Numerical methods for solving fuzzy equations: a survey. (English) Zbl 1464.65049

Jafari, Raheleh; Yu, Wen; Razvarz, Sina; Gegov, Alexander

Fuzzy Sets Syst. 404, 1-22 (2021).

Summary: In this paper, we study different numerical methods for solving fuzzy equations, dual fuzzy equations, fuzzy differential equations (FDEs) and fuzzy partial differential equations (PDEs). In this study, conditions that guarantee the existence of the roots of these equations are discussed. Also, this paper provides some discussion about the rates of convergence of each of the numerical methods. Finally, some numerical examples are given to illustrate the efficiency of these methods.

MSC:

65H99	Nonlinear algebraic or transcendental equations
26E50	Fuzzy real analysis

Keywords:

nonlinear systems; fuzzy number; fuzzy solution

References:

[1]	Jafari, R.; Yu, W., Uncertainty nonlinear systems control with fuzzy equations, Proc. IEEE Int. Conf. Syst. Man Cybern., 2885-2890 (2015)
[2]	Jafari, R.; Razvarz, S., Solution of fuzzy differential equations using fuzzy Sumudu transforms, Math. Comput. Appl., 23, 1-15 (2018) · Zbl 1390.34009
[3]	Jafari, R.; Razvarz, S.; Gegov, A., A new computational method for solving fully fuzzy nonlinear systems, (Computational Collective Intelligence: 10th International Conference. Computational Collective Intelligence: 10th International Conference, ICCCI 2018, Bristol, UK, September 5-7, 2018, Proceedings, Part I. Computational Collective Intelligence: 10th International Conference. Computational Collective Intelligence: 10th International Conference, ICCCI 2018, Bristol, UK, September 5-7, 2018, Proceedings, Part I, Lecture Notes in Computer Science, vol. 11055 (2018), Springer), 503-512
[4]	Buckley, J.; Hayashi, Y., Can fuzzy neural nets approximate continuous fuzzy functions, Fuzzy Sets Syst., 61, 1, 43-51 (1994) · Zbl 0837.68088
[5]	Gibson, J., An analysis of optimal modal regulation: convergence and stability, SIAM J. Control Optim., 19, 5, 686-707 (1981) · Zbl 0477.49019
[6]	Abbasbandy, S.; Ezzati, R., Newton’s method for solving a system of fuzzy nonlinear equations, Appl. Math. Comput., 175, 2, 1189-1199 (2006) · Zbl 1093.65049
[7]	Allahviranloo, T.; Otadi, M.; Mosleh, M., Iterative method for fuzzy equations, Soft Comput., 12, 10, 935-939 (2008) · Zbl 1145.65032
[8]	Gouyandeh, Z.; Allahviranloo, T.; Abbasbandy, S.; Armand, A., A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform, Fuzzy Sets Syst., 309, 81-97 (2017) · Zbl 1386.35465
[9]	Moghaddam, R. G.; Allahviranloo, T., On the fuzzy Poisson equation, Fuzzy Sets Syst., 347, 105-128 (2018) · Zbl 1397.35341
[10]	Kajani, M.; Asady, B.; Vencheh, A., An iterative method for solving dual fuzzy nonlinear equations, Appl. Math. Comput., 167, 1, 316-323 (2005) · Zbl 1082.65532
[11]	Waziri, M.; Majid, Z., A new approach for solving dual fuzzy nonlinear equations using Broyden’s and Newton’s methods, Adv. Fuzzy Syst., 2012, 1-5 (2012) · Zbl 1250.65069
[12]	Pederson, S.; Sambandham, M., The Runge-Kutta method for hybrid fuzzy differential equation, Nonlinear Anal. Hybrid Syst., 2, 2, 626-634 (2008) · Zbl 1155.93370
[13]	Khastan, A.; Ivaz, K., Numerical solution of fuzzy differential equations by Nyström method, Chaos Solitons Fractals, 41, 2, 859-868 (2009) · Zbl 1198.65113
[14]	Palligkinis, S.; Papageorgiou, G.; Famelis, I., Runge-Kutta methods for fuzzy differential equations, Appl. Math. Comput., 209, 1, 97-105 (2009) · Zbl 1161.65058
