Numerical solution of fuzzy linear Fredholm integro-differential equation by \\fuzzy neural network

Document Type: Research Paper


Department of Mathematics, Firoozkooh Branch, Islamic Azad Uni- versity, Firoozkooh, Iran


In this paper, a novel hybrid method based on learning algorithm
of fuzzy neural network and Newton-Cotes
methods with positive coefficient for the solution of linear Fredholm
integro-differential equation of the second kind
with fuzzy initial value is presented. Here neural network is
considered as a part of large field called neural computing or
soft computing. We propose a
learning algorithm from the cost function for adjusting fuzzy
weights. This paper is one of the first attempts to derive learning
algorithms from fuzzy neural networks with real input, fuzzy output,
and fuzzy weights. Finally, we illustrate our approach by numerical examples.


bibitem{aba} S. Abbasbandy, E. Babolian and M. Alavi, {it Numerical
method for solving linear fredholm fuzzy integral equations of
the second kind}, Chaos Solitons & Fractals, {bf 31} (2007), 138-146.

bibitem{abo} S. Abbasbandy and M. Otadi, {it Numerical solution of fuzzy
polynomials by fuzzy neural network}, Applied Mathematics and Computation, {bf 181}
(2006), 1084-1089.

bibitem{abom} S. Abbasbandy, M. Otadi and M. Mosleh, {it Numerical
solution of a system of fuzzy polynomials by fuzzy neural
network}, Information Sciences, {bf 178} (2008), 1948-1960.

bibitem{al1} G. Alefeld and J. Herzberger, {it Introduction to interval
computations}, Academic Press, New York, 1983.

bibitem{aaa1} T. Allahviranloo, E. Ahmady and N. Ahmady, {it Nth-order
fuzzy linear differential eqations},Information Sciences, {bf 178} (2008),

bibitem{aaa2} T. Allahviranloo, N. Ahmady and E. Ahmady, {it Numerical
solution of fuzzy differential eqations by predictor-corrector
method}, Information Sciences, {bf 177} (2007), 1633-1647.

bibitem{at} K. E. Atkinson, {it An introduction to numerical analysis},
New York, Wiley, 1987.

bibitem{bsa} E. Babolian, H. S. Goghary and S. Abbasbandy, {it Numerical
solution of linear fredholm fuzzy integral equations of the
second kind by adomian method}, Applied Mathematics and
Computation, {bf 161} (2005), 733-744.

bibitem{bm} P. Balasubramaniam and S. Muralisankar, {it Existence and uniqueness of fuzzy
solution for the nonlinear fuzzy integro-differential equations}, Applied mathematics letters, {bf 14} (2001), 455-462.

bibitem{bb} B. Bede, I. J. Rudas and A. L. Bencsik, {it First order linear
fuzzy differential eqations under generalized differentiability},
Information Sciences, {bf 177} (2007), 1648-1662.

bibitem{ber} J. F. Bernard, {it Use of rule-based system for process control}, IEEE Control System Management, {bf 8} (1988), 3-13.

bibitem{bf} J. J. Buckley and T. Feuring, {it Fuzzy differential equations}, Fuzzy Sets and Systems, {bf 110} (2000), 69-77.

bibitem{by} J. J. Buckley and Y. Hayashi, {it Can fuzzy neural nets
approximate continuous fuzzy functions?}, Fuzzy Sets and Systems, {bf 61}
(1994), 43-51.

bibitem{cz} S. L. Chang and L. A. Zadeh, {it On fuzzy mapping and control},
IEEE Transactions Systems Man and Cybernetics, {bf 2} (1972), 30-34.

bibitem{che} Y. C. Chen and C. C. Teng, {it A model reference control structure using a fuzzy neural network}, Fuzzy Sets and Systems, {bf 73} (1995), 291-312.

bibitem{com} W. Congxin and M. Ming, {it On embedding problem of fuzzy
number space}, Fuzzy Sets and Systems, {bf 44} (1991), 33-38. 

