# American Institute of Mathematical Sciences

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. [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. [3] A. B. Badiru and J. Cheung, Fuzzy Engineering Expert Systems with Neural Network Applications, Vol. 11, John Wiley & Sons, 2002. [4] E. Balagurusamy, Computer Oriented Statistical and Numerical Methods, Macmillan India Limited, 1988. [5] C. Bector and S. Chandra, Fuzzy numbers and fuzzy arithmetic, Fuzzy Mathematical Programming and Fuzzy Matrix Game, (2005), 39–56. [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. [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. [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. [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. [10] A. Cardone, E. 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. [11] J. Dijkman, H. 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. [12] D. Dumitrescu, B. Lazzerini and L. C. Jain, Fuzzy Sets & their Application to Clustering & Training, CRC Press, 2000. [13] R. Fullér, Introduction to Neuro-Fuzzy Systems, Advances in Soft Computing, Physica-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-7908-1852-9. [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. [15] K. Gurney, An Introduction to Neural Networks, CRC Press, 2014. [16] Y. Hayashi, J. J. Buckley and E. Czogala, Fuzzy neural network with fuzzy signals and weights, International Journal of Intelligent Systems, 8 (1993), 527-537. [17] D. O. Hebb and D. Hebb, The Organization of Behavior, Vol. 65, Wiley New York, 1949. [18] R. Hecht-Nielsen, Kolmogorov's mapping neural network existence theorem, IEEE Press, 3 (1987), 11-14. [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. [20] H. Ishibuchi, K. 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. [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. [22] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7. [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. [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. [25] D. Kriesel, A Brief Introduction on Neural Network, 2007. Available from: http://www.dkriesel.com/en/science. [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. [27] C. T. Leondes, Fuzzy Logic and Expert Systems Applications, Vol. 6, Elsevier, 1998. [28] T. Mach, Eigenvalue Algorithms for Symmetric Hierarchical Matrices, Thomas Mach, 2012. [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. [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. [31] J. McCarthy, Programs with Common Sense, RLE and MIT Computation Center, 1960. [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. [33] K. Mehrotra, C. K. Mohan and S. Ranka, Elements of Artificial Neural Networks, MIT Press, 1997.  doi: 10.7551/mitpress/2687.001.0001. [34] P. Milner, A brief history of the hebbian learning rule, Canadian Psychology/Psychologie Canadienne, 44 (2003), 5-9.  doi: 10.1037/h0085817. [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. [36] J. Moor, The dartmouth college artificial intelligence conference: The next fifty years, Ai Magazine, 27 (2006), 87-87. [37] F. Mora-Camino and C. A. N. Cosenza, Fuzzy dual numbers, in fuzzy dual numbers, Springer, (2018), 11–16. [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. [39] M. A. Nielsen, Neural Networks and Deep Learning, Vol. 25, Determination Press, 2015. [40] B. Pachpatte, On mixed Volterra-Fredholm type integral equations, Indian J. Pure Appl. Math., 17 (1986), 488-496. [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. [42] W. Pedrycz, Fuzzy Modelling: Paradigms and Practice, Vol. 7, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4613-1365-6. [43] T. A. Polk and C. M. Seifert, Cognitive Modeling, MIT Press, 2002. [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. [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. [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. [47] P. Sibi, S. 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. [48] R. Subramanian, Emergent AI, Social Robots and the Law: Security, Privacy and Policy Issues, Journal of International, Technology and Information Management, 26 (2017). [49] E. A. Wan, Time Series Prediction by using a Connectionist Network with Internal Delay Lines, Addison-Wesley Publishing Co, 1993. [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. [51] A.-M. Wazwaz, A First Course in Integral Equations, 2$^{nd}$ edition, World Scientific Publishing Company, Co. Pte. Ltd., Hackensack, NJ, 2015. [52] P. Werbos, Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. Ph. D. dissertation, Harvard University, 1975. [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. [54] L. A. Zadeh and F. sets, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

