Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method
The aim of this research is to investigate numerically the problem on solving Delay Differential Equations (DDEs) using Nakashima’s 2 Stages 4th Order Pseudo-Runge-Kutta Method. This research also discussed the problem on solving Ordinary Differential Equations (ODEs) using the same method. In this...
محفوظ في:
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| التنسيق: | أطروحة |
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2010
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my-utm-ep.21290 |
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uketd_dc |
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Universiti Teknologi Malaysia |
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UTM Institutional Repository |
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Q Science (General) Q Science (General) |
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Q Science (General) Q Science (General) Sabri, Nur Ain Ayunni Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method |
| description |
The aim of this research is to investigate numerically the problem on solving Delay Differential Equations (DDEs) using Nakashima’s 2 Stages 4th Order Pseudo-Runge-Kutta Method. This research also discussed the problem on solving Ordinary Differential Equations (ODEs) using the same method. In this research, we also developed the algorithm of Nakashima’s 2 Stages 4th Order Pseudo-Runge-Kutta Method and the algorithm of Nakashima’s 2 Stages 4th Order Pseudo-Runge-Kutta Method incorporated with Hermite Interpolation. Finally, we apply this method to a real life problem and we choose Food-Limited Model. In this research, we use Mathematica 7 software and Microsoft Excel to conduct the calculation. |
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Thesis |
| qualification_level |
Master's degree |
| author |
Sabri, Nur Ain Ayunni |
| author_facet |
Sabri, Nur Ain Ayunni |
| author_sort |
Sabri, Nur Ain Ayunni |
| title |
Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method |
| title_short |
Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method |
| title_full |
Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method |
| title_fullStr |
Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method |
| title_full_unstemmed |
Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method |
| title_sort |
solving ordinary differential equations (odes) and delay differential equations (ddes) using nakashima’s 2 stages 4th order pseudo-runge-kutta method |
| granting_institution |
Universiti Teknologi Malaysia, Faculty of Science |
| granting_department |
Faculty of Science |
| publishDate |
2010 |
| _version_ |
1747815420185804800 |
| spelling |
my-utm-ep.212902020-03-03T07:30:46Z Solving ordinary differential equations (ODES) and delay differential equations (DDES) using Nakashima’s 2 stages 4Th order Pseudo-Runge-Kutta method 2010 Sabri, Nur Ain Ayunni Q Science (General) QA75 Electronic computers. Computer science The aim of this research is to investigate numerically the problem on solving Delay Differential Equations (DDEs) using Nakashima’s 2 Stages 4th Order Pseudo-Runge-Kutta Method. This research also discussed the problem on solving Ordinary Differential Equations (ODEs) using the same method. In this research, we also developed the algorithm of Nakashima’s 2 Stages 4th Order Pseudo-Runge-Kutta Method and the algorithm of Nakashima’s 2 Stages 4th Order Pseudo-Runge-Kutta Method incorporated with Hermite Interpolation. Finally, we apply this method to a real life problem and we choose Food-Limited Model. In this research, we use Mathematica 7 software and Microsoft Excel to conduct the calculation. 2010 Thesis http://eprints.utm.my/id/eprint/21290/ masters Universiti Teknologi Malaysia, Faculty of Science Faculty of Science [1] Nakashima, M. On Pseudo-Runge-Kutta Methods with 2 and 3 stages. Publications RIMS, Kyoto University. 1982. 18,pp.895-909. [2] Nakashima, M. Implicit Pseudo-Runge-Kutta Processes. Publications RIMS, Kyoto University. 1984. 20,pp.39-56. [3] Byrne, G.D. Pseudo-Runge-Kutta Methods Involving Two Points. Jourrnal of the Association for Computing Machinery. 1966. Vol.13(1), pp.114-123. [4] Byrne, G.D. Parameters for Pseudo-Runge-Kutta Methods. Pittsburgh Publications, University of Pittsburgh. 1967. Vol.10(2). [5] William, B.G. Pseudo-Runge-Kutta Method of the Fifth Order. Journal of the Association for Computing Machinery. 1970. Vol.17(4), pp.613-628. [6] Nakashima, M. Implicit Pseudo-Runge-Kutta Methods. Siam Journal Numerical Analysis. 1991. Vol.28(6), pp.1790-1802. [7] Shintani, H. On Pseudo-Runge-Kutta Methods of the Third Kind. Hiroshima Mathematics Journal. 1981. Vol.11, pp.247-254. [8] Al-Mutib, A.N. One-Step Implicit Methods for Solving Delay Differential Equations. International Journal of Computer Mathematics. 1984. Vol.16, pp.157-168. [9] Ishak. F., Suleiman, M.B., and Omar. Z. Two-Point Predictor-Corrector Block Methods for Solving Delay Differential Equations. Matematika. 2008. Vol.24(2), pp.131-140. [10] Lwin, A.S. Solving Delay Differential Equations Using Explicit Runge-Kutta Methods. Master Thesis. Universiti Putra Malaysia. 2004 [11] Wong, W.S., Zhang, Y., and Li, S. Stability of Continuous Runge-Kutta Methods for Non-Linear Neutral Delay Differential Equations. Applied Mathematical Modeling. 2009. Vol.33, pp.3319-3329. [12] Ismail, F., and Suleiman, M.B. Solving Delay Differential Equations Using Intervalwise Partitioning by Runge-Kutta Method. Applied Mathematics and Computation. 2001. Vol,121, pp.37-53. [13] Ismail, F. et al .Numerical Treatment of Delay Differential Equations by Runge-Kutta Method using Hermite Interpolation. Matematika. 2002. Vol.18(2), pp.79-90. [14] Bellen, A., and Zennaro, M. Numerical Methods for Delay Differential Equations. United States: Oxford Science Publications. 2003. [15] Tenenbaum, M., and Pollard, H. Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering and the Sciences. United States of America: Dover Publications. [16] Huang, C. et al. D-Convergence of General Linear Methods for Stiff Delay Differential Equations. Computers and Mathematics with Applications. 2001. Vol.41, pp.627-639. [17] Verheyden, K., Luzyanina, T., and Roose, D. Efficient Computation of Characteristic Roots of Delay Differential Equations Using Linear Multistep Method. Journal of Computational and Applied Mathematics. 2008. Vol.214, pp.209-226. [18] Zhang, C., and Sun, G. Nonlinear Stability of Runge-Kutta Methods Applied to Infinite Delay Differential Equations. Mathematical and Computer Modeling. 2004. Vol.39, pp.495-503. [19] Hout, K.J. On The Stability of Adaptations Of Runge-Kutta Methods to System of Delay Differential Equations. Applied Numerical Mathematics. 1996. Vol.22, pp.237-250. [20] Bartoszewski, Z., and Jackiewicz, Z. Stability Analysis of Two-Step Runge- Kutta Method for Delay Differential Equations. Computers and Mathematics With Application. 2002. Vol.44, pp.83-93. [21] Butcher, J.C. Numerical Methods for Ordinary Differential Equation. 2nd ed. John Willey and Sons Ltd. 2008 [22] Berezonsky, L., and Braverman, E. On Oscilation of a Food-Limited Population Model With Time Delay. Abstract and Applied Analysis. 2003, pp.55-66. [23] Arino, O. et al. Delay Differential Equations and Applications. Berlin:Springer. pp.477-517. 2006. |