bibitem{dd} D. Dubois and H. Prade, {it Operations on fuzzy numbers}, International Journal of Systems Science, {bf 9} (1978), 613-626.

bibitem{dp} D. Dubois and H. Prade, {it Towards fuzzy differential
calculus}, Fuzzy Sets and Systems, {bf 8} (1982), 225-233.

bibitem{effati} S. Effati and M. Pakdaman, {it Artificial neural network approach for solving fuzzy differential equations}, Information Sciences, {bf 180} (2010), 1434-1457.

bibitem{wf} W. Fei, {it Existence and uniqueness of solution for fuzzy random differential equations with non-lipschitz
coefficients}, Information Sciences, {bf 177} (2007), 4329-4337.

bibitem{fmk} M. Friedman, M. Ma and A. Kandel, {it Numerical solutions
of fuzzy differential and integral equations}, Fuzzy Sets and
Systems, {bf 106} (1999), 35-48.

bibitem{gv} R. Goetschel and W. Voxman, {it Elementary fuzzy calculus},
Fuzzy Sets and Systems, {bf 18} (1986), 31-43.

bibitem{go} D. Gottlieb and S.A. Orszag, {it Numerical analysis of
spectral methods}, Theory and applications, CBMS-NSF Regional
Conference Series in Applied Mathematics, SIAM,
Philadelphia, {bf 26} (1977).
bibitem{hdb} M. T. Hagan, H. B. Demuth and M. Beale, {it Neural network
design}, PWS publishing company, Massachusetts, 1996.

bibitem{jb2} Y. Hayashi, J. J. Buckley and E. Czogala, {it Fuzzy neural
network with fuzzy signals and weights}, International Journal of Intelligent
Systems, {bf 8} (1993), 527-537.

bibitem{ha} S. Haykin,{it Neural networks: a comprehensive
foundation}, Prentice Hall, New Jersey, 1999.

bibitem{h} H. Hochstadt, {it Integral equations}, New York: Wiley,

bibitem{hsw} K. Hornick, M. Stinchcombe and H. White, {it Multilayer
feedforward networks are universal approximators}, Neural Networks,
{bf 2} (1989), 359-366.

bibitem{dr} H. Ishibuchi, K. Kwon and H. Tanaka, {it A learning algorithm of
fuzzy neural networks with triangular fuzzy weights}, Fuzzy Sets
and Systems, {bf 71} (1995), 277-293.

bibitem{imt} H. Ishibuchi, K. Morioka and I.B. Turksen, {it Learning by
fuzzified neural networks}, International Journal of Approximate Reasoning, {bf 13} (1995),

bibitem{ism} H. Ishibuchi and M. Nii, {it Numerical analysis of the
learning of fuzzified neural networks from fuzzy if-then rules},
Fuzzy Sets and Systems, {bf 120} (2001), 281-307.

bibitem{wc1} H. Ishibuchi, H. Okada and H. Tanaka, {it Fuzzy neural networks
with fuzzy weights and fuzzy biases}, Proceedings ICNN, {bf 93} (1993), 1650-1655.

bibitem{ito} H. Ishibuchi, H. Tanaka and H. Okada, {it Fuzzy neural
networks with fuzzy weights and fuzzy biases}, IEEE International Conferences on Neural Networks, (1993), 1650-1655.

bibitem{kal} O. Kaleva, {it Fuzzy differential equations}, Fuzzy Sets
and Systems, {bf 24} (1987), 301-317.

bibitem{kh} T. Khanna, {it Foundations of neural networks},
Addison-Wesly, Reading, MA, 1990.

bibitem{kcy} G. J. Klir, U. S. Clair, B. Yuan, {it Fuzzy set theory:
foundations and applications}, Prentice-Hall, 1997.