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##### 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. [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. [3] A. B. Badiru and J. Cheung, Fuzzy Engineering Expert Systems with Neural Network Applications, Vol. 11, John Wiley & Sons, 2002. [4] E. Balagurusamy, Computer Oriented Statistical and Numerical Methods, Macmillan India Limited, 1988. [5] C. Bector and S. Chandra, Fuzzy numbers and fuzzy arithmetic, Fuzzy Mathematical Programming and Fuzzy Matrix Game, (2005), 39–56. [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. [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. [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. [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. [10] A. Cardone, E. 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. [11] J. Dijkman, H. 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. [12] D. Dumitrescu, B. Lazzerini and L. C. Jain, Fuzzy Sets & their Application to Clustering & Training, CRC Press, 2000. [13] R. Fullér, Introduction to Neuro-Fuzzy Systems, Advances in Soft Computing, Physica-Verlag, Heidelberg, 2000. doi: 10.1007/978-3-7908-1852-9. [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. [15] K. Gurney, An Introduction to Neural Networks, CRC Press, 2014. [16] Y. Hayashi, J. J. Buckley and E. Czogala, Fuzzy neural network with fuzzy signals and weights, International Journal of Intelligent Systems, 8 (1993), 527-537. [17] D. O. Hebb and D. Hebb, The Organization of Behavior, Vol. 65, Wiley New York, 1949. [18] R. Hecht-Nielsen, Kolmogorov's mapping neural network existence theorem, IEEE Press, 3 (1987), 11-14. [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. [20] H. Ishibuchi, K. 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. [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. [22] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7. [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. [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. [25] D. Kriesel, A Brief Introduction on Neural Network, 2007. Available from: http://www.dkriesel.com/en/science. [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. [27] C. T. Leondes, Fuzzy Logic and Expert Systems Applications, Vol. 6, Elsevier, 1998. [28] T. Mach, Eigenvalue Algorithms for Symmetric Hierarchical Matrices, Thomas Mach, 2012. [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. [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. [31] J. McCarthy, Programs with Common Sense, RLE and MIT Computation Center, 1960. [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. [33] K. Mehrotra, C. K. Mohan and S. Ranka, Elements of Artificial Neural Networks, MIT Press, 1997.  doi: 10.7551/mitpress/2687.001.0001. [34] P. Milner, A brief history of the hebbian learning rule, Canadian Psychology/Psychologie Canadienne, 44 (2003), 5-9.  doi: 10.1037/h0085817. [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. [36] J. Moor, The dartmouth college artificial intelligence conference: The next fifty years, Ai Magazine, 27 (2006), 87-87. [37] F. Mora-Camino and C. A. N. Cosenza, Fuzzy dual numbers, in fuzzy dual numbers, Springer, (2018), 11–16. [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. [39] M. A. Nielsen, Neural Networks and Deep Learning, Vol. 25, Determination Press, 2015. [40] B. Pachpatte, On mixed Volterra-Fredholm type integral equations, Indian J. Pure Appl. Math., 17 (1986), 488-496. [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. [42] W. Pedrycz, Fuzzy Modelling: Paradigms and Practice, Vol. 7, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4613-1365-6. [43] T. A. Polk and C. M. Seifert, Cognitive Modeling, MIT Press, 2002. [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. [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. [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. [47] P. Sibi, S. 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. [48] R. Subramanian, Emergent AI, Social Robots and the Law: Security, Privacy and Policy Issues, Journal of International, Technology and Information Management, 26 (2017). [49] E. A. Wan, Time Series Prediction by using a Connectionist Network with Internal Delay Lines, Addison-Wesley Publishing Co, 1993. [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. [51] A.-M. Wazwaz, A First Course in Integral Equations, 2$^{nd}$ edition, World Scientific Publishing Company, Co. Pte. Ltd., Hackensack, NJ, 2015. [52] P. Werbos, Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. Ph. D. dissertation, Harvard University, 1975. [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. [54] L. A. Zadeh and F. sets, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.
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