August  2021, 4(3): 209-219. doi: 10.3934/mfc.2021013

Solving fuzzy volterra-fredholm integral equation by fuzzy artificial neural network

1. 

Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, Tehran, Iran

2. 

Department of Mathematics, Faculty of Basic Science, West Tehran Branch, Islamic Azad University, Tehran, Iran

* Corresponding author

Received  March 2021 Revised  June 2021 Published  August 2021 Early access  August 2021

The volterra-fredholm integral equation in all forms are arose from physics, biology and engineering problems which is derived from differential equation modelling. On the other hand, the trained programming algorithm by the fuzzy artificial neural networks has effective solution to find the best answer. In this article we try to estimate the equation and its answer by developed fuzzy artificial neural network to fuzzy volterra-fredholme integral. Our attempts would lead to benchmark other extended forms of this type of equation.

Citation: Seiyed Hadi Abtahi, Hamidreza Rahimi, Maryam Mosleh. Solving fuzzy volterra-fredholm integral equation by fuzzy artificial neural network. Mathematical Foundations of Computing, 2021, 4 (3) : 209-219. doi: 10.3934/mfc.2021013
References:
[1]

G. Adomian, Solution of physical problems by decomposition, Comput. Math. Appl., 27 (1994), 145-154.  doi: 10.1016/0898-1221(94)90132-5.  Google Scholar

[2]

M. Y. Ali, A. Sultana and A. Khan, Comparison of fuzzy multiplication operation on triangular fuzzy number, IOSR J. Math (IOSR-JM), 12, (2016), 35–41. Google Scholar

[3]

A. B. Badiru and J. Cheung, Fuzzy Engineering Expert Systems with Neural Network Applications, Vol. 11, John Wiley & Sons, 2002. Google Scholar

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E. Balagurusamy, Computer Oriented Statistical and Numerical Methods, Macmillan India Limited, 1988. Google Scholar

[5]

C. Bector and S. Chandra, Fuzzy numbers and fuzzy arithmetic, Fuzzy Mathematical Programming and Fuzzy Matrix Game, (2005), 39–56. Google Scholar

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S. S. Behzadi, Solving fuzzy nonlinear Volterra-Fredholm integral equations by using homotopy analysis and Adomiandecomposition methods, J. Fuzzy Set Valued Anal., 2011 (2011), 1-13.   Google Scholar

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H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal., 27 (1990), 987-1000.  doi: 10.1137/0727057.  Google Scholar

[8]

A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere Publishing Corp. Washington, D. C.; Distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 1975.  Google Scholar

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J. J. Buckley and Y. Hayashi, Can fuzzy neural nets approximate continuous fuzzy functions?, Fuzzy Sets and Systems, 61 (1994), 43-51.  doi: 10.1016/0165-0114(94)90283-6.  Google Scholar

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A. CardoneE. Messina and A. Vecchio, An adaptive method for Volterra-Fredholm integral equations on the half line, J. Comput. Appl. Math., 228 (2009), 538-547.  doi: 10.1016/j.cam.2008.03.036.  Google Scholar

[11]

J. DijkmanH. V. Haeringen and S. D. Lange, Fuzzy numbers, Journal of Mathematical Analysis and Applications, 92 (1983), 301-341.  doi: 10.1016/0022-247X(83)90253-6.  Google Scholar

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R. Fullér, Introduction to Neuro-Fuzzy Systems, Advances in Soft Computing, Physica-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-7908-1852-9.  Google Scholar

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I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operator, Vol. 63, Birkhäuser, 2013. doi: 10.1007/978-3-0348-8558-4_1.  Google Scholar

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D. O. Hebb and D. Hebb, The Organization of Behavior, Vol. 65, Wiley New York, 1949. Google Scholar

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R. Hecht-Nielsen, Kolmogorov's mapping neural network existence theorem, IEEE Press, 3 (1987), 11-14.   Google Scholar

[19]

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U. S. A., 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.  Google Scholar

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[22]

O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7.  Google Scholar

[23]

J. M. Keller and D. J. Hunt, Incorporating fuzzy membership functions into the perceptron algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1985), 693-699.  doi: 10.1109/TPAMI.1985.4767725.  Google Scholar

[24]

E. Khan and P. Venkatapuram, Neufuz: Neural network based fuzzy logic design algorithms, Second IEEE International Conference on Fuzzy Systems, (1993), 647–654, https://ieeexplore.ieee.org/abstract/document/327412. Google Scholar

[25]

D. Kriesel, A Brief Introduction on Neural Network, 2007. Available from: http://www.dkriesel.com/en/science. Google Scholar

[26]

A. R. Krommer and C. W. Ueberhuber, Computational Integration, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971460.  Google Scholar

[27]