bibitem{kbrh} P. V. Krishnamraju, J. J. Buckley, K. D. Relly and Y.
Hayashi, {it Genetic learning algorithms for fuzzy neural nets},
IEEE International Conference on Fuzzy
Systems, (1994), 1969-1974.

bibitem{lag} I. E. Lagaris and A. Likas, {it Artificial neural networks for solving ordinary and partial differential equations}, IEEE Transactions on Neural Networks, September, {bf 9}textbf{(5)} (1998).

bibitem{lam} J. D. Lamber, {it Computational methods in ordinary
differential equations}, John Wiley & Sons, New York, 1983.

bibitem{lf} A. Lapedes and R. Farber, {it How neural nets work?}, Neural Information Processing Systems, AIP, 1988,
bibitem{lk} H. Lee and I. S. Kang, {it Neural algorithms for solving
differential equations}, Journal of Computational Physics, {bf 91}
(1990), 110-131.

bibitem{tlee} T. Leephakpreeda, {it Novel determination of
differential-equation solutions: universal approximation method},
Computational and Applied Mathematics, {bf 146} (2002), 443-457.

bibitem{leng} G. Leng, G. Prasad and T. M. McGinnity, {it An on-line algorithm for creating self-organizing fuzzy neural networks}, Neural Networks, {bf 17} (2004), 1477-1493.

bibitem{lin1} D. Lin and X. Wang, {it Observer-based decentralized fuzzy neural sliding mode control for interconnected unknown chaotic systems via network structure adaptation}, Fuzzy Sets and Systems, {bf 161} (2010), 2066-2080.

bibitem{lin3} D. Lin and X. Wang, {it Self-organizing adaptive fuzzy neural control for the synchronization of uncertain chaotic systems with random-varying parameters}, Neurocomputing, {bf 74} (2011), 2241-2249.

bibitem{lin2} D. Lin, X. Wang, F. Nian and Y. Zhang, {it Dynamic fuzzy neural networks modeling and adaptive backstepping tracking control of uncertain chaotic systems}, Neurocomputing, {bf 73} (2010), 2873-2881.

bibitem{li} R. P. Lippmann, {it An introduction to computing with
neural nets}, IEEE ASSP Magazine, (1987), 4-22.

bibitem{mas} A. Malek and R. Shekari Beidokhti, {it Numerical solution
for high order differential equations using a hybrid neural
network-Optimization method}, Applied Mathematics and Computation, {bf 183} (2006),

bibitem{mf} A. J. Meade Jr and A. A. Fernandez, {it The numerical solution
of linear ordinary differential equations by feedforward neural
networks}, Mathematical and Computer Modelling, {bf 19}textbf{(12)} (1994), 1-25.

bibitem{mef} A. J. Meade Jr and A. A. Fernandez, {it Solution of nonlinear
ordinary differential equations by feedforward neural networks},
Mathematical and Computer Modelling, {bf 20}textbf{(9)} (1994), 19-44.

bibitem{mbcrb} M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. Román-Flores and R. C.
Bassanezi, {it Fuzzy differential equations and the extention
principle}, Information Sciences, {bf 177} (2007), 3627-3635.

bibitem{mto} M. Mosleh, T. Allahviranloo and M. Otadi, {it Evaluation of fully fuzzy regression models by fuzzy neural
network}, Neural Comput and Applications, {bf 21} (2012), 105 - 112.

bibitem{mom1} M. Mosleh and M. Otadi, {it Minimal solution of fuzzy linear system of differential equations}, Neural Comput and Applications, {bf 21} (2012), 329-336.

bibitem{mom} M. Mosleh and M. Otadi, {it Simulation and evaluation of fuzzy differential equations by fuzzy neural network}, Applied Soft Computing, {bf 12} (2012), 2817–2827.

