Hybrid Block Method for Direct Integration of First, Second and Third Order IVPs

Year 2021, Volume: 18 Issue: 1, 1 - 8, 01.05.2021

Adeyemi Olagunju Emmanuel Adeyefa

Abstract

Construction of numerical methods for the solution of initial value problems (IVPs) in ordinary differential equations (ODEs) has been considered overwhelmingly in literature. However, the use of a single numerical method for the integration of ODEs of more than one order has not been commonly reported. In this paper, we focus on the development of a numerical method capable of obtaining the numerical solution of first, second and third-order IVPs. The method is formulated from continuous schemes obtained via collocation and interpolation techniques and applied in a block-by-block manner as a numerical integrator for first, second and third-order ODEs. The convergence properties of the method are discussed via zero-stability and consistency. Numerical examples are included and comparisons are made with existing methods in the literature.

Keywords

Consistency, Convergence, Zero-Stability

References

• [1] J. D. Lambert, Computational methods for ordinary differential equations. John Wiley, New York, 1973.
• [2] J. D. Lambert, Numerical methods for ordinary differential systems. New York John Willey and Sons, 1991.
• [3] D. O. Awoyemi, “A class of continuous linear multistep methods for general second-order initial value problems in ordinary differential equations,” International Journal of Computational Mathematics, vol. 72, pp. 29-37, 1999.
• [4] L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep IVPs and BVP methods. Gordon and Breach Science Publishers, 1998.
• [5] D. O. Awoyemi, “A P-stable linear multistep method for solving general third-order ordinary differential equations,” International Journal of Computational Mathematics, vol. 80, no. 8, pp. 987-993, 2003.
• [6] D. O. Awoyemi, “Algorithmic collocation approach for direct solution of fourth-order initial-value problems of ordinary differential equations,” International Journal of Computational Mathematics, vol. 82, no. 3, pp. 321-329, 2005.
• [7] S. J. Kayode, “An order six zero-stable method for direct solution of fourth-order ordinary differential equations,” American Journal of Applied Sciences, vol. 5, no. 11, pp. 1461-1466, 2008.
• [8] S. N. Jator, “Numerical integrators for fourth-order initial and boundary value problems,” International Journal of Pure and Applied Mathematics, vol. 47, no. 4, pp. 563 -576, 2008.
• [9] J. Vigo-Aguiar, H. Ramos, “Variable stepsize implementation of multistep methods for y^('')=f(x,y,y'),” Journal of Computational and Applied Mathematics, vol. 192, pp. 114-131, 2006.
• [10] U. Mohammed, “A six-step block method for the solution of fourth-order ordinary differential equations,” The Pacific Journal of Science and Technology, vol. 11, no. 1, pp. 258-265, 2010.
• [11] S. N. Jator, “A sixth-order Linear Multistep Method for direct solution of y^('')=f(x,y,y'),” International Journal of Pure and Applied Mathematics, vol. 40, no. 1, pp. 407-472, 2007.
• [12] S. J. Kayode, M. K. Duromola, B. Bolarinwa, “Direct Solution of Initial Value Problems of Fourth Order Ordinary Differential Equations Using Modified Implicit Hybrid Block Method,” Journal of Scientific Research and Reports, vol. 3, no. 21, pp. 2792 – 2800, 2014.
• [13] D. O. Awoyemi, S. J. Kayode, L. O. Adoghe, “A six-step continuous multistep method for the solution of general fourth-order initial value problems of ordinary differential equations,” Journal of Natural Sciences Research, vol. 5, no. 5, pp. 131-138, 2015.
• [14] L. E. Yap, F. Ismail, “Block hybrid collocation method with application to fourth order differential equations,” Mathematical Problems in Engineering, Article ID 561489, 2015. doi:10.1155/2015/561489.
