DIRECT INTEGRATION OF GENERAL FOURTH ORDER ORDINARY DIFFERENTIAL EQUATIONS USING FIFTH ORDER RUNGE-KUTTA METHOD
Abstract
In this work, the fifth-order Runge-Kutta method for the solution of first order Ordinary Differential Equations (ODEs) was modified for direct integration of general fourthorder ODEs via the idea as those invented by Nystr¨om. The theory of Nystr¨om was adopted in the derivation of the method. The method has an explicit structure for efficient implementation,self-starting produces simultaneously approximation of the solution of special and general fourth-order ODEs. The proposed method is direct, does not involve the reduction of higher order ODEs to system of first-order ODEs as in most recent related articles, convergence analysis of the method was presented, was tested with Numerical experiment to illustrate its efficiency, could be extended to solve higher-order differential equations, simple to implement and the approximate results are in good agreement with other methods mentioned in literature.
References
Tanner, L. H., The spreading of silicone oil drops on horizontal surfaces, Phys. D: Appl. Phys. 12 1473-1484, 1979.
Vigo - Aguinar J. Ramos H, Dissipative chebyshev exponential fitted method for numerical solution of second order differential equation, J. comp. Appl. Math. 158 187-211, 2003.
Butcher J.C and Wright W.M., The Construction of Practical General Linear Methods, BIT 43 695-721, 2003.
Agam S.A and Irhebbhude, Modification of the Fourth Order Runge-Kutta Method for Third Order Ordinary Differential Equations, Abacus 38 (2) 87-95, 2011.
Butcher J.C. General Linear Methods, Acta Numerical 157-256, 2005.
Hairer E. andWarner G., A Theory for Nystrom Method, Numerical Mathematical, 25 383-400, 1976.
Butcher J.C and Hojjat G., Second Derivative Methods with Runge-Kutta Stability, Num. alg. 40 415-429, 2005.