bibitem{moa1} M. Mosleh, M. Otadi and S. Abbasbandy, {it Evaluation of fuzzy regression models by fuzzy neural network}, Journal of Computational and Applied
Mathematics, {bf 234} (2010), 825-834.
bibitem{moa2} M. Mosleh, M. Otadi and S. Abbasbandy, {it Fuzzy polynomial regression with fuzzy neural networks}, Applied Mathematical Modelling, {bf 35} (2011), 5400-5412.

bibitem{oma1} M. Otadi and M. Mosleh, {it Simulation and evaluation of dual fully fuzzy linear systems by fuzzy neural network}, Applied Mathematical Modelling, {bf 35} (2011), 5026-5039.

bibitem{oma2} M. Otadi, M. Mosleh and S. Abbasbandy, {it Numerical solution of fully fuzzy linear systems
by fuzzy neural network}, Soft Computing, {bf 15} (2011), 1513-1522.

bibitem{oms} M. Otadi, M. Mosleh, S. Saidanlu and N. A. Aris, {it Fuzzy hyperbolic regression with fuzzy neural networks}, Australian Journal of Basic and Applied Sciences, {bf 5}textbf{(10)} (2011), 838-847.

bibitem{pse} G. Papaschinopoulos, G. Stefanidou and P. Efraimidis,
{it Existence, uniquencess and asymptotic behavior of the solutions
of a fuzzy differential equation with piecewise constant
argument}, Information Sciences, {bf 177} (2007), 3855-3870.

bibitem{pi} P. Picton, {it Neural Networks}, Second edition, Palgrave,
Great Britain, 2000.

bibitem{pr} M. L. Puri and D. Ralescu, {it Fuzzy random variables}, Journal of Mathematical Analysis and Applications, {bf 114} (1986), 409-422.

bibitem{rrl} R. Rodriguez-Lopez, {it Comparison results for fuzzy
differential eqations}, Information Sciences, {bf 178} (2008), 1756-1779.

bibitem{ru} D. E. Rumelhart and J. L. McClelland, {it Parallel distributed processing}, MIT Press,
Cambridge, MA, 1986.

bibitem{sc} R. J. Schalkoff, {it Artificial neural networks},
McGraw-Hill, New York, 1997.

bibitem{seik} S. Seikkala, {it On the fuzzy initial value problem},
Fuzzy Sets and Systems, {bf 24} (1987), 319-330.

bibitem{st} J. Stanley, {it Introduction to neural networks}, Sierra Mardre, 1990.

bibitem{sb} J. Store and R. Bulirsch, {it Introduction to numerical
analysis}, Springer-Verlag, New York, 1993.

bibitem{tung} W. L. Tung and C. Quek, {it A generic self-organizing fuzzy neural network}, IEEE Transactions on Neural networks, {bf 13} (2002), 1075-1086.

bibitem{yuan} X. Wang and J. Zhao, {it Cryptanalysis on a parallel keyed hash function based on chaotic neural network}, Neurocomputing, {bf 73} (2010,) 3224-3228.

bibitem{xin} W. Xingyuan, X. Bing and Z. Huaguang, {it A multi-ary number communication system based on hyperchaotic system of 6th-order cellular neural network}, Communications in Nonlinear Science and Numerical Simulation, {bf 15} (2010), 124-133.

bibitem{kin} L. A. Zadeh, {it The concept of a liguistic variable and its
application to approximate reasoning} Information Sciences, {bf 8}
(1975), 199-249, 301-357; {bf 9} (1975), 43-80.

bibitem{laz} L. A. Zadeh, {it Is there a need for fuzzy logic?}, Information
Sciences, {bf 178} (2008), 2751-2779.

bibitem{zhang1} H. G. Zhang and D. R. Liu, {it Fuzzy modeling and fuzzy control}, Boston, 2006.

bibitem{zhang2} H. G. Zhang and Y. B. Quan, {it Modeling, identification and control of a class of nonlinear systems}, IEEE Transactions on Fuzzy Systems, {bf 9} (2001), 349-354.