Non-linear singular systems using RK–Butcher algorithms

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.

[1]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[2]  David J. Evans,et al.  Weighted fifth-order Runge-Kutta formulas for second-order differential equations , 1998, Int. J. Comput. Math..

[3]  J. Butcher.  Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods , 2005 .

[4]  David J. Evans,et al.  A comparison of extended runge-kutta formulae based on variety of means to solve system of ivps , 2001, Int. J. Comput. Math..

[5]  David J. Evans,et al.  Optimal control of singular systems using the rk–butcher algorithm , 2004, Int. J. Comput. Math..

[6]  Wen-June Wang,et al.  State analysis of time-varying singular bilinear systems via Haar wavelets , 2000 .

[7]  David J. Evans,et al.  A new fifth order weighted runge kutta formula , 1996, Int. J. Comput. Math..

[8]  David J. Evans,et al.  A fourth order Runge-Kutta RK(4, 4) method with error control , 1999, Int. J. Comput. Math..

[9]  L. Shampine,et al.  Computer solution of ordinary differential equations : the initial value problem , 1975 .

[10]  R. Alexander,et al.  Runge-Kutta methods and differential-algebraic systems , 1990 .

[11]  David J. Evans,et al.  A Fourth Order Embedded Runge-Kutta RKACeM(4,4) Method Based on Arithmetic and Centroidal Means with Error Control , 2002, Int. J. Comput. Math..

[12]  Jing-Yue Lin,et al.  Existence and uniqueness of solutions for non-linear singular (descriptor) systems , 1988 .

[13]  David J. Evans,et al.  Analysis of non-linear singular system from fluid dynamics using extended runge-kutta methods , 2000, Int. J. Comput. Math..

[14]  David J. Evans,et al.  Analysis of second order multivariate linear system using single term walsh series technique and runge kutta method , 1999, Int. J. Comput. Math..

[15]  Morris Bader.  A new technique for the early detection of stiffness in coupled differential equations and application to standard Runge-Kutta algorithms , 1998 .

[16]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[17]  K. Murugesan,et al.  Numerical solution of an industrial robot arm control problem using the RK-Butcher algorithm , 2004, Int. J. Comput. Appl. Technol..

[18]  K. Murugesan,et al.  Analysis of nonlinear singular systems via stws method , 1990, Int. J. Comput. Math..

[19]  Mohsen Razzaghi,et al.  Solution of time-varying singular nonlinear systems by single-term Walsh series , 2003 .

[20]  David J. Evans,et al.  New runge kutta starters for multistep methodsStarters for multistep methods , 1999, Int. J. Comput. Math..

[21]  Morris Bader.  A comparative study of new truncation error estimates and intrinsic accuracies of some higher order Runge-Kutta algorithms , 1987, Comput. Chem..

[22]  David J. Evans,et al.  Analysis of different second order systems via runge-kutta method , 1999, Int. J. Comput. Math..

[23]  David J. Evans.  A new 4th order runge-kutta method for initial value problems with error control , 1991, Int. J. Comput. Math..

[24]  L. Shampine,et al.  Numerical Solution of Ordinary Differential Equations. , 1995 .