Nelder-Mead Algorithm in Solving Ordinary Differential Equations Whose Solutions Possess Singularities
Keywords:Ordinary Differential Equation; Initial Value Problems; Singularities; Optimization; Nelder-Mead Algorithm
This research considers Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) whose solutions possess singularities. Here, we represent the theoretical solution by a rational function as it is more convenient representing a function close to a singularity by a rational function. The process of transforming the IVP to a constrained optimization problem and application of Nelder-Mead algorithm in obtaining approximate solution is presented in this work. Accuracy and efficiency of this scheme is demonstrated on two numerical examples. The proposed approach produced better results compared with existing methods discussed in literature.
(1) Bakre, O. F., Wusu, A. S., Akanbi M. A., Solving ordinary differential equations with evolutionary algorithms, Open Journal of Optimization. Vol. 4 (2015), 69-73.
(2) Bakre, O. F., Wusu, A. S. and Akanbi, M. A. An Explicit Single-Step Nonlinear Numerical Method for First Order Initial Value Problems (IVPs), Journal of Applied Mathematics and Physics, 8(2020): 1729-1735.
(3) Fatunla, S.O. Nonlinear multistep methods for initial value problems. An international Journal of Computers and Mathematics with Applications. Vol. 8 No.(3) pp:231–239, (1982).
(4) Fatunla S.O., Numerical treatment of singular initial value problems. An international Journal of Computers and Mathematics with Applications. 128 (1986) 1109–1115.
(5) Fatunla S.O., Numerical Methods for IVPs in ODEs, Academic Press, New York, (1991).
(6) Ikhile, M. N. O. Coefficients for studying one–step rational schemes for IVPs in ODEs, Computers and Mathematics with Applications, Vol.(41) No.(12), pp: 769–781, (2001).
(7) Junaid A., Raja A.Z., Qureshi I.M. Evolutionary Computing Approach for the Solution of Initial Value Problems in Ordinary Diffential Equations, World Academic of Science, Engineering and Tecnology, Vol. 31 (2009).
(8) Lambert, J.D., Computational methods in ordinary differential equations. Academic Press (1973).
(9) Lambert, J. D., Shaw, B. On the numerical solution of y^'=f(x,y) by a class of formulae based on rational approximation, Mathematics of Computation, Vol.(19), pp: 456–462, (1965).
(10) Luke, Y.L. , Fair, W. and Wimp, J. Predictor corrector formulas based on rational interpolants. J.Comput.Appl.Math. 1(1)(1975) 3–12.
(11) Mastorakis N. E., Numerical Solution of Non-Linear Ordinary Differential Equations via Collocation Method (Finite Elements) and Genetic Algorithms, Proceedings of the 6th WSEAS Int. Conf. on Eolutionary Computing.
Lisbon, Portugal. June 16-18, (2005), 36-42.
(12) Mastorakis N. E. Unstable Ordinary Differential Equations: Solution via Genetic Algorithms and the method of Nelder-Mead, Proceedings of the 6th WSEAS Int. Conf. on Systems Theory & Scientific Computation. Elounda, Greece. August 21-23, (2006) pp: 1-6.
(13) Niekerk, Van F. D., Non-linear One-Step methods for initial value problems. , Journal of Computational and Applied Mathematics 13(4) (1987) 367–371.
(14) Niekiek, Van F. D. Rational One Step Methods for Initial Value Problems, Comput. Math. Applic., Vol.(16) No.(12), pp: 1035–1039, (1988).
(15) Nkatse, T., Tshelametse, R. Analysis of Derivative Free Rational Scheme, MATEMATIKA, Vol.(31) No.(2), pp: 135–142, (2015).
(16) Okagbue, H. I., Adamu, M. O., Anake, T. A., Wusu A. S. Nature inspired quantile estimates of the Nakagami distribution, Telecommunication Systems, Vol.(72), pp:517–541, (2019)..
(17) Okagbue, H. I., Adamu, M. O., Anake, T. A., Wusu A. S. Quantile Approximation of the Erlang Distribution using Differential Evolution Algorithm, International Journal of Advanced Trends in Computer Science and Engineering, Vol.(9), No.(3) pp:2746–2755, (2020)..
(18) Omar A. A., Zaer A., Shaher M. ,Nabil S., Solving singular two-point boundary value problems using continuous genetic algorithm, Abst. Appl. Anal. (2012).
(19) Tasneem, A., Asif, A. S., Sania, Q. Development of a Nonlinear Hybrid Numerical Method, Advances in Differential Equations and Control Process, Vol.(19) No.(3), pp: 275–285, (2018).
(20) Teh, Y. Y., Nazeeruddin Yaacob One–Step Exponential–rational Methods for the Numerical Solution of First Order Initial Value Problems, Sains Malaysiana, Vol.(42) No.(6), pp: 456–462, (2013).
(21) Teh Yuan Ying, Zurni Omar and Kamarun Hizam Mansor Modified Exponential-rational Methods for the Numerical Solution of First Order Initial Value Problems Sains Malaysiana 43(12)(2014): 1951-1959.
(22) Wusu, A. S., Akanbi M. A., Solving oscillatory/periodic ordinary differential equations with differential evolution algorithms, Communications in Optimization Theory (2016), Article ID 7
(23) Wusu, A. S., Olabanjo, O. A., Aribisala, B. S. Application of Differential Evolution in the Solution of Stiff System of Ordinary Differential Equations, Transactions on Machine Learning and Artificial Intelligence, Vol.(8), No.(1)
(24) George, D. M., On the appliaction of genetic algorithms to differential equations, Romanian Journal of Economic Forecasting, Vol. 2 (2006).