Effectiveness of a One-fifth Hybrid Block Approach for Second Order Ordinary Differential Equations

Year 2023, Volume: 1 Issue: 2, 86 - 95, 17.12.2023

Oluwasayo Esther Taıwo Lateef Olakunle Moshood Muideen Odunayo Ogunnıran

https://izlik.org/JA48ER93UC

Abstract

In this paper, a class of hybrid block methods for solving second order ordinary differential equations directly was developed. This class was obtained by interpolation and collocation techniques. The methods were analyzed based on the qualitative properties of linear multi-step methods and were found to be zero-stable, consistent and convergent with good region of absolute stability. The proposed methods were analyzed quantitatively and implemented on second order ordinary differential initial value problems. An improved performance of the new methods over existing methods in the literature was shown by solving five numerical examples. The results were presented in tabular form.

Keywords

Block method , hybrid method , initial value problems , one-fifth step , convergence , stability

References

• 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) , 24(2A), 58-77. google scholar
• Adesanya, A. O., Anake, T. A. & Oghonyon, G. J. (2009). ontinuous implicit method for the solution of general second order ordinary differential Equations Journal of Nigerian Association of Mathematical Physics, 15, 71-78. google scholar
• Adesanya, A. O., Odekunıe, M. Rand Alkali, M. A. (2012). Order six blockpredictor corrector for the solutionof y00 = f (x, y, y0), Canadian google scholar
• J. on Science and Engineering Mathematics, 3(4), 180-185. google scholar
• Adeyefa, E. O. (2013). Collocation approach for continuous hybrid block methods for second order ordinary differential equations with Chebyshev basis function. Unpublished doctoral thesis, University of Ilorin, Ilorin, Nigeria. google scholar
• Adeyeye, O. & Omar, Z. (2016). Maximal Order Block Method For The Solution Of Second Order Ordinary Differential Equations, IAENG International Journal of Applied Mathematics 46(4). google scholar
• Anake, T. A. (2011). Continuous implicit hybrid one-step Methods for solution of initial value problems of general second ordinary differential equation Ph.D. Thesis,(Unpublished) Covenant University, Ota Nigeria. google scholar
• Areo, E. A., Ademiluyi R. A. and Babalola, P. O.(2008). Accurate collocation multi-step methods for integration of first order ordinary differential equations. International Journal Computer Mathematics 2(1), 15-27. google scholar
• Areo E. A. and Rufai M. A. (2016). Develop a new uniform fourth order one-third step continuous block method for the direct solution. British journal of Mathematics and computer science 15(4), 1-12. google scholar
• Areo, E. and Omojoıa, M. T. (2017). One-twelveth step continuous blockmethodfor the solutionof y000 = f (x, y, y0, y00) Internationaljournal of pure and applied mathematics. google scholar
• Awoyemi, D. O. (1992). On some continuous linear multi-step methods for initial value problems Unpublished doctoral thesis, University of Ilorin, Ilorin Nigeria. google scholar
• Butcher, J. C. (1965). A modified multi-step method for numerical integration of ordinary differential equations Journal of the ACM,12, 124-135. google scholar
• Dahlquist, G. (1959). Stability and error bound in the numerical integration of ODEs Transcript/130/, Royal Institute of Technology, Stockholom. google scholar
• Dahlquist, G. (1963). A special stability problem for linear multi-step methods BIT 3, 27-43. google scholar
• Ehigie, J. O., Okunuga S. A. and Sofoluwe, A. B. (2011). A class of 2-step continuous hybrid implicit linear multi-step methods for IVP’s Journal of Nigerian Mathematical Society 30, 145-161. google scholar
• Fotta, A. U., Alabi, T. J., & Abdulqadir, B. (2015). Block method with one hybrid point for the solution of first order initial value problems of ordinary differential equations. International journal of pure and applied mathematics. google scholar
• Mohammed, U. and Adeniyi, R. B. (2014). A Three Step Implicit Hybrid Linear Multistep Method for the Solution of Third Order Ordinary Differential Equations. Gen. Math. Notes, 25(1), 62-74. google scholar
• Muhammed, U., Ma’ali A. I., and Badeggi, A. Y. (2014). Derivation of block hybrid method for the solution of first order initial value problems in ODEs" Pacific Journal of Science and Technology 15(1), 93-98. google scholar
• Olabode, B.T. (2013). Block Multistep Method for Direct Solution of Third Order Ordinary Differential Equations. FUTA Journal of Research in Science 2, 194-200. google scholar
• Rajabi, M. Ismail, F., Senu, N. (2016). Linear 3 and 5-step methods using Taylor series expansion for solving special 3rd order ODEs. google scholar
• Ramos H., Mehta S. and Vigo-Aguiter J. (2016). A unified approach for k-step block Falkner methods for solving general second-order problems in ODEs. Journal of Computation and Applied Mathematics google scholar
• Rufai M. K., Duramola M. K. and Ganiyu.A. A. (2016). Derivation of one-sixth hybrid block method for solving general first order ordinary differential equations IOSR Journal of Mathematics12(5), 20-27. google scholar
• Sahi, R. K., Jator, S. N. and Khan, N. A. (2013). Continuous Fourth Derivative Method for Third Order Boundary Value Problems. International Journal of Pure and Applied Mathematics, International Journal of Pure and Applied Mathematics, 85(5), 907-923. google scholar