Sains Malaysiana 41(2)(2012): 253–260

High Order Explicit Hybrid Methods for Solving Second-Order Ordinary Differential Equations

(Kaedah Hibrid Tak Tersirat Peringkat Tinggi bagi Menyelesaikan Persamaan Pembezaan

Biasa Peringkat-Dua)

F. Samat*, F. Ismail & M. Suleiman

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia

43400 UPM Serdang, Selangor D.E. Malaysia

Received: 18 March 2011 / Accepted: 19 August 2011

ABSTRACT

Two explicit hybrid methods with algebraic order seven for the numerical integration of second-order ordinary differential equations of the form y̋ = f (x, y) are developed. The algebraic order of these methods is the highest in comparison with other explicit hybrid methods of the same class. Numerical comparisons carried out show the advantage of the new methods.

Keywords: Algebraic order; explicit hybrid method; second-order ordinary differential equations

ABSTRAK

Dua kaedah hibrid tak tersirat dengan peringkat algebra tujuh untuk pengamiran berangka persamaan pembezaan biasa peringkat-dua berbentuk y̋ = f (x, y) dibangunkan. Peringkat algebra bagi kaedah-kaedah ini adalah yang tertinggi berbanding dengan kaedah hibrid tak tersirat lain dalam kelas yang sama. Perbandingan berangka yang dilakukan menunjukkan kelebihan bagi kaedah-kaedah baru ini.

Kata kunci: Kaedah hibrid tak tersirat; peringkat algebra; persamaan pembezaan biasa peringkat-dua

REFERENCES

Brusa, L. & Nigro, L. 1980. A one-step method for direct integration of structural dynamic equations. International Journal for Numerical Methods in Engineering 15: 685-699.

Cash, J.R. 1981. High order P-stable formulae for the numerical integration of periodic initial value problems. Numerische Mathematik 37: 355-370.

Chan, R.P.K., Leone, P. & Tsai, A. 2004. Order conditions and symmetry for two-step hybrid methods. International Journal of Computer Mathematics 81(12): 1519-1536

Chawla, M.M. 1984. Numerov made explicit has better stability. BIT 24: 117-118.

Coleman, J.P. 2003. Order conditions for a class of two-step methods for y̋ = f (x,y). Journal of Numerical Analysis 23: 197-220.

Fatunla, S.O., Ikhile M.N.O. & Otunta, F.O. 1999. A class of P-stable linear multistep numerical methods. International Journal of Computer Mathematics 72: 1-13.

Franco, J. M. 2006. A class of explicit two-step hybrid methods for second-order IVPs. Journal of Computational .and Applied Mathematics 187: 41-57.

Hairer, E. 1979. Unconditionally stable methods for second order differential equations, Numerische Mathematik 32: 373-379.

Tsitouras, Ch. 2002. Explicit two-step methods for second-order linear IVPs. Computers and Mathematics with Applications 43: 943-949.

Tsitouras, Ch. 2003. Families of explicit two-step methods for integration of problems with oscillating solutions. Applied Mathematics and Computation 135: 169-178.

Tsitouras, Ch. 2006. Stage reduction on P-stable Numerov type methods of eighth order. Journal of Computational and Applied.Mathematics 191: 297-305.

van Der Houwen, P.J & Sommeijer, B.P. 1987. Explicit Runge-Kutta (-Nystrom) methods with reduced phase errors for computing oscillating solutions. SIAM Journal of Numerical Analysis 24: 595-617.

*Corresponding author; email: faieza77@yahoo.com