On Review of the Convergence Analyses of the Runge Kutta Fixed Point Iterative Methods

Main Article Content

Eziokwu, C. Emmanuel
Nwosu Chidinma
Nnochiri Ifeoma

Abstract

Runge Kutta alongside K. Heun and E. J. Nystrom early in the nineteenth century extensively developed and consequently expanded the so called numerical methods of solution for the Ordinary Differential equations of various orders. Since then, work on the method has never ceased. However in this paper, we review and make stronger the fact that these methods are not only just a numeric method of solution but a very efficient iterative method. We outline in section one all the various Runge Kutta iterative methods and in section two their convergence while in section 3, the convergence analysis was illustrated with numeric examples which confirmed that only consistent and stable iterative Runge Kutta methods are convergent. Hence the objective of this research work is to establish that every Runge Kutta method which is not consistent and stable cannot be said to be convergent.

Keywords:
Fixed point iteration, Runge Kutta methods, consistency, stability and convergence

Article Details

How to Cite
Emmanuel, E. C., Chidinma, N., & Ifeoma, N. (2020). On Review of the Convergence Analyses of the Runge Kutta Fixed Point Iterative Methods. Asian Journal of Pure and Applied Mathematics, 2(1), 20-44. Retrieved from https://globalpresshub.com/index.php/AJPAM/article/view/809
Section
Review Article

References

Ababneh OY, Ahmed R, Ismail S. Design of new implicit Runge Kutta methods for stiff problems. Faculty of Science and Technology, University Kebangsan Malaysia; 2009.

Abul Hassan Siddiqi. Applied functional analysis: Numerical methods, Eavelet methods and image processing. King Fahd University, Saudi Arabia; 2004.

Butcher JC. The numerical analysis of ordinary differential equations. John Wiley and Sons, New York, USA; 2003.

Butcher JC. The numerical analysis of ordinary differential equations, Runge Kutta and general linear methods. John Wiley and Sons New York, USA; 1987.

Cash JR, Girdlestone. Variable step Runge Kutta Nystrom methods for the numerical solution of reversible systems. Department of Mathematics, Imperial College London South Kensington London SW7 2AZ England; 2006.

Gupta GK, Sacks-Davis R, Tischer PE. A review of recent developments in solving ODEs computing surveys. 1985;17:6-9.

Iheagwam VA, Onwuatu J. Fundamentals of numerical analysis. Alphabet Nigeria Publishers; 1996.

Jeffery A. Advanced engineering mathematics. University of Newcastle Upon-Tyne, Harcourt Academic Press; 2009.

Lambert JD. Numerical methods for ordinary differential systems. New York, Wiley; 1992.

Akanbi MA. A third order Euler method for numerical solution of ODE. Journal of Engineering and Applied Sciences. Department of Mathematics Lagos State University. 2010;5.

Ogbonna N. Lecture note on numerical analysis. Mathematics Department, Michael Okpara University of Agriculture, Umudike; 2013.

Didier G. Numerical methods to solve ordinary differential equation. Journal of Mathematics. 2009;10:53-64.

Dekker H, Verwer JG. Stability of Runge Kutta methods for stiff non-linear differential equation. Worth Holland, Amsterdam Netherlands; 1984.

Ogunrinde RB, Fadugba SE, Okunlola JT. On some numerical methods for solving initial value problems in ordinary differential equations. Journal of Mathematics. 2012;1:25-31.

Fatunla SO. Numerical methods for initial value problem in ordinary differential equations. Academic Press, New York; 1998.

Hairer E, Wanner G. Solving ordinary differential equations for non-stiff problems. 2nd Ed., Springer-Verglag Berlin; 1996.

Krezig E. Advanced engineering mathematics. John Wiley, New York; 1997.

Charles Chidume. Geometric properties of Banach spaces and nonlinear iterations. CTP, Trieste, Italy Psringer; 2009.

Evan DJ. A new fourth order Runge Kutta method for initial value problems with error control. Intern J. Comp. Mathematics; 1991.