Geodesic
Zero Dissipative DIRKN Pairs of Order 5(4) for Solving Special Second Order IVPs
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 53 (2014) no. 2, pp. 53-69 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

For initial value problem (IVPs) in ordinary second order differential equations of the special form $y^{\prime \prime }=f\left(x,y\right)$ possessing oscillating solutions, diagonally implicit Runge–Kutta–Nystrom (DIRKN) formula-pairs of orders 5(4) in 5-stages are derived in this paper. The method is zero dissipative, thus it possesses a non-empty interval of periodicity. Some numerical results are presented to show the applicability of the new method compared with existing Runge–Kutta (RK) method applied to the problem reduced to first-order system.
For initial value problem (IVPs) in ordinary second order differential equations of the special form $y^{\prime \prime }=f\left(x,y\right)$ possessing oscillating solutions, diagonally implicit Runge–Kutta–Nystrom (DIRKN) formula-pairs of orders 5(4) in 5-stages are derived in this paper. The method is zero dissipative, thus it possesses a non-empty interval of periodicity. Some numerical results are presented to show the applicability of the new method compared with existing Runge–Kutta (RK) method applied to the problem reduced to first-order system.
Classification : 65L05
Keywords: Initial value problems; Runge–Kutta–Nystrom pairs; zero dissipative 
@article{AUPO_2014_53_2_a3,
     author = {Imoni, S. O. and Ikhile, M. N. O.},
     title = {Zero {Dissipative} {DIRKN} {Pairs} of {Order} 5(4) for {Solving} {Special} {Second} {Order} {IVPs}},
     journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
     pages = {53--69},
     year = {2014},
     volume = {53},
     number = {2},
     mrnumber = {3331006},
     zbl = {1311.65098},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUPO_2014_53_2_a3/}
}
TY  - JOUR
AU  - Imoni, S. O.
AU  - Ikhile, M. N. O.
TI  - Zero Dissipative DIRKN Pairs of Order 5(4) for Solving Special Second Order IVPs
JO  - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY  - 2014
SP  - 53
EP  - 69
VL  - 53
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/AUPO_2014_53_2_a3/
LA  - en
ID  - AUPO_2014_53_2_a3
ER  - 
%0 Journal Article
%A Imoni, S. O.
%A Ikhile, M. N. O.
%T Zero Dissipative DIRKN Pairs of Order 5(4) for Solving Special Second Order IVPs
%J Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
%D 2014
%P 53-69
%V 53
%N 2
%U http://geodesic.mathdoc.fr/item/AUPO_2014_53_2_a3/
%G en
%F AUPO_2014_53_2_a3
Imoni, S. O.; Ikhile, M. N. O. Zero Dissipative DIRKN Pairs of Order 5(4) for Solving Special Second Order IVPs. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 53 (2014) no. 2, pp. 53-69. http://geodesic.mathdoc.fr/item/AUPO_2014_53_2_a3/

[1] Allen, R. C., Wing, G. M.: An invariant imbedding algorithm for the solution of inhomogeneous linear two-point boundary value-problems. J. Computer Physics 14 (1974), 40–45. | DOI | MR | Zbl

[2] Butcher, J. C., Chen, D. J.: A new Type of Singly-implicit Runge–Kutta Method. Applied Numerical Mathematics 34 (2000), 179–188. | DOI | MR | Zbl

[3] Dormand, J. R., El-Mikkawy, M. E. A., Prince, P. J.: Families of Runge–Kutta Nystrom Formulas. IMA J. Num. Analysis 7 (1987), 235–250. | DOI | MR

[4] El-Mikkawy, M., El-Dousky, R. A.: New optimized non-FSAL Embedded Runge–Kutta Nystrom algorithms of orders 6 and 4 in six stages. Applied Mathematics and computer 145 (2003), 33–43. | DOI | MR

[5] Fehlberg, E.: Classical eight and lower-order Runge–Kutta–Nystrom formulae with step size control for the special second-order differential equations. Technical Report, R-381, NASA, 1972.

[6] Fillipi, S., Graf, J.: New Runge–Kutta–Nystrom Formulae pairs of order $8(7)$, $9(8)$, $10(9)$ and $11(10)$ for differential equation of the form $y^{\prime \prime } =f(x,y)$. J. computing Appl. Math. 14 (1986), 362–370. | MR

[7] Ferracina, L., Spijker, M. N.: Strong stability of singly-diagonally implicit Runge–Kutta methods. Applied Numerical Mathematics 58 (2008), 1675–1686. | DOI | MR | Zbl

[8] Hairer, E., Norsett, S. P., Wanner, G.: Solving Ordinary Differential Equations I, Non Stiff problems. springer-Verlag, Berlin, 1993. | MR

[9] Imoni, S. O., Otunta, F. O., Ramamohan, T. R.: Embedded implicit Runge–Kutta–Nystrom method for solving second order differential equations. International Journal of Computer Mathematics 83, 11 (2006), 777–784. | DOI | MR | Zbl

[10] Imoni, S. O., Otunta, F. O., Ikhile, M. N. O.: Embedded diagonally implicit Runge–Kutta–Nystrom method of order two and three for special second order differential equations. The Journal of the Mathematical Association of Nigeria (ABACUS) 34, 213 (2007), 363–373.

[11] Kanagarajam, K., Sambath, M.: Runge–Kutta–Nystrom method of order three for solving Fuzzy Differential Equations. Computational methods in Applied Mathematics 10, 2 (2010), 195–203. | MR

[12] Papakostas, S. N., Tsitouras, Ch.: High phase-lag order Runge–Kutta–Nystrom pairs. SIAM J. Sci. Comput. 21 (2002), 747–763. | DOI | MR

[13] Qinghong, Li, Yongzhong, S.: Explicit one-step p-stable methods for second order periodic initial value problems. Numerical Mathematics, Journal of Chinese Universities (English series) 15, 3 (2006), 237–247. | MR | Zbl

[14] Sharp, P. W., Fine, J. M.: A contrast of direct and transformed Nystrom pairs. J. Comput. Appl. Math 42 (1992), 293–308. | DOI | MR | Zbl

[15] Sharp, P. W., Fine, J. M., Burrage, K.: Two-stage and three-stage diagonally implicit Runge–Kutta–Nystrom methods of orders three and four. IMA Journal Numerical Analysis 10 (1990), 489–504. | DOI | MR | Zbl

[16] Sommeijer, B. P.: A note on a diagonally implicit Runge–Kutta–Nystrom Method. J. Comput. Appl. Math. 19 (1987), 395–399. | DOI | MR | Zbl

[17] Simos, T. E., Famelis, I. T., Tsitouras, Ch.: Zero dissipative, explicit Numerov-type methods for second order IVPs with oscillating solutions. Numerical Algorithms 34 (2003), 27–40. | DOI | MR | Zbl

[18] Simos, T. E., Dimas, E., Sideridis, A. B.: A Runge–Kutta–Nystrom method for the numerical integration of special second-order periodic initial-value problems. J. Computational and Applied Maths. 51 (1994), 317–326. | DOI | MR | Zbl

[19] Van der Houwen, P. J., Sommeijer, B. P.: Diagonally Implicit Runge–Kutta–Nystrom methods for oscillatory problems. SIAM J. Numerical Analysis 26, 2 (1989), 414–429. | DOI | MR | Zbl