Sains
Malaysiana 39(5)(2010): 851–857
Performance of Euler-Maruyama, 2-Stage
SRK and 4-Stage SRK in Approximating the Strong Solution of Stochastic Model
(Keberkesanan
Kaedah Euler-Maruyama, Stokastik Runge-Kutta Peringkat 2, Stokastik
Runge-Kutta
Peringkat 4 dalam Mencari Penyelesaian Penghampiran Model Stokastik)
Norhayati Rosli*, Arifah
Bahar, Yeak Su Hoe & Haliza Abdul Rahman
Department of Mathematics, Faculty
of Science
Universiti Teknologi Malaysia,
81310 UTM Skudai,
Johor, Malaysia
Madihah Md. Salleh
Faculty of Bioscience and
Bioengineering, Universiti Teknologi Malaysia
81310 UTM Skudai, Johor, Malaysia
Diserahkan: 13 Ogos 2009 / Diterima:
19 Mac 2010
ABSTRACT
Stochastic
differential equations play a prominent role in many application areas
including finance, biology and epidemiology. By incorporating random elements
to ordinary differential equation system, a system of stochastic differential
equations (SDEs) arises. This leads to a
more complex insight of the physical phenomena than their deterministic
counterpart. However, most of the SDEs do not have an analytical solution where numerical method is
the best way to resolve this problem. Recently, much work had been done in
applying numerical methods for solving SDEs. A very general class of Stochastic Runge-Kutta, (SRK) had been studied and 2-stage SRK with order convergence of 1.0 and 4-stage SRK with order convergence of 1.5 were discussed. In this study, we
compared the performance of Euler-Maruyama, 2-stage SRK and 4-stage SRK in approximating
the strong solutions of stochastic logistic model which describe the cell
growth of C. acetobutylicum P262. The MS-stability functions of these schemes were calculated and
regions of MS-stability are given. We also
perform the comparison for the performance of these methods based on their
global errors.
Keywords:
2-stage stochastic Runge-Kutta; 4-stage stochastic Runge-Kutta; Euler-Maruyama;
stochastic differential equations
ABSTRAK
Persamaan
pembezaan stokastik memainkan peranan penting dalam kebanyakan bidang seperti
kewangan, biologi dan epidemiologi. Dengan menggabungkan elemen rawak terhadap
sistem persamaan pembezaan biasa, pesamaan pembezaan stokastik muncul. Ini
membawa kepada fenomena fizikal yang lebih kompleks berbanding dengan persamaan
deteministik yang setara dengannya. Walau bagaimanapun, persamaan pembezaan
stokastik tidak mempunyai penyelesaian analitik dan kaedah penyelesaian
berangka merupakan cara terbaik untuk mengatasi masalah ini. Pada abad ini,
banyak usaha telah dilakukan untuk mencari penyelesaian hampiran persamaan
pembezaan stokastik. Bentuk am kelas Stokastik Runge-Kutta, SRK telah dikaji dan secara khusunya SRK peringkat 2 dengan pangkat penumpuan 1.5 dan SRK peringkat 4 dengan pangkat penumpuan 2.0 telah dibincangkan.
Dalam kajian ini, kami melakukan perbandingan bagi melihat keberkesanan kaedah
Euler-Maruyama, SRK peringkat 2
dan SRK peringkat 4 bagi mencari
penyelesaian hampiran ke atas model logistik stokastik yang menerangkan kadar
pertumbuhan sel C. acetobutylicum P262. Keberkesanan kaedah tersebut telah
dibandingkan berdasarkan analisis stabiliti min kuasa dua dan ralat sejagat.
Kata kunci:
Euler-Maruyama; persamaan pembezaan stokastik; stokastik Runge-Kutta peringkat
2; stokastik Runge-Kutta peringkat 4
RUJUKAN
Arifah B. 2005. Applications of
Stochastic Differential Equations and Stochastic Delay Differential Equations
in Population Dynamics. PhD Thesis, University of Strathclyde.
Burrage K. & Burrage P.M. 1996. High Strong Order Explicit
Runge-Kutta Methods for Stochastic Ordinary Differential Equations. Applied
Numerical Mathematics 22: 81-101.
Haliza A.R., Arifah B, Mohd Khairul
Bazli, Norhayati R. & Madihah M.S. 2009. Parameter Estimation via Levenberg Marquardt of Stochastic
Differential Equations, 2nd International Conference and Workshops on Basic and Applied
Sciences and Regional Annual Fundamental Science Seminar 44-48.
Milstein G.N. 1974. Approximate Integration of Stochastic
Differential Equations. Theory Probability Applied 19: 557-562.
Oksendal, B. 2003. Stochastic
Differential Equations: An Introduction with Applications. New York:
Springer-Verlag.
Rumelin, W. 1982. Numerical
treatment of stochastic differential equations. SIAM J. Numer. Analysis 19:
604-613.
Saito, Y. & Mitsui, T. 1996. Stability
analysis of numerical schemes for stochastic differential equations. SIAM J.
Numer. Anal. 33(6): 2254-2267.
*Pengarang untuk surat-menyurat;
e-mail: norhayati@ump.edu.my