Artificial Neural Network for Solving Fuzzy Differential Equations under Generalized H – Derivation

Mazin H. Suhhiem^1,

¹Department of Statistics, University of Sumer, Alrifaee, Iraq

Cite this paper:
Mazin H. Suhhiem. Artificial Neural Network for Solving Fuzzy Differential Equations under Generalized H – Derivation. International Journal of Partial Differential Equations and Applications. 2017; 5(1):1-9. doi: 10.12691/ijpdea-5-1-1

Abstract

The aim of this work is to present a novel approach based on the artificial neural network for finding the numerical solution of first order fuzzy differential equations under generalized H-derivation. The differentiability concept used in this paper is the generalized differentiability since a fuzzy differential equation under this differentiability can have two solutions. The fuzzy trial solution of fuzzy initial value problem is written as a sum of two parts. The first part satisfies the fuzzy condition, it contains no adjustable parameters. The second part involves feed-forward neural networks containing adjustable parameters. Under some conditions the proposed method provides numerical solutions with high accuracy.

Keywords:
fuzzy differential equation artificial neural network generalized H-derivation error function trial solution

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References:

[1]	Mosleh M., Otadi M, “Simulation and Evaluation of Fuzzy Differential Equations by Fuzzy Neural Network”, Applied Soft Computing, 12, 2817-2827, 2012.
[2]	Buckley J. J., Feuring T., “Fuzzy Differential Equations”, Fuzzy Sets and Systems, 110, 69-77, 2000.
[3]	Lee H., Kang I. S., “Neural Algorithms For Solving Differential Equations”, Journal of Computational Physics, 91, 110-131,1990.
[4]	Meade A. J., Fernandes A. A., “The Numerical Solution of Linear Ordinary Differential Equations by Feed-Forward Neural Networks”, Mathematical and Computer Modelling, Vol. 19, No. 12 , 1-25, 1994.
[5]	Meade A. J., Fernandes A. A., “Solution of Nonlinear Ordinary Differential Equations by Feed-Forward Neural Networks”, Mathematical and Computer Modelling, Vol. 20, No. 9, 19-44, 1994.
[6]	Lagaris I. E., Likas A., et al., “Artificial Neural Networks For Solving Ordinary and Partial Differential Equations”, Journal of Computational Physics, 104, 1-26, 1997.
[7]	Lagaris I. E., Likas A., et al., “Artificial Neural Networks For Solving Ordinary and Partial Differential Equations”, IEEE Transaction on Neural Networks, Vol. 9, No. 5, 987-1000, 1998.
[8]	Liu B., Jammes B., “Solving Ordinary Differential Equations by Neural Networks”, Warsaw, Poland, 1999.
[9]	Alli H., Ucar A., et al., “The Solutions of Vibration Control Problems Using Artificial Neural Networks”, Journal of the Franklin Institute, 340, 307-325, 2003.
[10]	Tawfiq L. N. M., “On Design and Training of Artificial Neural Network For Solving Differential Equations”, Ph.D. Thesis, College of Education Ibn AL-Haitham, University of Baghdad, Iraq, 2004.
[11]	Malek A., Shekari R., “Numerical Solution For High Order Differential Equations by Using a Hybrid Neural Network Optimization Method”, Applied Mathematics and Computation, 183, 260-271, 2006.
[12]	Pattanaik S., Mishra R. K., “Application of ANN For Solution of PDE in RF Engineering”, International Journal on Information Sciences and Computing, Vol. 2, No. 1, 74-79, 2008.
[13]	Baymani M., Kerayechian A., et al., “Artificial Neural Networks Approach For Solving Stokes Problem”, Applied Mathematics, 1, 288-292, 2010.
[14]	Oraibi Y. A., “Design Feed-Forward Neural Networks For Solving Ordinary Initial Value Problem”, M.Sc. Thesis, College of Education Ibn Al-Haitham, University of Baghdad, Iraq, 2011.
[15]	Ali M. H., “Design Fast Feed-Forward Neural Networks to Solve Two Point Boundary Value Problems”, M.Sc. Thesis, College of Education Ibn Al-Haitham, University of Baghdad, Iraq, 2012.
[16]	Hussein A. A. T., “Design Fast Feed-Forward Neural Networks to Solve Singular Boundary Value Problems”, M.Sc. Thesis, College of Education Ibn Al-Haitham, University of Baghdad, Iraq, 2013.
[17]	Tawfiq L. N. M., Al-Abrahemee K. M. M., “Design Neural Network to Solve Singular Perturbation Problems”, Applied and Computational Mathematics, Vol. 3, No. 3, 1-5, 2014.
[18]	Effati S., Pakdaman M., “Artificial Neural Network Approach For Solving Fuzzy Differential Equations”, Information Sciences, 180, 1434-1457, 2010.
[19]	Mosleh M., Otadi M., “Fuzzy Fredholm Integro-Differential Equations with Artificial Neural Networks”, Communications in Numerical Analysis, Article ID cna-00128, 1-13, 2012.
[20]	Ezadi S., Parandin N., et al., “Numerical Solution of Fuzzy Differential Equations Based on Semi-Taylor by Using Neural Network”, Journal of Basic and Applied Scientific Research, 3(1s), 477-482, 2013.
[21]	Suhhiem M. H., “Fuzzy Artificial Neural Network For Solving Fuzzy and Non-Fuzzy Differential Equations”, Ph.D. Thesis, College of Sciences, AL-Mustansiriyah University, Iraq, 2016.
[22]	Cano Y. C., Flores H. R., “On New Solutions of Fuzzy Differential Equations”, Chaos, Solitons and Fractals, 38, 112-119, 2008.
[23]	Cano Y. C., Flores H. R., et al., “Fuzzy Differential Equations with Generalized Derivative”, Fuzzy Sets and Systems, 160, 1517-1527, 2008.