C. T. Leondes, Fuzzy Logic and Expert Systems Applications, Vol. 6, Elsevier, 1998. Google Scholar

[28]

T. Mach, Eigenvalue Algorithms for Symmetric Hierarchical Matrices, Thomas Mach, 2012. Google Scholar

[29]

A. Malek and R. S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural network-optimization method, Appl. Math. Comput., 183 (2006), 260-271.  doi: 10.1016/j.amc.2006.05.068.  Google Scholar

[30]

K. Maleknejard and M. Hadizadeh, A new computational method for Volterra-Fredholm integral equations, Comput. Math. Appl., 37 (1999), 1-8.  doi: 10.1016/S0898-1221(99)00107-8.  Google Scholar

[31]

J. McCarthy, Programs with Common Sense, RLE and MIT Computation Center, 1960. Google Scholar

[32]

W. S. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5 (1943), 115-133.  doi: 10.1007/BF02478259.  Google Scholar

[33] K. MehrotraC. K. Mohan and S. Ranka, Elements of Artificial Neural Networks, MIT Press, 1997.  doi: 10.7551/mitpress/2687.001.0001.  Google Scholar
[34]

P. Milner, A brief history of the hebbian learning rule, Canadian Psychology/Psychologie Canadienne, 44 (2003), 5-9.  doi: 10.1037/h0085817.  Google Scholar

[35]

F. Mirzaee and A. A. Hoseini, Numerical solution of nonlinear Volterra-Fredholm integral equations using hybrid of block-pulse functions and Taylor series, Alexandria Engineering Journal, 52 (2013), 551-555.  doi: 10.1016/j.aej.2013.02.004.  Google Scholar

[36]

J. Moor, The dartmouth college artificial intelligence conference: The next fifty years, Ai Magazine, 27 (2006), 87-87.   Google Scholar

[37]

F. Mora-Camino and C. A. N. Cosenza, Fuzzy dual numbers, in fuzzy dual numbers, Springer, (2018), 11–16. Google Scholar

[38]

M. Mosleh, Fuzzy neural network for solving a system of fuzzy differential equations, Applied Soft Computing, 13 (2013), 3597-3607.  doi: 10.1016/j.asoc.2013.04.013.  Google Scholar

[39]

M. A. Nielsen, Neural Networks and Deep Learning, Vol. 25, Determination Press, 2015. Google Scholar

[40]

B. Pachpatte, On mixed Volterra-Fredholm type integral equations, Indian J. Pure Appl. Math., 17 (1986), 488-496.   Google Scholar

[41]

J. Y. Park and J. U. Jeong, On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations, Fuzzy Sets and Systems, 115 (2000), 425-431.  doi: 10.1016/S0165-0114(98)00341-8.  Google Scholar

[42]

W. Pedrycz, Fuzzy Modelling: Paradigms and Practice, Vol. 7, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4613-1365-6.  Google Scholar

[43] T. A. Polk and C. M. Seifert, Cognitive Modeling, MIT Press, 2002.   Google Scholar
[44]

V. Raju and R. Jayagopal, An arithmetic operations of icosagonal fuzzy number using Alpha cut, International Journal of Pure and Applied Mathematics, 120 (2018), 137-145.   Google Scholar

[45]

A. L. Samuel, Some studies in machine learning using the game of checkers, IBM J. Res. Develop., 3 (1959), 211-229.  doi: 10.1147/rd.33.0210.  Google Scholar

[46]

S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.  doi: 10.1016/0165-0114(87)90030-3.  Google Scholar

[47]

P. SibiS. A. Jones and P. Siddarth, Analysis of different activation functions using back propagation neural networks, Journal of Theoretical and Applied Information Technology, 47 (2013), 1264-1268.   Google Scholar

[48]

R. Subramanian, Emergent AI, Social Robots and the Law: Security, Privacy and Policy Issues, Journal of International, Technology and Information Management, 26 (2017). Google Scholar

[49]

E. A. Wan, Time Series Prediction by using a Connectionist Network with Internal Delay Lines, Addison-Wesley Publishing Co, 1993. Google Scholar

[50] P. P. Wang and S. K. Chang, Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, Plenum Press, New York-London, 1980.   Google Scholar
[51]

A.-M. Wazwaz, A First Course in Integral Equations, 2$^{nd}$ edition, World Scientific Publishing Company, Co. Pte. Ltd., Hackensack, NJ, 2015.  Google Scholar

[52]

P. Werbos, Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. Ph. D. dissertation, Harvard University, 1975.  Google Scholar

[53]

J. Yuan and S. Yu, Privacy preserving back-propagation neural network learning made practical with cloud computing, IEEE Transactions on Parallel and Distributed Systems, 25 (2013), 212-221.   Google Scholar

[54]

L. A. Zadeh and F. sets, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

show all references

References:
[1]

