A Neural Computational Intelligence Method Based on Legendre Polynomials for Fuzzy Fractional Order Differential Equation

[1] Z. Li, D. Chen, J. Zhu, and Y. Liu, Nonlinear dynamics of fractional order Duffing system, Chaos, Solitons & Fractals, 81 (2015) 111–116.Search in Google Scholar

[2] S. Pourdehi, A. Azami, and F. Shabaninia, Fuzzy Kalman-type filter for interval fractional-order systems with finite-step auto-correlated process noises, Neurocomputing, 159 (2015) 44–49.Search in Google Scholar

[3] A. Boulkroune, A. Bouzeriba, and T. Bouden, Fuzzy generalized projective synchronization of incommensurate fractional-order chaotic systems, Neurocomputing, 173(3) (2016) 606-614.10.1016/j.neucom.2015.08.003Search in Google Scholar

[4] Y. Ji, L. Su, and J. Qiu, Design of fuzzy output feedback stabilization for uncertain fractional-order systems, Neurocomputing, 173(3) (2016) 1683-1693.10.1016/j.neucom.2015.09.041Search in Google Scholar

[5] Y. Zheng, Fuzzy prediction-based feedback control of fractional order chaotic systems, Opt. - Int. J. Light Electron Opt., 126(24) (2015) 5645-5649.10.1016/j.ijleo.2015.08.164Search in Google Scholar

[6] S. T. S. Chakraverty, A New Approach to Fuzzy Initial Value Problem by Improved Euler Method, Fuzzy information and Engineering, 4(3) (2012) 293–312.10.1007/s12543-012-0117-xSearch in Google Scholar

[7] V. H. Ngo, Fuzzy fractional functional integral and differential equations, Fuzzy Sets Syst., 280 (2015) 58–90.Search in Google Scholar

[8] N. Van Hoa, Fuzzy fractional functional differential equations under Caputo gH-differentiability, Commun. Nonlinear Sci. Numer. Simul., 22 (2015) 1134–1157.Search in Google Scholar

[9] T. Senthilkumar, Robust stabilization and H ∞ control for nonlinear stochastic T–S fuzzy Markovian jump systems with mixed time-varying delays and linear fractional uncertainties, Neurocomputing, 173(3) (2016) 1615–1624.10.1016/j.neucom.2015.09.033Search in Google Scholar

[10] H. Delavari, R. Ghaderi, A. Ranjbar, and S. Momani, Fuzzy fractional order sliding mode controller for nonlinear systems, Commun. Nonlinear Sci. Numer. Simul., 15(4) (2010) 963–978.10.1016/j.cnsns.2009.05.025Search in Google Scholar

[11] S. Salahshour, T. Allahviranloo, and S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17(3) (2012) 1372–1381.10.1016/j.cnsns.2011.07.005Search in Google Scholar

[12] S. Tapaswini, S. Chakraverty, and D. Behera, Numerical solution of the imprecisely defined inverse heat conduction problem, Chin. Phys. B, 24(5) (2015) 05020310.1088/1674-1056/24/5/050203Search in Google Scholar

[13] A. Ahmadian, M. Suleiman, and S. Salahshour, An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations, Abstract and appl. analysis, 2013 Article ID 50590310.1155/2013/505903Search in Google Scholar

[14] T. Allahviranloo,, N. Ahmady, and E. Ahmady, Numerical solution of fuzzy differential equations by predictor–corrector method, Information Sciences, 177(7) (2007) 1633-1647.10.1016/j.ins.2006.09.015Search in Google Scholar

[15] M.Z. Ahmad, M.K. Hasan, and B.D. Baets, Analytical and numerical solutions of fuzzy differential equations, Information Sciences, 236 (2013) 156–167.Search in Google Scholar

[16] A. Khastan, and K. Ivaz, Numerical solution of fuzzy differential equations by Nyström method, Chaos, Solitons & Fractals, 41(2) (2009) 859–868.10.1016/j.chaos.2008.04.012Search in Google Scholar

