A New Integrator of Runge-Kutta Type for Directly Solving General Third-order ODEs with Application to Thin Film Flow Problem

Authors

F. A. Fawzi, Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia\\ Department of Mathematics, Faculty of Computer Science and Mathematics, University of Tikrit, Salah ad Din, IraqFollow
N. Senu, Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, MalaysiaFollow
F. Ismail, Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, MalaysiaFollow
Z. A. Majid, Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, MalaysiaFollow

Author Country (or Countries)

Malaysia

Abstract

In this study, a five-stage fourth-order Runge-Kutta type method for directly solving general third-order ordinary differential equations (ODEs) of the form y′′′ = f (x, y, y′, y′′) which is denoted as RKTGG method is constructed. The order conditions of RKTGG method up to order four are derived. Based on the order conditions developed, five-stage fourth-order explicit Runge-Kutta type method is constructed. Zero-stability of the current method is shown. The various type of general third-order ODEs has been solved using new method and numerical comparisons are made when the same problem is reduced to the first-order system of equations which are solved using existing Runge-Kutta methods. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is also discussed. Numerical results show that the new method is more efficient in terms of accuracy and number of function evaluations.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/amis/120412

Recommended Citation

A. Fawzi, F.; Senu, N.; Ismail, F.; and A. Majid, Z. (2018) "A New Integrator of Runge-Kutta Type for Directly Solving General Third-order ODEs with Application to Thin Film Flow Problem," Applied Mathematics & Information Sciences: Vol. 12: Iss. 4, Article 12.
DOI: http://dx.doi.org/10.18576/amis/120412
Available at: https://digitalcommons.aaru.edu.jo/amis/vol12/iss4/12