G. Adomian, Solution of physical problems by decomposition, Comput. Math. Appl., 27 (1994), 145-154.  doi: 10.1016/0898-1221(94)90132-5.  Google Scholar

[2]

M. Y. Ali, A. Sultana and A. Khan, Comparison of fuzzy multiplication operation on triangular fuzzy number, IOSR J. Math (IOSR-JM), 12, (2016), 35–41. Google Scholar

[3]

A. B. Badiru and J. Cheung, Fuzzy Engineering Expert Systems with Neural Network Applications, Vol. 11, John Wiley & Sons, 2002. Google Scholar

[4]

E. Balagurusamy, Computer Oriented Statistical and Numerical Methods, Macmillan India Limited, 1988. Google Scholar

[5]

C. Bector and S. Chandra, Fuzzy numbers and fuzzy arithmetic, Fuzzy Mathematical Programming and Fuzzy Matrix Game, (2005), 39–56. Google Scholar

[6]

S. S. Behzadi, Solving fuzzy nonlinear Volterra-Fredholm integral equations by using homotopy analysis and Adomiandecomposition methods, J. Fuzzy Set Valued Anal., 2011 (2011), 1-13.   Google Scholar

[7]

H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal., 27 (1990), 987-1000.  doi: 10.1137/0727057.  Google Scholar

[8]

A. E. Bryson and Y. C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Hemisphere Publishing Corp. Washington, D. C.; Distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 1975.  Google Scholar

[9]

J. J. Buckley and Y. Hayashi, Can fuzzy neural nets approximate continuous fuzzy functions?, Fuzzy Sets and Systems, 61 (1994), 43-51.  doi: 10.1016/0165-0114(94)90283-6.  Google Scholar

[10]

A. CardoneE. Messina and A. Vecchio, An adaptive method for Volterra-Fredholm integral equations on the half line, J. Comput. Appl. Math., 228 (2009), 538-547.  doi: 10.1016/j.cam.2008.03.036.  Google Scholar

[11]

J. DijkmanH. V. Haeringen and S. D. Lange, Fuzzy numbers, Journal of Mathematical Analysis and Applications, 92 (1983), 301-341.  doi: 10.1016/0022-247X(83)90253-6.  Google Scholar

[12] D. DumitrescuB. Lazzerini and L. C. Jain, Fuzzy Sets & their Application to Clustering & Training, CRC Press, 2000.   Google Scholar
[13]

R. Fullér, Introduction to Neuro-Fuzzy Systems, Advances in Soft Computing, Physica-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-7908-1852-9.  Google Scholar

[14]

I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operator, Vol. 63, Birkhäuser, 2013. doi: 10.1007/978-3-0348-8558-4_1.  Google Scholar

[15] K. Gurney, An Introduction to Neural Networks, CRC Press, 2014.   Google Scholar
[16]

Y. HayashiJ. J. Buckley and E. Czogala, Fuzzy neural network with fuzzy signals and weights, International Journal of Intelligent Systems, 8 (1993), 527-537.   Google Scholar

[17]

D. O. Hebb and D. Hebb, The Organization of Behavior, Vol. 65, Wiley New York, 1949. Google Scholar

[18]

R. Hecht-Nielsen, Kolmogorov's mapping neural network existence theorem, IEEE Press, 3 (1987), 11-14.   Google Scholar

[19]

J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U. S. A., 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554.  Google Scholar

[20]

H. IshibuchiK. Morioka and I. Turksen, Learning by fuzzified neural networks, International Journal of Approximate Reasoning, 13 (1995), 327-358.  doi: 10.1016/0888-613X(95)00060-T.  Google Scholar

[21]

H. Ishibuchi, H. Tanaka and H. Okada., Fuzzy neural networks with fuzzy weights and fuzzy biases, IEEE International Conference on Neural Networks, (1993), 1650–1655. doi: 10.1109/ICNN.1993.298804.  Google Scholar

[22]

O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7.  Google Scholar

[23]

J. M. Keller and D. J. Hunt, Incorporating fuzzy membership functions into the perceptron algorithm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1985), 693-699.  doi: 10.1109/TPAMI.1985.4767725.  Google Scholar

[24]

E. Khan and P. Venkatapuram, Neufuz: Neural network based fuzzy logic design algorithms, Second IEEE International Conference on Fuzzy Systems, (1993), 647–654, https://ieeexplore.ieee.org/abstract/document/327412. Google Scholar

[25]

D. Kriesel, A Brief Introduction on Neural Network, 2007. Available from: http://www.dkriesel.com/en/science. Google Scholar

[26]

A. R. Krommer and C. W. Ueberhuber, Computational Integration, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. doi: 10.1137/1.9781611971460.  Google Scholar

[27]

C. T. Leondes, Fuzzy Logic and Expert Systems Applications, Vol. 6, Elsevier, 1998. Google Scholar

[28]

