{"title":"Evolutionary Computing Approach for the Solution of Initial value Problems in Ordinary Differential Equations","authors":"A. Junaid, M. A. Z. Raja, I. M. Qureshi","volume":31,"journal":"International Journal of Electrical and Computer Engineering","pagesStart":1477,"pagesEnd":1481,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/12849","abstract":"
An evolutionary computing technique for solving initial value problems in Ordinary Differential Equations is proposed in this paper. Neural network is used as a universal approximator while the adaptive parameters of neural networks are optimized by genetic algorithm. The solution is achieved on the continuous grid of time instead of discrete as in other numerical techniques. The comparison is carried out with classical numerical techniques and the solution is found with a uniform accuracy of MSE ≈ 10-9 .<\/p>\r\n","references":"[1] D. Kahaner, C. Moler, S. Nash, \"Numerical Methods and Software,\"\r\nPrentice-Hall, New Jersey, 1989.\r\n[2] K. Kunz, R. Luebbers, \"Finite difference time domain method for\r\nelectromagnetic,\" Boca Raton, CRC Press. 1993.\r\n[3] J.R Dormand, P.J Prince, \"A family of embedded Runge-Kutta\r\nformulae,\" Comp. Appl. Math. J., vol. 6, pp. 19, 1980.\r\n[4] MJ. Jang, CL. Chen, YC. Liu, \"On solving the initial-value problems\r\nusing the differential transform method,\" J. Comp. Appl. Math. J., vol.\r\n115, pp. 145-160, 2000.\r\n[5] L.F. Shampine, M. W. Reichelt, \"The MATLAB ode suite,\" SIAM\r\nScientific Computing J., vol.18, pp. 1-22, 1997.\r\n[6] M.H Li, G.J Wang, \"Solving differential equation with constant\r\ncoefficient by using radial basis function,\" Shen Yang Institute of\r\nChemical Technology J., vol.20, pp. 68-72, 2006.\r\n[7] A.J Meada, A.A Fernandez, \"The numerical solution of linear ordinary\r\ndifferential equation by feedforward neural network,\" Math Compute\r\nModeling J., vol.19, pp. 1-25, 1994.\r\n[8] C. Monterola, C. Saloma, \"Characterizing the dynamics of constrained\r\nphysical systems with unsupervised neural network,\" Phys Rev E 57,\r\npp.1247R-1250R, 1998.\r\n[9] B.Ph van Milligen, V.Tribaldos, J.A.Jimenez, \"Neural network\r\ndifferential equation and plasma equilibrium solver,\" Physical Review\r\nLetters 75, pp. 3594-3597, 1995.\r\n[10] L.Qin, M.Yang, moving mass attitude law based on neural network, in\r\nproc. 6th Int. Conf. machine learning and cybernetics, ICMLC, 5, art.\r\nNo. 4370622, 2007, pp.2791-2795.\r\n[11] J.H. Holland, \"Adaptation in natural and artificial systems,\" Ann arbor,\r\nMI, University of Michigan press, 1975.\r\n[12] C.R. Houck, J.A. Joines, M.G. Kay, \"A genetic algorithm for function\r\noptimization,\" A matlab implementation, Technical Report NCSU-IE\r\nTR 95-09, North Carolina State University, Raleigh NC,1995.\r\n[13] Zbigniew Michalewicz, \"Genetic algorithms + data structure=\r\nEvolution programs,\" 2nd ed, New York: Springer-verlag, Berlin, 1994.\r\n[14] R. Hetch-Nielsen, \"Kolmogorov-s mapping neural network existence\r\ntheorem,\" 1st 1987 IEEE Int. Conf. Neural networks, San Diego CA, 3,\r\npp. 11.\r\n[15] K.I. Funahashi, \"On the approximate realization of continuous\r\nmappings by neural networks,\" Neural Networks J., vol. 2.issue 3,\r\npp.183-192, 1989.\r\n[16] C. Chen, \"Degree of approximation by superposition of sigmoidal\r\nfunction,\" Analysis in theory & appl J., vol. 9, no 3, pp. 17-28, 1993.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 31, 2009"}