A Numerical Technique for Direct Solution of Special Fourth Order Ordinary Differential Equation Via Hybrid Linear Multistep Method

Year 2021, Volume: 18 Issue: 1, 47 - 60, 01.05.2021

Tiamiyu Abd'gafar , Falade Kazeem Iyanda , Quadri Rauf , Sıkırulaı Akande

Abstract

We propose and present a self-starting numerical approximation with a higher order
of accuracy for direct solution of a special fourth-order ordinary differential equation
(ODE) using a Hybrid Linear Multistep Method (HLMM). The technique utilizes the
collocation and interpolation approach with six-step numbers and two off-step points
using power series as the basis function. Error constants and basic properties proved
the convergence of the method. Numerical experiments involving both linear, nonlinear, and linear systems of fourth-order initial value problems appearing in modeling
of physical phenomenon from various areas of applied sciences were used to demonstrate the effectiveness and efficiency of the proposed method. The results revealed that
the proposed method is an excellent choice for approximating general fourth-order ODE
and shows the impact of choices of step sizes in the numerical solution of the problem
considered. In addition, the proposed HLMM outperformed existing methods in the
literature in terms of accuracy

Keywords

HLMM, Numerical approximation, Fourth-order ordinary differential equation, Self-starting

References

• [1] Akgul, A., & Akg ¨ ul, E. K. ”A novel method for solutions of fourth-order fractional boundary value problems”, ¨ Fractal and Fractional, vol. 3(2), pp. 1–13, 2019.
• [2] A. Boutayeb and C. Abdelaziz, ”A mini-review of numerical methods for high-order problems”, Int. J. Comput. Math., no. April, 2007.
• [3] Y. Wang, ”Some Fourth Order Differential Equations Modeling Suspension Bridges”, 2015.
• [4] S. H. Saker, R. P. Agarwal, and D. O. Regan, “Properties of solutions of fourth-order differential equations with boundary conditions”, J. of Inequalities and Applications, pp. 1–15, 2013.
• [5] S. Jator, “Numerical integrators for fourth order initial and boundary value problems”, Int. J. Pure Appl. Math., no. January 2008, 2015.
• [6] K. Hussain, F. Ismail, and N. Senu, “Solving Directly Special Fourth-Order Ordinary Differential Equations Using Runge-Kutta Type Method”, J. Comput. Appl. Math., 2016, doi: 10.1016/j.cam.2016.04.002.
• [7] J. D. Lambert, “Numerical Methods for Ordinary Differential Systems”, p. 284, 1972.
• [8] J. C. Butcher, ”Numerical Methods for Ordinary Differential Equations”. 2008.
• [9] E. A. Areo and E. O. Omole, “Solutions of fourth order ordinary differential equations”, Arch. Appl. Sci. Res., vol. 7, no.10, pp. 39–49, 2015.
• [10] Z. Omar, J. O. Kuboye, and B. Ax, “Fourth Order Ordinary Differential Equations,” J. Math. Fund.Science, vol. 48, no. 2, pp. 94–105, 2016, doi: 10.5614/j.math.fund.sci.2016.48.2.1.
• [11] A. B. Familua and E. O. Omole, “Five Points Mono Hybrid Point Linear Multistep Method for,” Asian Res. J. Math., vol. 3, no. 1, pp. 1–17, 2017, doi: 10.9734/ARJOM/2017/31190.
• [12] J. Talwar and R. K. Mohanty, “A Class of Numerical Methods for the Solution of Fourth-Order Ordinary Differential Equations in Polar Coordinates,” Adv. Numer. Anal., vol. 2012, 2012, doi: 10.1155/2012/626419.
• [13] A. T. Cole and A. T. Tiamiyu, “Hybrid Block Method for Direct Solution of General Fourth Order Ordinary Differential Equations using Power Series,” Int. Conf. Math. Anal. Optim. Theory Appl. (ICAPTA 2019), no. March, pp. 500–513, 2019.
• [14] K. State, “A Six-Step Continuous Multistep Method For The Solution Of,” J. Nat. Sci. Res., vol. 5, no. 5, pp. 131–139,2015.
• [15] B. T. Olabode, “A Six-Step Scheme for the Solution of Fourth Order Ordinary Differential Equations A Six-Step Scheme for the Solution of Fourth Order Ordinary Differential Equations .,” Pacific J. Sci. Technol., January, 2016.
• [16] Omar, Z., & Abdelrahim, R., ”Direct solution of fourth order ordinary differential equations using a one step hybrid block method of order five”, International Journal of Pure and Applied Mathematics, vol. 109(4), pp. 763–777, 2016, https://doi.org/10.12732/ijpam.v109i4.1
• [17] Waeleh, N., & Majid, Z. A., ”A 4-Point Block Method for Solving Higher Order Ordinary Differential Equations Directly”, 2016.
• [18] G. G. Dahlquist, “A Special Stability Problem for Linear Multistep Methods”, BIT Numer. Math., vol. 3, 1963.
• [19] G. Wanner, “Dahlquist’s Classical Papers on Stability”, BIT Numer. Math., no. September, pp. 671–683, 2006, doi:10.1007/s10543-006-0072-1.