T. Mach, Eigenvalue Algorithms for Symmetric Hierarchical Matrices, Thomas Mach, 2012. Google Scholar

[29]

A. Malek and R. S. Beidokhti, Numerical solution for high order differential equations using a hybrid neural network-optimization method, Appl. Math. Comput., 183 (2006), 260-271.  doi: 10.1016/j.amc.2006.05.068.  Google Scholar

[30]

K. Maleknejard and M. Hadizadeh, A new computational method for Volterra-Fredholm integral equations, Comput. Math. Appl., 37 (1999), 1-8.  doi: 10.1016/S0898-1221(99)00107-8.  Google Scholar

[31]

J. McCarthy, Programs with Common Sense, RLE and MIT Computation Center, 1960. Google Scholar

[32]

W. S. McCulloch and W. Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5 (1943), 115-133.  doi: 10.1007/BF02478259.  Google Scholar

[33] K. MehrotraC. K. Mohan and S. Ranka, Elements of Artificial Neural Networks, MIT Press, 1997.  doi: 10.7551/mitpress/2687.001.0001.  Google Scholar
[34]

P. Milner, A brief history of the hebbian learning rule, Canadian Psychology/Psychologie Canadienne, 44 (2003), 5-9.  doi: 10.1037/h0085817.  Google Scholar

[35]

F. Mirzaee and A. A. Hoseini, Numerical solution of nonlinear Volterra-Fredholm integral equations using hybrid of block-pulse functions and Taylor series, Alexandria Engineering Journal, 52 (2013), 551-555.  doi: 10.1016/j.aej.2013.02.004.  Google Scholar

[36]

J. Moor, The dartmouth college artificial intelligence conference: The next fifty years, Ai Magazine, 27 (2006), 87-87.   Google Scholar

[37]

F. Mora-Camino and C. A. N. Cosenza, Fuzzy dual numbers, in fuzzy dual numbers, Springer, (2018), 11–16. Google Scholar

[38]

M. Mosleh, Fuzzy neural network for solving a system of fuzzy differential equations, Applied Soft Computing, 13 (2013), 3597-3607.  doi: 10.1016/j.asoc.2013.04.013.  Google Scholar

[39]

M. A. Nielsen, Neural Networks and Deep Learning, Vol. 25, Determination Press, 2015. Google Scholar

[40]

B. Pachpatte, On mixed Volterra-Fredholm type integral equations, Indian J. Pure Appl. Math., 17 (1986), 488-496.   Google Scholar

[41]

J. Y. Park and J. U. Jeong, On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations, Fuzzy Sets and Systems, 115 (2000), 425-431.  doi: 10.1016/S0165-0114(98)00341-8.  Google Scholar

[42]

W. Pedrycz, Fuzzy Modelling: Paradigms and Practice, Vol. 7, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4613-1365-6.  Google Scholar

[43] T. A. Polk and C. M. Seifert, Cognitive Modeling, MIT Press, 2002.   Google Scholar
[44]

V. Raju and R. Jayagopal, An arithmetic operations of icosagonal fuzzy number using Alpha cut, International Journal of Pure and Applied Mathematics, 120 (2018), 137-145.   Google Scholar

[45]

A. L. Samuel, Some studies in machine learning using the game of checkers, IBM J. Res. Develop., 3 (1959), 211-229.  doi: 10.1147/rd.33.0210.  Google Scholar

[46]

S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.  doi: 10.1016/0165-0114(87)90030-3.  Google Scholar

[47]

P. SibiS. A. Jones and P. Siddarth, Analysis of different activation functions using back propagation neural networks, Journal of Theoretical and Applied Information Technology, 47 (2013), 1264-1268.   Google Scholar

[48]

R. Subramanian, Emergent AI, Social Robots and the Law: Security, Privacy and Policy Issues, Journal of International, Technology and Information Management, 26 (2017). Google Scholar

[49]

E. A. Wan, Time Series Prediction by using a Connectionist Network with Internal Delay Lines, Addison-Wesley Publishing Co, 1993. Google Scholar

[50] P. P. Wang and S. K. Chang, Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, Plenum Press, New York-London, 1980.   Google Scholar
[51]

A.-M. Wazwaz, A First Course in Integral Equations, 2$^{nd}$ edition, World Scientific Publishing Company, Co. Pte. Ltd., Hackensack, NJ, 2015.  Google Scholar

[52]

P. Werbos, Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. Ph. D. dissertation, Harvard University, 1975.  Google Scholar

[53]

J. Yuan and S. Yu, Privacy preserving back-propagation neural network learning made practical with cloud computing, IEEE Transactions on Parallel and Distributed Systems, 25 (2013), 212-221.   Google Scholar

[54]

L. A. Zadeh and F. sets, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

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