THREE-STEP SECOND DERIVATIVE HYBRID BLOCK BACKWARD DIFFERENTIATION FORMULAE FOR SOLVING SYSTEM OF DIFFERENTIAL ALGEBRAIC EQUATIONS (DAES)
Keywords:
Differentiation-index, backward differentiation formula, Hybrid Block, Physical model, L- stability and A-stability
Abstract
This paper presents a new Second Derivative Hybrid Block Backward Differentiation Formulae (SDHBBDF) for solving series of engineering problems that are represented by some sets of Dierential-Algebraic Equations (DAEs). The main and complimentary methods, were developed by collocation and interpolation techniques that are combined as a set of block equations. The analysis of the method showed that it is consistent, convergent, and satisfied the L-stability condition. The SDHBBDF was implemented on some physical problems of DAEs with broad intervals and the numerical results demonstrated that the method is accurate, efficient and suitable for solving DAEs. Moreso, the method compared favorably well with some excellent methods in the literature.
References
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[27] J. Sand. On implicit Euler for high-order high-index DAEs. Applied Numerical Mathematics. 42 (2002), 411424.
[28] G.R. Duan. Analysis and design of Descriptor Linear System, Advances in Mechanics and Mathematics, Springer Science and Business Media. Advances in Mechanics and Mathematics, Springer Science and Business Media Springer 14 (2010).
[29] Hairer, Ernst and Wanner, Gerhard. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Verlag Series in Comput. Math., doi:10.1007/978-3-662-09947-6, 14, (1996).
[2] Cash, JR. On the exponential fitting of composite, multiderivative linear multistep methods. SIAM Journal on Numerical Analysis 18 (1981), 808-821.
[3] Zaboon, R. A. and Abd, G. F.. Solution of Time-Varying Index-2 Linear Differential Algebraic Control Systems Via a Variational Formulation Technique. Iraqi Journal of Science. 62 (2021), 3656-3671, DOI: 10.24996/ijs.2021.62.10.24.
[4] Radhi A. Zaboon, Ghazwa F. Abd. Approximate Solution of a Linear Descriptor Dynamic Control System via a Non-Classical Variation AL Approach. IOSR Journal of Mathematics (IOSR-JM). 11 (2015), ISSN: 2278-5728, www.iosrjournals.org pages 01-11.
[5] Soltanin, F. Karbassi, S. M. Dehghan, M.. Solution of the differential algebraic equations via homotopy perturbation method and their engineering applications. International Journal of Computer Mathematics, vol.87, no.9,(2019), 1950-1974.
[6] Ercan Celik and Mustafa Bayram. The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs). Journal of the Franklin Institute Vol. 342 (2005), 1-6.
[7] Ercan Celik and Mustafa Bayram. On the numerical solution of chemical differential algebraic equations by Pade series Applied Mathematics and Computation. Matematik Bolumu Fen-Edebiyat Fakultesi, Ataturk Universitesi, 25240 Erzurum, Turkey. Vol. 153 (2004), 13-17.
[8] Chein-Shan Liu. Solving Nonlinear Differential Algebraic Equations by an Implicit GL(n;R) Lie-Group Method Hindawi Publishing Corporation. Journal of Applied Mathematics. Vol. 2013 (2013), doi:http://dx.doi.org/10.1155/2013/987905. 8 pages.
[9] Benhammouda B and Vazquez-Leal H. Analytical solution of a nonlinear index three DAEs system modelling a slidercrank mechanism.. Discrete Dyn Nat Soc.
[10] Brenan KE. Stability and convergence of di erence approximations for higher index differential algebraic systems with applications in trajectory control. Ph.D. thesis, Department of Mathematics, University of California, Los Angeles. Vol. 2013 (1983), 8 pages.
[11] elik. E, Bayram M, Yeloglu T. Solution of differential algebraic equations (DAEs) by Adomian Decomposition Method. Int J. Pure Appl Math Sci. Vol. 3, (2006),93-100 .
