DEVELOPMENT OF NEW HARMONIC EULER USING NONSTANDARD FINITE DIFFERENCE TECHNIQUE FOR SOLVING STIFF PROBLEMS
DOI:
https://doi.org/10.11113/jt.v77.6546Keywords:
Harmonic Euler, nonstandard, stiffAbstract
Solving stiff problem always required very tiny size of meshes if it is solved via traditional numerical algorithm. Using insufficient of mesh size, will triggered instabilities. In this paper, we develop an algorithm applying Harmonic Mean on Euler method to solve the stiff problems. The main purpose of this paper is to discuss the improvement of Harmonic Euler using Nonstandard Finite Difference (NSFD). The combination of these methods can provide new advantages that Euler method could offer. Four set of stiff problems are solved via three schemes, i.e. Harmonic Euler, Nonstandard Harmonic Euler and Nonstandard EO with Harmonic Euler. Findings show that both nonstandard schemes produce high accuracy results.
References
Moaddy, K., Hashim, I., Alomari, A. K., and Momani, S., A 2011. New Hybrid Non-standard Finite Difference-Adomian Scheme for Solution of Nonlinear Equations. Sains Malaysiana. 40: 515-519.
E. M. R. 2005. Advances in the Application of Nonstandard Finite Differences Schemes. Singapore. World Scientific.
Manning, P. M., and Margrave, G. F. 2006. Introduction to Non-standard Finite Difference Modelling. 1-10.
E. M. R., 1999. An Introduction to Nonstandard Finite Difference Scheme. Journal of Computational Acoustics. 7(1): 39-58.
Mickens, R. E. 2001. Nonstandard Finite Difference Schemes for Differential Equations. Journal of Differential Equations and Applications. 8: 823-847.
Mickens. R. E and Smith, A., 1990. Finite-Difference Models of Ordinary Differential Equations: Influence of Denominator Functions. Journal of Franklin Institute. 327: 143-149.
Ibijola, E. A., and Obayomi. 2011. Derivation of New Non-standard Finite Difference Schemes. 2: 1-7.
Ibijola, E. A., and Obayomi, A. A. 2012. Numerical Scehemes Based on Non-standard Methods for Initial Value Problems in Ordinary Differential Equations. Journal of Emerging Trends in Engineering and Applied Sciences. 3: 49-55.
Ibijola, E. A., and Obayomi, A. A. 2012. Derivation of New Non- standard Finite Difference Schemes for Non-autonomous Ordinary Differential Equation. American Journal of Scientific and Industrial Research. 122-127.
Ibijola. E. A, L., J. M. S., and Ade-Ibijola, O. A. 2008. On Nonstandard Finite Difference Schemes for Initial Value Problems in Ordinary Differential Equations. International Journal of Physical Science. 3: 59-64.
Gurski, K. F. 2013. A Simple Construction of Nonstandard Finite- difference Schemes for Small Nonlinear Systems Applied to SIR models Computers and Mathematics. 2165-2177.
Erdogan, U., and Ozis, T., 2011. A Smart Nonstandard Finite Difference Scheme for Second Order Nonlinear Boundary Value Problems. Journal of Computational Physics. 230: 6464-6474.
Fadugba. S., O. B., and Onkulola. T, 2012. Euler's Method for Solving Initial Value Problems in Ordinary Differential Equation. The Pacific Journal of Science and Technology. 13: 152-158.
Nurhafizah Moziyana, M. Y., Mohammad Khatim, H., and Masura, R. 2015. Comparison New Algorithm Modified Euler in Ordinary Differential Equation Using Scilab Programming. Lecture Notes on Software Engineering. 3(3): 199-202.
Nurhafizah Moziyana, M. Y., and Mohammad Khatim, H. 2014. Perbandingan Kaedah Euler dengan Tiga Kaedah Euler Terubahsuai pada Masalah Nilai Awal. Faculty of Information Science and Technology. 1-13.
Henrici, P. 1962. Discrete Variable Methods in Ordinary Differential Equation. John Wiley & Sons.
Kreyszig, E. 2006. Advanced Engineering Mathematics. 9ed. John Wiley and Sons. 19-20 and 906-907.
Zarina, B. I., Mohamed, S., and Khairil, I. 2009. Penyelesaian Persamaan Pembezaan Biasa Kaku Menggunakan Kaedah Blok Formulasi Beza Ke Belakang. MATEMATIKA. 25(1): 9-14.
Burden, R. L., and Faires, J. D. 2005. Numerical Analysis. 8 ed. Thomson.
Marini, A. B., Norleyza. J., and Sufian. I. 2004. Pengaturcaraan C. Prentice Hall.
Universiti Teknologi Malaysia, Ensiklopedia Sains dan Teknologi, in Teknologi Maklumat. 2005, Universiti Teknologi Malaysia, Dewan Bahasa dan Pustaka and Kementerian Pelajaran Malaysia. 4-5.
Zarina, B. I., Mohamed, S., Khairil, I. and Zanariah, M. 2005. Block Method for Generalised Multistep Adams and Backwards Differentiation Formulae In Solving First Order ODEs. MATEMATIKA. 21(1): 25-11.
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