Non-linear singular systems using RK–Butcher algorithms

Abstract

The Runge–Kutta (RK)–Butcher algorithm is used to study time-invariant and time-varying non-linear singular systems. The results (discrete solutions) obtained using the RK method based on the arithmetic mean (RKAM), single-term Walsh series (STWS) and RK–Butcher algorithms are compared with the exact solutions of the non-linear singular systems for the time-invariant and time-varying cases. It is found that the solution obtained using the RK–Butcher algorithm is closer to the exact solutions of the non-linear singular systems. Stability regions for the RKAM, STWS and RK–Butcher algorithms are presented. Error graphs for discrete and exact solutions are presented in a graphical form to highlight the efficiency of this method. The RK–Butcher algorithm can easily be implemented using a digital computer and the solution can be obtained for any length of time for both time-invariant and time-varying cases for these non-linear singular systems, which is an added advantage of this algorithm.

Keywords:

• Time-invariant non-linear singular systems
• Time-varying non-linear singular systems
• STWS algorithm
• RKAM algorithm
• RK–Butcher algorithm

Acknowledgements

One of the authors Dr. K. Murugesan would like to thank Professor Jong Yeoul Park, Depart-ment of Mathematics, Pusan National University, Pusan 609-735, Republic of Korea, for his kind help during the stay in Pusan National University, and also, he would like to thank the Korean Science and Engineering Foundation (KOSEF), Republic of Korea, for the financial assistance provided to him under the Brain Pool Program.