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A Hybrid Neural Network and Gravitational Search Algorithm (HNNGSA) Method to Solve well known Wessinger's Equation

Authors: M. Ghalambaz, A.R. Noghrehabadi, M.A. Behrang, E. Assareh, A. Ghanbarzadeh, N.Hedayat


This study presents a hybrid neural network and Gravitational Search Algorithm (HNGSA) method to solve well known Wessinger's equation. To aim this purpose, gravitational search algorithm (GSA) technique is applied to train a multi-layer perceptron neural network, which is used as approximation solution of the Wessinger's equation. A trial solution of the differential equation is written as sum of two parts. The first part satisfies the initial/ boundary conditions and does not contain any adjustable parameters and the second part which is constructed so as not to affect the initial/boundary conditions. The second part involves adjustable parameters (the weights and biases) for a multi-layer perceptron neural network. In order to demonstrate the presented method, the obtained results of the proposed method are compared with some known numerical methods. The given results show that presented method can introduce a closer form to the analytic solution than other numerical methods. Present method can be easily extended to solve a wide range of problems.

Keywords: Neural Networks, Gravitational Search Algorithm (GSR), Wessinger's Equation

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[1] J.A. Khan, R.M.A. Zahoor, I.M. Qureshi, Swarm intelligence for the problem of non-linear ordinary differential equations and its application to well known Wessinger's equation. European Journal of scientific research. 2009; 34(4): 514-525.
[2] I.E. Lagris, A. Likas, D.I. Fotiadis. Artificial neural networks for solving ordinary and partitial differential equations. IEEE Transactions on Neural Networks. 1998; 9 (5): 987-1000.
[3] D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 26, SIAM, Philadelphia, 1977.
[4] H. Lee, I.S. Kang, Neural algorithms for solving differential equations, Journal of Computational Physics 1990; 91: 110-131.
[5] A.J. Meade Jr, A.A. Fernandez, The numerical solution of linear ordinary differential equations by feedforward neural networks, Mathematical and Computer Modelling. 1994; 19 (12): 1-25.
[6] A. Malek, R.S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural networkÔÇöOptimization method. Applied Mathematics and Computation. 2006; 183: 260-271.
[7] D.T. Pham, E. Koc, A. Ghanbarzadeh, S. Otri. Optimisation of the weights of multi-layered perceptrons using the bees algorithm. Proceedings of 5th International Symposium on Intelligent Manufacturing Systems. Sakarya University, Department of Industrial Engineering, 2006; pp. 38-46, May 29-31.
[8] A.S. Yilmaz, Z. Ozer, Pitch angle control in wind turbines above the rated wind speed by multi-layer percepteron and Radial basis function neural networks. Expert Systems with Applications. 2009; 36: 9767- 9775.
[9] D.T. Pham, X. Liu, Neural Networks for Identification, Prediction and Control. Springer Verlag, London. 1995.
[10] R.P. Lippmann, An introduction to computing with neural nets, IEEE ASSP Magazine 1987; 4-22.
[11] K. Hornick, M. Stinchcombe, H. white, Multilayer feedforward networks are universal approximators, Neural Networks 1989; 2 (5): 359-366.
[12] M.A. Behrang, E. Assareh, A. Ghanbarzadeh, A.R. Noghrehabadi, The potential of different artificial neural network (ANN) techniques in daily global solar radiation modeling based on meteorological data. Solar Energy 2010; 84: 1468-1480.
[13] E. Rashedi, H. Nezamabadi-pour, S. Saryazdi, GSA: A Gravitational Search Algorithm. Information Sciences. 2009; 179: 2232-2248.
[14] E. Rashedi, H. Nezamabadi-pour, S. Saryazdi, Filter modeling using gravitational search algorithm. Energy policy; doi:10.1016/j.engappai.2010.05.007.