Abstract

Conventionally, the most used method of solving fourth-order initial value problems of ordinary differential is to first reduce to a system of first-order differential equations. This approach affects the effectiveness and convergence of the numerical method due to the transformation. This paper comprises the derivation, analysis, and implementation of a new hybrid block method for direct solution of fourth-order equations. The method is derived by collocation and interpolation of an assumed basis function. The basic properties of the block method, including zero stability, error constants, consistency, order, and convergence, were analyzed. From the analysis, the block method derived was found to be zero-stable, consistent, and convergent. Errors were computed for the proposed method, and they were proven to produce approximations that agree with exact solutions and as such this shows improvement with those of existing works.

Keywords

Hybrid methods, block method, linear stability, errors

References

• D. O. Awoyemi, S. J. Kayode, L. O. Adoghe, A six-step continuous multi-step method for the solution of general fourth order initial value problems of ordinary differential equations, Journal of Natural Sciences Research 5 (5) (2015) 131–139.
• A. M. Badmus, A new eighth order implicit block algorithms for the direct solution of second order ordinary differential equations, American Journal of Computational Mathematics 4 (4) (2014) 376–386.
• A. B. Familua, E. O. Omole, Five points mono hybrid point linear multi-step method for solving nth order ordinary differential equations using power series function, Asian Research Journal of Mathematics 3 (1) (2017) 1–17.
• Y. Haruna, R. B. Adeniyi, M. O. Ogunniran, A two-step hybrid block method with four off-step points on singular initial and boundary value problems, Turkish Journal of Mathematics and Computer Science 13 (2) (2021) 248–260.
• M. O. Ogunniran, G. C. Olaleye, O. A. Taiwo, A. Shokri, K. Nonlaopon, Generalization of a class of uniformly optimized k-step hybrid block method for solving two-point boundary value problems, Results in Physics 44 106147 12 pages.
• A. T. Cole, A. Tiamiyu, Hybrid block method for direct solution of general fourth order ordinary differential equations using power series function, in: Y. Zeren (Ed.), 6th International Conference on Mathematical Analysis and Optimization: Theory and Applications (ICMAOTA), Abuja, 2019, pp. 55–68.
• M. K. Duromola, An accurate five off-step points implicit block method for direct solution of fourth order differential equations, Open Access Library Journal 3 (2016) e2667.
• U. Mohammed, A six step block method for solution of fourth order ordinary differential equations, The Pacific Journal of Science and Technology 11 (2010) 258–265.
• G. D. Habtamu, J. K. Masho, Numerical solution of fourth order ordinary differential equations using fifth order Runge-Kutta method, Asian Journal of Science and Technology 8 (2) (2017) 4332–4339.