• [15] K. Hussain, F. Ismail, N. Senua, “Solving directly special fourth-order ordinary differential equations using Runge Kutta type method,” Journal of Computational and Applied Mathematics, vol. 306, pp. 179-199, 2016.
• [16] F. Ismail, Y. L. Ken, M. Othman, “Explicit and implicit 3-point block methods for solving special second order ordinary differential equations directly,” International Journal of Mathematical Analysis, vol. 3, pp. 239-254, 2009.
• [17] H. Ramos, S. Mehta, J. Vigo-Aguiar, “A unified approach for the development of k-Step block Falkner-type methods for solving general second-order initial-value problems in ODEs,” Journal of Computational and Applied Mathematics, Article in Press, 2016.
• [18] E. O. Adeyefa, “Orthogonal-based hybrid block method for solving general second-order initial value problems,” Italian Journal of Pure and Applied Mathematics, vol. 37, pp. 659-672, 2017.
• [19] M. O. Ogunniran, Y. Haruna, R. Adeniyi, M. O. Olayiwola, “Optimized three-step hybrid block method for stiff problems in ODEs,” Cankaya University Journal of Science and Technology, vol. 17, no. 2, pp. 80-95, 2020.
• [20] M. O. Ogunniran, O. A. Tayo, Y. Haruna, A. F. Adebisi, “Linear stability analysis of Runge-Kutta Methods for Singular Lane-Emden Equations,” Journal of Nigerian Society of Physical Sciences, vol. 2, pp. 134-140, 2020.
• [21] G. Singh, A. Garg, V. Kanwar, H. Ramos, “An efficient optimized adaptive step-size hybrid block method for integrating differential systems,” Applied Mathematics and Computation, vol. 362, 124567, 2019.
• [22] R. B. Adeniyi, M. O. Alabi, “A collocation method for direct numerical integration of initial value problems in higher-order ordinary differential equations,” Analele Stiintifice Ale Universitatii AL. I. Cuza Din Iasi (SN), Matematica, Tomul, vol. 2, pp. 311-321, 2011.
• [23] J. O. Kuboye, “Block methods for direct solution of higher-order ordinary differential equations using interpolation and collocation approach,” Ph.D. thesis, Universiti Utara, Malaysia, 2015. http://etd.uum.edu.my/id/eprint/5789.
• [24] J. O. Kuboye, Z. Omar, O. E. Abolarin, R. Abdelrahim, “Generalized hybrid block method for solving second-order ordinary differential equations directly,” Res Rep Math, vol. 2, no. 2, 2018.
• [25] T. A. Anake, A. O. Adesanya, G. J. Oghonyon, M. C. Agarana, “Block Algorithm for General Third Order Ordinary Differential Equations,” ICASTOR Journal of Mathematical Sciences, vol. 7, no. 2, pp. 127-136, 2013.
• [26] S. O. Fatunla, Numerical Methods for initial value problems in ordinary differential equations. Academic Press Inc. Harcourt Brace, Jovanovich Publishers, New York, 1988.
• [27] S. O. Fatunla, “Block method for second-order initial value problem (IVP),” International Journal of Computer Mathematics, vol. 41, pp. 55-63, 1991.
• [28] G. Dahlquist, “Some properties of linear multistep and one leg method for ordinary differential equations,” Department of computer science, Royal Institute of Technology, Stockholm, 1979.
• [29] G. Ajileye, S. A. Amoo, O. D. Ogwumu, “Hybrid Block Method Algorithms for Solution of First Order Initial Value Problems in Ordinary Differential Equations,” Journal of Applied and Computational Mathematics, vol. 7, no. 2, pp.1-4, 2018.
• [30] J. Sunday, M. R. Odekunle, A. O. Adesanya, “Order six block integrator for the solution of first-order ordinary differential equations,” International Journal of Mathematics and Soft Computing, vol. 3, pp. 87-96, 2013.
• [31] U. Mohammed, R. B. Adeniyi, “Derivation of five-step block Hybrid Backward Differential Formulas (HBDF) through the continuous multi-step collocation for solving second order differential equation,” Pacific Journal of Science and Technology, vol. 15, no. 2, pp. 89 - 95, 2014.
• [32] U. Mohammed, “A class of implicit five-step block method for general second-order ordinary differential equations,” Journal of Nigeria Mathematical Society (JNMS), vol. 30, pp. 25 – 39, 2011.