Application of Differential Evolution in the Solution of Stiff System of Ordinary Differential Equations
DOI:
https://doi.org/10.14738/tmlai.81.7510Keywords:
Ordinary Differential Equation; Initial Value Problems; Stiff System; Optimization; Differential Evolution.Abstract
In recent times, the adaptation of evolutionary optimization algorithms for obtaining optimal solutions of many classical problems is gaining popularity. In this paper, optimal approximate solutions of initial--valued stiff system of first--order Ordinary Differential Equation (ODE) are obtained by converting the ODE into constrained optimization problem. The later is then solve via differential evolution algorithm. To illustrate the efficiency of the proposed approach, two numerical examples were considered. This approach showed significant improvement on the accuracy of the results produced compared with existing methods discussed in literature.
References
(1) Abhulimen C.E. and F.O. Otunta, A Sixth Order Multiderivative Multistep Methods for Stiff Systems of Differential Equations. International Journal of Numerical Mathematics (IJNM), 2006. 2(1): p. 248–268
(2) Abhulimen C.E. and F.O. Otunta, A new class of exponential fitting for initial value problems in ordinary differential equations. Journal of Nigerian Mathematical Society, 28, 2009.
(3) Abhulimen C.E., A class of exponentially-fitted third derivative methods for solving stiff differential equations. International Journal of Physical Science, 2008. 3(8): p. 188–193
(4) Adesanya A.O., R. O. Onsachi, and M. R. Odekunle, New algorithm for first order stiff initial value problems, FASCICULI MATHEMATICI, 2017. 58: p. 1–8.
(5) Bakre O. F., A. S. Wusu and M. A. Akanbi, Solving ordinary differential equations with evolutionary algorithms. Open Journal of Optimization, 2015. 4: p. 69–73
(6) Cash J.R., On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae. SIAM J. Numerical Math. , 1980. 34: p. 235–246
(7) Cash J.R., On exponentially fitting of composite multiderivative Linear Methods. SIAM J. Numerical Anal. 1981. 18(5): p. 808–821
(8) Cash J.R., Second Derivative Extended Backward Differentiation Formulas for the Numerical Solution of stiff Systems. SIAM J. Numerical Anal., 1981. 18: p. 21–36
(9) Curtiss C.F. and J.D. Hirschfelder, Integration of stiff equations. Proc. Nat. Acad. Sci., 1952. 38: p. 235–243
(10) Ehigie J.O., S.A. Okunuga, and A.B. Sofoluwe, A class of exponentially fitted second derivative extended backward differentiation formula for solving stiff problems. FASCICULI MATHEMATICI, 2013. 51: p. 71–84
(11) George D.M., On the appliaction of genetic algorithms to differential equations. Romanian Journal of Economic Forecasting, 2006. 3(2): p. 5–9
(12) Jackson L.W. and S.K. Kenue, A Fourth Order Exponentially Fitted Method. SIAM J. Numer. Anal., 1974. 11: p. 965–978
(13) Jator S.N. and R. K. Sahi, Boundary value technique for initial value problems based on adams–type second derivative methods. International Journal of Mathematical Education in Science and Technology, First published on: 07 June 2010 (iFirst), 2010. p. 1–8,
(14) Junaid A., A. Z. Raja, and I. M. Qureshi, Evolutionary computing approach for the solution of initial value problems in ordinary diffential equations. World Academic of Science, Engineering and Tecnology, 2009. 55: p. 578–581
(15) Lambert J.D., Computational Methods in ODEs. John Wiley & Sons, New York, 1973.
(16) Liniger W.S. and R.A. Willoughby, Efficient Integration methods for Stiff System of ODEs. SIAM J. Numerical Anal., 1970. 7: p. 47–65
(17) Mastorakis N.E., Unstable ordinary differential equations: Solution via genetic algorithms and the method of nelder-mead. In Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation. Elounda, Greece., August 21–23, 2006. p. 1–6
(18) Mastorakis N.E., Numerical solution of non-linear ordinary differential equations via collocation method (finite elements) and genetic algorithms. In Proceedings of the 6th WSEAS Int. Conf. on Eolutionary Computing. Lisbon, Portugal., Springer Berlin Heidelberg, June 16–18, 2005. p. 36–42.
(19) Okunuga S.A., A Class of Multiderivative Composite Formula for Stiff Initial Value Problems. Advances in Modeling & Analysis, 1999. 35(2): p. 21–32
(20) Okunuga S.A., A Fourth Order Composite Two-Step Method for Stiff Problems. International Journal of Computer Mathematics, 1999. 72(1): p.39–47
(21) Omar A.A., A. Zaer, M. Shaher, and S. Nabil, Solving singular two-point boundary value problems using continuous genetic algorithm. Abstract and Applied Analysis, 2012. Article ID 205391, vol. 2012: p.1-25
(22) Wikipedia. Differential evolution — wikipedia, the free encyclopedia, 2019. [Online; accessed 1 December 2019].
(23) Wusu A.S. and Akanbi M.A., Solving oscillatory/periodic ordinary differential equations with differential evolution algorithms. Communications in Optimization Theory, 2016. 2016: p. 1–8
