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Comparative Study of Matlab ODE Solvers for the Korakianitis and Shi Model
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Comparative Study of Matlab ODE Solvers for the Korakianitis and Shi Model

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Abstract:

Changing parameters of the Korakianitis and Shi heart valve model over a cardiac cycle has led to the investigation of appropriate numerical technique(s) for good speed and accuracy. Two sets of parameters were selected for the numerical test. For the seven MATLAB ODE solvers, the computed results, computational cost and execution time were observed for varied error tolerance and initial time steps. The results were evaluated with descriptive statistics; the Pearson correlation and ANOVA at The dependence of the computed result, accuracy of the method, computational cost and execution time of all the solvers, on relative tolerance and initial time steps were ascertained. Our findings provide important information that can be useful for selecting a MATLAB ODE solver suitable for differential equation with time varying parameters and changing stiffness properties.

Info:

Periodical:
Bulletin of Mathematical Sciences and Applications (Volume 19)
Pages:
31-44
DOI:
https://doi.org/10.18052/www.scipress.com/BMSA.19.31
Citation:
E. Emagbetere et al., "Comparative Study of Matlab ODE Solvers for the Korakianitis and Shi Model", Bulletin of Mathematical Sciences and Applications, Vol. 19, pp. 31-44, 2017
Online since:
August 2017
Authors:
Eyere Emagbetere, Oluleke O. Oluwole, Tajudeen A.O. Salau
Keywords:
Mathematical Model, MATLAB ODE Suite, Numerical Solvers, Ordinary Differential Equation
Export:
RIS, BibTeX, Text
Distribution:
Open Access

Creative Commons License
This work is licensed under a
Creative Commons Attribution 4.0 International License

References:

[1] T. Koriatinits, Y. Shi, Concentrated parameter model for the human cardiovascular system including heart valve dynamics and atrioventricular interaction, Med. Eng. Phys. 28(7) (2006) 623–628.

DOI: https://doi.org/10.1016/j.medengphy.2005.10.004

[2] M.N. Spijker, Stiffness in numerical initial value problems, Journal of Computational and Applied Mathematics. 72(2) (1996) 393–406.

DOI: https://doi.org/10.1016/0377-0427(96)00009-x

[3] T. Korakianitis, Y. Shi, Numerical simulation of cardiovascular dynamics with healthy and diseased heart valve, Journal of Biomechanics. 39 (2006) 1964 - (1982).

DOI: https://doi.org/10.1016/j.jbiomech.2005.06.016

[4] Y. Shi, Lumped-parameter modelling of cardiovascular system dynamics under different healthy and diseased conditions. Sheffield, UK: Thesis submitted to Department of Cardiovascular Science, Faculty of Medicine, Dentistry and Health, University of Sheffield, (2013).

[5] J.C. Butcher, Numerical methods for ordinary differential equations in the 20th century, Journal of Computational and Applied Mathematics. 125(1) (2000) 1–29.

DOI: https://doi.org/10.1016/s0377-0427(00)00455-6

[6] T.E. Hull, W.H. Enright, A.E. Sedgwick, Comparing numerical methods for ordinary differential equations, SIAM Journal on Numerical Analysis. 9(4) (1972) 603-637.

DOI: https://doi.org/10.1137/0709052

[7] F.T. Krogh, On testing a subroutine for the numerical integration of ordinary differential equations, Journal of the ACM (JACM). 20(4) (1973) 545-562.

DOI: https://doi.org/10.1145/321784.321786

[8] G.D. Byrne, A.C. Hindmarsh, Stiff ODE solvers: a review of current and coming attractions, Journal of Computational Physics. 70(1) (1987) 1-62.

DOI: https://doi.org/10.1016/0021-9991(87)90001-5

[9] T.D. Bui, A.K. Oppenheim, D.T. Pratt, Recent advances in methods for numerical solution of O.D.E. initial value problems, Journal of Computational and Applied Mathematics. 11 (1984) 283-296.

DOI: https://doi.org/10.1016/0377-0427(84)90003-7

[10] H.W. Enright, T.E. Hull, B. Lindberg, Comparing numerical methods for stiff systems of ODE's, BIT Numerical Mathematics. 15(1) 1975 10-48.

DOI: https://doi.org/10.1007/bf01932994

[11] L. F. Shampine, Stiff and nonstiff differential equation solvers, II: Detecting stiffness with Runge-Kutta methods, ACM Transactions on Mathematical Software (TOMS). 3(1) (1977) 44-53.

DOI: https://doi.org/10.1145/355719.355722

[12] L.F. Shampine, Evaluation of a test set for stiff ODE solvers, ACM Transactions on Mathematical Software (TOMS). 7(4) (1981) 409-420.

DOI: https://doi.org/10.1145/355972.355973

[13] L. Petzold, Automatic selection of methods for solving stiff and non-stiff systems of ordinary differential equations, SIAM Journal on Scientific and Statistical Computing, 4(1) (1983) 136-148.

DOI: https://doi.org/10.1137/0904010

[14] W.H. Enright, J.D. Pryce, Two FORTRAN packages for assessing initial value methods, ACM Transactions on Mathematical Software (TOMS). 13(1) (1987) 1-27.

DOI: https://doi.org/10.1145/23002.27645

[15] L.F. Shampine, R.M. Corless, Initial value problems for ODEs in problem solving environments, Journal of Computational and Applied Mathematics. 125 (2000) 31-40.