[15]	Tapaswini, S.; Chakraverty, S., Euler-based new solution method for fuzzy initial value problems, Int. J. Artif. Intell. Soft Comput., 4, 1, 58-79 (2014) · Zbl 1311.34008
[16]	Ahmadi, M.; Kiani, N.; Mikaeilvand, N., Laplace transform formula on fuzzy nth-order derivative and its application in fuzzy ordinary differential equations, Soft Comput., 18, 12, 2461-2469 (2014) · Zbl 1336.34007
[17]	Guo, S.; Mei, L.; Zhou, Y., The compound \(\frac{ G^\prime}{ G} \)-expansion method and double non-traveling wave solutions of \((2 + 1)\)-dimensional nonlinear partial differential equations, Comput. Math. Appl., 69, 8, 804-816 (2015) · Zbl 1443.35125
[18]	Chehlabi, M.; Allahviranloo, T., Positive or negative solutions to first-order fully fuzzy linear differential equations under generalized differentiability, Appl. Soft Comput., 70, 359-370 (2018)
[19]	Chehlabi, M.; Allahviranloo, T., Concreted solutions to fuzzy linear fractional differential equations, Appl. Soft Comput., 44, 108-116 (2016)
[20]	Buckley, J.; Eslami, E., Neural net solutions to fuzzy problems: the quadratic equation, Fuzzy Sets Syst., 86, 3, 289-298 (1997) · Zbl 0914.68172
[21]	Jafarian, A.; Jafari, R.; Khalili, A.; Baleanud, D., Solving fully fuzzy polynomials using feed-back neural networks, Int. J. Comput. Math., 92, 4, 742-755 (2015) · Zbl 1347.65094
[22]	Jafari, R.; Yu, W., Fuzzy control for uncertainty nonlinear systems with dual fuzzy equations, J. Intell. Fuzzy Syst., 29, 3, 1229-1240 (2015) · Zbl 1362.93084
[23]	Mosleh, M., Evaluation of fully fuzzy matrix equations by fuzzy neural network, Appl. Math. Model., 37, 9, 6364-6376 (2013) · Zbl 1426.65055
[24]	Tahavvor, A.; Yaghoubi, M., Analysis of natural convection from a column of cold horizontal cylinders using artificial neural network, Appl. Math. Model., 36, 7, 3176-3188 (2012)
[25]	Effati, S.; Pakdaman, M., Artificial neural network approach for solving fuzzy differential equations, Inf. Sci., 180, 8, 1434-1457 (2010) · Zbl 1185.65114
[26]	Yazdi, H.; Pourreza, R., Unsupervised adaptive neural-fuzzy inference system for solving differential equations, Appl. Soft Comput., 10, 1, 267-275 (2010)
[27]	Lee, H.; Kang, I., Neural algorithms for solving differential equations, J. Comput. Phys., 91, 1, 110-131 (1990) · Zbl 0717.65062
[28]	Dissanayake, M.; Phan-Thien, N., Neural-network based approximations for solving partial differential equations, Int. J. Numer. Methods Biomed. Eng., 10, 3, 195-201 (1994) · Zbl 0802.65102
[29]	He, S.; Reif, K.; Unbehauen, R., Multilayer neural networks for solving a class of partial differential equations, Neural Netw., 13, 3, 385-396 (2000)
[30]	Montelora, C.; Saloma, C., Solving the nonlinear Schrodinger equation with an unsupervised neural network: estimation of error in solution, Opt. Commun., 222, 1-6, 331-339 (2003)
[31]	Sukavanam, N.; Panwar, V., Computation of boundary control of controlled heat equation using artificial neural networks, Int. Commun. Heat Mass Transf., 30, 8, 1137-1146 (2003)
[32]	Lodwick, W.; Dubois, D., Interval linear systems as a necessary step in fuzzy linear systems, Fuzzy Sets Syst., 281, 227-251 (2015) · Zbl 1368.15029
[33]	Shary, S., Solving the tolerance problem for interval linear equations, Interval Comput., 2, 4-22 (1994)
[34]	Nguyen, H., A note on the extension principle for fuzzy sets, J. Math. Anal. Appl., 64, 369-380 (1978) · Zbl 0377.04004
[35]	Jafari, R.; Yu, W.; Li, X., Fuzzy differential equations for nonlinear system modeling with Bernstein neural networks, IEEE Access, 4, 9428-94367 (2016)