[17] E. ElJaoui, S. Melliani, and L.S. Chadli, Solving second-order fuzzy differential equations by the fuzzy Laplace transform method, Advances in diff. Eqn., (2015) 2015: 66.10.1186/s13662-015-0414-xSearch in Google Scholar

[18] N. A. Khan, F. Riaz, and O. A. Razzaq, A comparison between numerical methods for solving fuzzy fractional differential equations, Non Linear Engineering, 3(3) (2014) 155-162.10.1515/nleng-2013-0029Search in Google Scholar

[19] N. A. Khan, , O. A. Razzaq, and F. Riaz, Numerical Simulations for Solving Fuzzy Fractional Differential Equations by Max-Min Improved Euler Methods, JACSM, 7(1) (2015) 53-83.10.1515/jacsm-2015-0010Search in Google Scholar

[20] H. Sadoghi, M. Pakdaman, and H. Modaghegh, Unsupervised kernel least mean square algorithm for solving ordinary differential equations, Neurocomputing, 74(12-13) (2011) 2062–2071.10.1016/j.neucom.2010.12.026Search in Google Scholar

[21] S. Mall and S. Chakraverty, Numerical solution of nonlinear singular initial value problems of Emden – Fowler type using Chebyshev Neural Network method, Neurocomputing, 149 (2015) 975–982.Search in Google Scholar

[22] A. J. Meade, The Numerical Solution of Linear Ordinary Differential Equations by Feedforward Neural Networks, 19(12) (1994) 1-25.10.1016/0895-7177(94)90095-7Search in Google Scholar

[23] A. J. Meade and A. A. Fernandez, Solution of Nonlinear Ordinary Differential Equations by Feedforward Neural Networks, Math.Comp.Model., 20(9) (1994) 19–44.10.1016/0895-7177(94)00160-XSearch in Google Scholar

[24] I. E. Lagaris, A. C. Likas, and D. I. Fotiadis, Artifical Neural Networks for Solving Ordinary and Partial Differential Equations, Neural Networks, IEEE Trans., 9(5) (1998) 1–26.10.1109/72.71217818255782Search in Google Scholar

[25] I. E. Lagaris, A. C. Likas, and D. G. Papageorgiou, Neural-Network Methods for Boundary Value Problems with Irregular Boundaries, 11(5) (2000) 1041–1049.10.1109/72.87003718249832Search in Google Scholar

[26] D. R. Parisi, C. Mariani, and M. A. Laborde, Solving differential equations with unsupervised neural networks, Chem. Eng. Process. Process Intensif., 42 (2003) 715–721.Search in Google Scholar

[27] 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.Search in Google Scholar

[28] H. J. Zimmermann, Fuzzy set theory and its application, Dordrecht, Kluwer Academi Publishers, 1991.10.1007/978-94-015-7949-0Search in Google Scholar

[29] D. Dubois, and H. Prade, Fundamentals of Fuzzy Sets, Kluwer Academic Publishers, USA, 2000.10.1007/978-1-4615-4429-6Search in Google Scholar

[30] B.R. Sontakke and A.S. Shaikh, Properties of Caputo operator and its applications to linear fractional differential equations, International Journal of Engineering Research and Applications, 5(5) (2015) 22-29.Search in Google Scholar

[31] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal International, 13(5) (1967) 529-56810.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar

[32] S. Salahshour, T. Allahviranloo, and S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun Nonlinear Sci Numer Simulat, 17 (2012) 1372–1381.Search in Google Scholar

[33] G. J. Klir, and B. Yuan, FUZZY SETS AND FUZZY LOGIC Theory and Applications, Prentice Hall P T R Upper Saddle River, New Jersey 07458Search in Google Scholar

[34] D. Dubois, and H. Prade, Operations on fuzzy numbers, INT. J. SYSTEMS SCI., 9(6) (1978) 613-626.10.1080/00207727808941724Search in Google Scholar

[35] S. Ledesma, G. Aviña, and R. Sanchez, Practical considerations for simulated annealing implementation, In: Tan CM, editor. Simulated Annealing, InTech, 2008.10.5772/5560Search in Google Scholar