[12] Brenan, K.E Campbell, S.L. Petzold L.R.. Numerical Solution of Initial Value Problems in Differential Algebraic Equations. Elsevier, New York.
[13] Ascher, U.M. and Petzold, L.R.. Computer Method for Ordinary Differential Equations and Differential-Algebraic Equations.. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. 14 (1998), doi:10.1137/1.9781611971392. 28-49
[14] Hosseini, M. M.. Numerical solution of linear differential-algebraic equations. Applied mathematics and computation. 162 (2005), 7-14.
[15] Henrici, P. Discrete Variable Methods in Ordinary Differential Equations. John Wiley and Sons, New York. 14 (1962)
[16] Gear, C. and Petzold, Linda. ODE Methods For the Solution of Differential/Algebraic Systems. SIAM Journal on Numerical Analysis. 21 (1982), doi:10.1137/0721048.
[17] Duan, N. and Sun, K.. Finding semi-analytic solutions of power system differential algebraic equations for fast transient stability simulation. url: arXiv:1412.0904 (2014).
[18] Ascher, U.M, and Lin, P.. Sequential regularization methods for higher index DAEs with constraint singularities: The linear index-2 case.. SIAM journal on numerical analysis. 33 (1996), 1921-1940.
[19] Ascher, U., and Lin, P.. Sequential regularization methods for nonlinear higher-index DAEs. SIAM Journal on Scienti c Computing. SIAM Journal on Scientific Computing. 14 (1997), 160-181.
[20] Gear, C. and Petzold, Linda. ODE Methods For the Solution of Differential-Algebraic Systems. SIAM Journal on Numerical Analysis,doi:10.1137/0721048 21 (1984), 757{774.
[21] Ascher, U.. On numerical differential algebraic problems with application to semiconductor device simulation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. 26 (1989), 517-538.
[22] Simeon, Bernd and Fhrer, Claus and Rentrop, Peter. Differential-algebraic Equations in Vehicle System Dynamics. Surv. Math. Ind. 1 (1991), 1-37.
[23] Petzold L.R. Numerical solution of differential-algebraic equations. Adv. Numer. Anal. Vol. 4 (1995).
[24] S.L. Campbell . A computational method for general higher index singular systems of differential equations. IMACS Trans. Sci. Comput. Vol. 89 (1989), 555560.
[25] E. Celik and M. Bayram. On the numerical solution of differential-algebraic equation by Pade series. Appl. Math. Comput. 137 (2003), 151160.
[26] L.M. Skvorsov. A Fifth Order Implicit Method for the Numerical Solution of Differential-Algebraic Equations. ISSN 0965 5425, Computational Mathematics and Mathematical Physics. 55 (2015), 962968.
[27] J. Sand. On implicit Euler for high-order high-index DAEs. Applied Numerical Mathematics. 42 (2002), 411424.
[28] G.R. Duan. Analysis and design of Descriptor Linear System, Advances in Mechanics and Mathematics, Springer Science and Business Media. Advances in Mechanics and Mathematics, Springer Science and Business Media Springer 14 (2010).
[29] Hairer, Ernst and Wanner, Gerhard. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Verlag Series in Comput. Math., doi:10.1007/978-3-662-09947-6, 14, (1996).
Published
2024-07-15
How to Cite
SONEYE, R. A., AKINFENWA, O. A., OSILAGUN, J. A., & OKUNUGA, S. A. (2024). THREE-STEP SECOND DERIVATIVE HYBRID BLOCK BACKWARD DIFFERENTIATION FORMULAE FOR SOLVING SYSTEM OF DIFFERENTIAL ALGEBRAIC EQUATIONS (DAES). Unilag Journal of Mathematics and Applications, 3, 81-101. Retrieved from http://lagjma.unilag.edu.ng/article/view/2146
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Copyright (c) 2023 Ramoni Adebola SONEYE, Oluseye Aremu AKINFENWA, Johnson Adekunle OSILAGUN, Solomon Adebola OKUNUGA
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