Sains Malaysiana 43(8)(2014): 1259-1262

 

A Third Order Nakashima Type Implicit Pseudo Runge-Kutta Method

for Delay Differential Equations

(Kaedah Runge-Kutta Nakashima Jenis 'Pseudo' Bertahap Tiga untuk Persamaan Perbezaan Lengah)

 

Lim Tian Hwee*

Foundation in Business, University of Reading Malaysia, Level 7, Menara Kotaraya, Jalan Trus
80000 Johor Bahru, Johor, Malaysia

 

Received: 24 July 2012/Accepted: 6 November 2013

 

ABSTRACT

 

A third order Nakashima type implicit Pseudo Runge-Kutta method is presented. The free parameter was determined by minimizing the error bound. The stability region of the method was presented. Some problems on delay differential equations are tested to compare the accuracy of the proposed method with third order RADAU I.

 

Keywords: Delay differential equations; implicit pseudo Runge-Kutta method; third order

 

ABSTRAK

 

Satu kaedah Runge-Kutta Nakashima tersirat jenis “pseudo” bertahap tiga telah diterbitkan. Parameter bebas telah ditentukan dengan meminimumkan batas ralat. Rantau kestabilan kaedah tersebut juga dipersembahkan. Beberapa soalan persamaan pembezaan lengah telah diuji untuk dibanding kejituan kaedah yang diteerbitkan dengan RADAU I bertahab tiga.

 

Kata kunci: Bertahap tiga; kaedah Runge-Kutta jenis 'pseudo' tersirat; persamaan perbezaan lengah

 

REFERENCES

 

Bellen, A. & Zennaro, M. 2003. Numerical Methods for Delay Differential Equations. Oxford: Clarendon Press.

Karoui, A. 1992. On the numerical solution of delay differential equations. Master Thesis. University of Ottawa, Ontario (Unpublished).

Ismail, F., Al-Khasawneh, R.A., Aung, S.L. & Suleiman, M. 2002. Numerical treatment of delay differential equations by Runge-Kutta method using Hermite interpolation. Matematika UTM 18(2): 79-90.

Lotkin, M. 1951. On the accuracy of Runge-Kutta methods. Mathematical Tables and Other Aids to Computation 5(35): 128-133.

Nakashima, M. 1982. On Pseudo-Runge-Kutta Methods with 2 and 3 stages. Publ. RIMS, Kyoto Univ. 18: 895-909.

Orbele, H.J. & Pesch, H.J. 1981. Numerical treatment of delay differential equations by Hermite interpolation. Numer. Math. 31: 235-255.

Ralston, A. 1962. Runge-Kutta methods with minimum error bounds. Math. Comp. 16: 431-437.

Shintani, H. 1981. On pseudo-Runge-Kutta Methods of the third kind. Hiroshima Math. J. 11: 247-254.

Yaacob, N., Sabri, A.A., Samsudin, N. & Lim, T.H. 2011. Application of Nakashima's 2 stages 4thorder Pseudo-Runge-Kutta method and 3 stages 5th order Pseudo-Runge-Kutta Method to Delay Problems. UTM Technical Report, LT/M BIL.6/2011.

 

 

*Corresponding author; email: t.h.lim@reading.ac.uk

 

 

 
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