DOI: https://doi.org/10.1016/s0377-0427(00)00456-8

[16] D. Petcu, Software issues in solving initial value problems for ordinary differential equations, Creative Math. 13 (2004) 97-110.

[17] A. Sandu et al., Benchmarking stiff ode solvers for atmospheric chemistry problems-I. Implicit vs Explicit, Atmospheric Environment. 31(19) (1997) 3151-3166.

DOI: https://doi.org/10.1016/s1352-2310(97)00059-9

[18] L.A. Nejad, A comparison of stiff ODE solvers for astrochemical kinetics problems, Astrophysics and Space Science. 299(1) (2005) 1-29.

DOI: https://doi.org/10.1007/s10509-005-2100-z

[19] S. Abelman, K.C. Patidar, Comparison of some recent numerical methods for initial-value problems for stiff ordinary differential equations, Computers & Mathematics with Applications. 55(4) (2008) 733-744.

DOI: https://doi.org/10.1016/j.camwa.2007.05.012

[20] S. Dallas, K. Machairas, A Comparison of ODE solvers for dynamical systems with impacts, Journal of Computational and Nonlinear Dynamics, Special Issue on Dynamics of Systems with Impacts. (2017). doi: 10. 1115/1. 4037074.

DOI: https://doi.org/10.1115/1.4037074

[21] A. Ryuichi, N. Michihiro, V. Remi, Behind and beyond the MATLAB ODE Suite. CRM-2651, (2000).

[22] M.A. Natick, Using MATLAB version 5. 1, The MathWorks, (1997).

[23] J.F. Shampine, M.W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput. 18(1) (1997) 1-22.

[24] F. Scheid, Schaum's outline of theory and problems of numerical analysis, Second Edition, McGraw-Hill Companies Inc., 1988, pp.184-240.

[25] J.D. Lambert, Numerical methods for ordinary differential equations: the initial value problem, Chichester, John Wiley & Sons, Inc., (1991).

[26] I. Farago, Numerical methods for ordinary differential equations, Eotvos Lorand University, (2013).

[27] E. Kreyszig, Advance engineering mathematics, John Wiley & Sons, Inc., (2006).

[28] J.R. Dormand, P.J. Prince, A family of embedded Runge-Kutta formulae, Journal of Computational Mathematics. 6(1) (1980) 19-26.

DOI: https://doi.org/10.1016/0771-050x(80)90013-3

[29] E. Fehlberg, Klassische Runge-Kutta-formula vierter und niedrigerer ordnung mit schrittweiten-kontrolle und ihre anwendung auf waermeleitungsprobleme, Computing. 6(1) (1970) 61-71.

DOI: https://doi.org/10.1007/bf02241732

[30] L.F. Shampine, Some practical Runge-Kutta formulas, Math. Comp. 46(173) (1986) 135-150.

DOI: https://doi.org/10.1090/s0025-5718-1986-0815836-3

[31] P. Bogacki, L.F. Shampine, An efficient Runge-Kutta (4, 5) pair, Computer Maths. Applied. 32(6) (1996) 15-28.

DOI: https://doi.org/10.1016/0898-1221(96)00141-1

[32] L. Brugnano, D. Trigiante, Solving differential problems by multistep, initial and boundary value methods, Gordon and Breach, Amsterdam, (1998).

[33] F. Lavernano, F. Mazzia, Solving ordinary differential equations by generalised Adams method: properties and implementation techniques, Applied Numerical Mathematics. 28(2-4) (1998) 107-126.

DOI: https://doi.org/10.1016/s0168-9274(98)00039-7

[34] Y.Y. Lui, Numerical methods for differential equations, Hongkong: Lecture Note, Department of Mathematics, City University of Hongkong, Kowloon, (2016).

[35] E. Hairer, G. Wanner, Solving differential equations II, stiff and differential algebraic problems, Springer-Verlag, Berlin, 1991, pp.5-8.

DOI: https://doi.org/10.1007/978-3-662-09947-6_3

[36] J.D. Verwer, A second-order Rosenbrock method applied to photochemical dispersion problems, SIAM Journal of Scientific Computation. 20 (1999) 1456-1480.

DOI: https://doi.org/10.1137/s1064827597326651

[37] E. Hairer, G. Wanner, On the instability of the BDF formulas, SIAM Journal of Numerical Anal. 20(6) (1983) 1206-1209.

DOI: https://doi.org/10.1137/0720090

[38] D. Omale, P.B. Ojih, M.O. Ogwo, Mathematical analysis of stiff and non-stiff initial value problems of ordinary differential equation using MATLAB, International Journal of Scientific & Engineering Research. 5(9) (2014) 49-59.

[39] S.A. Belmana, K.C. Patidarb, Comparison of some recent numerical methods for initial-value problems for stiff ordinary differential equations, Computers and Mathematics with Applications. 55 (2007) 733–744.

DOI: https://doi.org/10.1016/j.camwa.2007.05.012

[40] S.A.M. Yatim et al., A quantitative comparison of numerical method for solving stiff ordinary differential equations, Mathematical Problems in Engineering. (2011) 1-12.

[41] E. Cocherova, Modification of ordinary differential equations MATLAB solver, Radioengineering. 12(4) (2003) 63-66.

[42] J. Hall, A. Guyton, Textbook of medical physiology, Eleventh Edition, Philadelphia, Pennyslvania: Elsevier Saunders, (2006).

[43] R.M. Bernie, M.N. Levy, Cardiovascular physiology, St Louis, MO, Mosby, (2001).

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