Numerical strategies for the system of first-order IVPS using the RK–Butcher algorithm

References

• Alexander , R. K. and Coyle , J. J. 1990 . Runge–Kutta methods for differential-algebric systems . SIAM Journal on Numerical Analysis , 27 ( 3 ) : 736 – 752 .
   Web of Science ®Google Scholar
• Evans , D. J. 1991 . A new 4th order Runge–Kutta method for initial value problems with error control . International Journal of Computational Mathematics , 139 : 217 – 227 .
   Web of Science ®Google Scholar
• Evans , D. J. and Yaakub , A. R. 1996 . A new fifth order weighted Runge–Kutta formula . International Journal of Computational Mathematics , 59 : 227 – 243 .
   Web of Science ®Google Scholar
• Evans , D. J. and Yaakub , A. R. 1998 . Weighted fifth order Runge–Kutta formulas for second order differential equations . International Journal of Computational Mathematics , 70 : 233 – 239 .
   Web of Science ®Google Scholar
• Hung , C. 2000 . Dissipativity of Runge–Kutta methods for dynamical systems with delays . IMA Journal of Numerical Analysis , 20 : 153 – 166 .
   Web of Science ®Google Scholar
• Murugesan , K. , Paul Dhayabaran , D. , Henry Amirtharaj , E. C. and Evans , D. J. 1999 . Analysis of different second order systems via Runge–Kutta method . International Journal of Computational Mathematics , 70 : 477 – 493 .
   Web of Science ®Google Scholar
• Murugesan , K. , Paul Dhayabaran , D. , Henry Amirtharaj , E. C. and Evans , D. J. 1999 . Analysis of different second order multivariable linear system using single term Walsh Series technique and Runge–Kutta method . International Journal of Computational Mathematics , 72 : 367 – 374 .
   Web of Science ®Google Scholar
• Murugesan , K. , Paul Dhayabaran , D. , Henry Amirtharaj , E. C. and Evans , D. J. 2000 . Analysis of non-linear singular system from fluid dynamics using extended Runge–Kutta methods . International Journal of Computational Mathematics , 76 : 239 – 266 .
   Web of Science ®Google Scholar
• Murugesan , K. , Paul Dhayabaran , D. , Henry Amirtharaj , E. C. and Evans , D. J. 2001 . A comparison of extended Runge–Kutta formulae based on variety of means to solve system of IVPs . International Journal of Computational Mathematics , 78 : 225 – 252 .
   Web of Science ®Google Scholar
• Murugesan , K. , Paul Dhayabaran , D. , Henry Amirtharaj , E. C. and Evans , D. J. 2002 . A fourth order embedded Runge–Kutta RKACeM(4,4) method based on arithmetic and centrodial means with error control . International Journal of Computational Mathematics , 79 ( 2 ) : 247 – 269 .
   Web of Science ®Google Scholar
• Shampine , L. F. and Watts , H. A. 1977 . The Art of Writing a Runge–Kutta Code, Part I, Mathematical Software , Vol. 3 , 3 257 – 275 . London : Academic Press .
   Google Scholar
• Shampine , L. F. and Watts , H. A. 1979 . The Art of Writing a Runge–Kutta Code, Part II, Applied Mathematical Computations Vol. 5 , 93 – 121 .
   Google Scholar
• Yaakub , A. R. and Evans , D. J. 1999 . A fourth order Runge–Kutta RK (4,4) method with error control . International Journal of Computational Mathematics , 71 : 383 – 411 .
   Web of Science ®Google Scholar
• Yaakub , R. and Evans , D. J. 1999 . New Runge–Kutta starters of multi-step methods . International Journal of Computational Mathematics , 71 : 99 – 104 .
   Web of Science ®Google Scholar
• Butcher , J. C. 1964 . On Runge–Kutta processes of higher order . Journal of the Australian Mathematical Society , 4 : 179
   Google Scholar
• Butcher , J. C. 1987 . The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods , UK : John Wiley & Sons .
   Google Scholar
• Butcher , J. C. 1990 . On order reduction for Runge–Kutta methods applied to differential-algebraic systems and to stiff systems of ODE’s . SIAM Journal on Numerical Analysis , 27 : 447 – 456 .
   Web of Science ®Google Scholar
• Shampine , L. F. 1985 . Some practical Runge–Kutta formulas . Mathematical Computations , 46 : 135 – 150 .
   Google Scholar
• Fehlberg , E. 1970 . Klassische Runge–Kutta-Formeln vierter und niedriger Ordnung mit Schrittweitenkontolle und ihre Anwendung auf Warmeleitungsprobleme , 6 : 61 – 71 . Computing
   Google Scholar
• Forsythe , G. E. , Malcolm , M. A. and Moler , C. D. 1977 . Computer Methods for Mathematical Computations , 135 Englewood Cliffs, NJ : Prentice-Hall .
   Google Scholar
• Bader , M. 1987 . A comparative study of new truncation error estimates and intrinsic accuracies of some higher order Runge–Kutta algorithms . Computational Chemistry , 11 : 121 – 124 .
   Google Scholar
• Bader , M. 1998 . A new technique for the early detection of stiffness in coupled differential equations and application to standard Runge–Kutta algorithms . Theoretical Chemistry Accounts , 99 : 215 – 219 .
   Web of Science ®Google Scholar
• Shampine , L. F. and Gordon , M. K. 1975 . Computer Solutions of Ordinary Differential Equations , 23 San Francisco, CA : W.H. Freeman .
   Google Scholar