[36]	Bede, B.; Stefanini, L., Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230, 119-141 (2013) · Zbl 1314.26037
[37]	Newton, I., The Method of Fluxions and Infinite Series, 3, 43-47 (1671), Henry Woodfall: Henry Woodfall London, ISBN 9781498167482, retrieved from
[38]	Abbasbandy, S.; Asady, B., Newton’s method for solving fuzzy nonlinear equations, Appl. Math. Comput., 159, 2, 349-356 (2004) · Zbl 1060.65049
[39]	Moore, R., Interval Analysis (1966), Prentice-Hall: Prentice-Hall Englewood Cliffs
[40]	Sevastjanov, P.; Dymova, L.; Bartosiewicz, P., A new approach to normalization of interval and fuzzy weights, Fuzzy Sets Syst., 198, 34-45 (2012) · Zbl 1251.94051
[41]	Abbasbandy, S.; Jafarian, A., Steepest descent method for solving fuzzy nonlinear equations, Appl. Math. Comput., 174, 1, 669-675 (2006) · Zbl 1089.65042
[42]	Faires, J.; Burden, R. L.; Burden, A., Numerical Analysis (2015), PWS-Kent: PWS-Kent Boston
[43]	Okada, S.; Gen, M., Fuzzy multiple choice knapsack problem, Fuzzy Sets Syst., 67, 71-80 (1994)
[44]	Lin, F., Simulating fuzzy numbers for solving fuzzy equations with constraints using genetic algorithm, Int. J. Innov. Comput. Inf. Control, 6, 1, 239-253 (2010)
[45]	Brudaru, O.; Leon, F.; Buzatu, O., Genetic algorithm for solving fuzzy equations, (8th International Symposium on Automatic Control and Computer Science. 8th International Symposium on Automatic Control and Computer Science, Iasi (2004))
[46]	Cantu-Paz, E., Implementing Fast and Flexible Parallel Genetic Algorithms, Practical Handbook of Genetic Algorithms, vol. 3 (1998), CRC Press: CRC Press Boca Raton
[47]	Delgado, M.; Vila, M.; Voxman, W., On a canonical representation of fuzzy numbers, Fuzzy Sets Syst., 93, 1, 125-135 (1998) · Zbl 0916.04004
[48]	Rouhparvar, H., Solving fuzzy polynomial equation by ranking method, (First Joint Congress on Fuzzy and Intelligent Systems (2007), Ferdowsi University of Mashhad: Ferdowsi University of Mashhad Iran), 1-6
[49]	Ab, R. N.; Lazim, A.; Termimi, A. A.; Noorani, A., Solutions of interval type-2 fuzzy polynomials using a new ranking method, AIP Conf. Proc., 1682, 1, Article 020012 pp. (2015)
[50]	Cybenko, G., Approximation by superposition of sigmoidal activation function, Math. Control Signals Syst., 2, 4, 303-314 (1989) · Zbl 0679.94019
[51]	Buckley, J.; Eslami, E.; Hayashi, Y., Solving fuzzy equations using neural nets, Fuzzy Sets Syst., 86, 3, 271-278 (1997) · Zbl 0918.68100
[52]	Buckley, J.; Qu, Y., Solving linear and quadratic fuzzy equations, Fuzzy Sets Syst., 35, 1, 43-59 (1990) · Zbl 0713.04004
[53]	Buckley, J.; Feuring, T.; Hayashi, Y., Solving fuzzy equations using evolutionary algorithms and neural nets, Soft Comput., 6, 2, 116-123 (2002) · Zbl 1001.68102
[54]	Allahviranloo, T.; Ahmadi, N.; Ahmadi, E., Numerical solution of fuzzy differential equations by predictor-corrector method, Inf. Sci., 177, 7, 1633-1647 (2007) · Zbl 1183.65090
[55]	Ma, M.; Friedman, M.; Kandel, A., Numerical solutions of fuzzy differential equations, Fuzzy Sets Syst., 105, 1, 133-138 (1999) · Zbl 0939.65086
[56]	Bede, B., Note on numerical solutions of fuzzy differential equations by predictor corrector method, Inf. Sci., 178, 7, 1917-1922 (2008) · Zbl 1183.65092
[57]	Paripour, M.; Hajilou, E.; Heidari, H., Application of Adomian decomposition method to solve hybrid fuzzy differential equations, J. Taibah Univ. Sci., 9, 1, 95-103 (2015)
[58]	Pirzada, U.; Vakaskar, D., Solution of fuzzy heat equations using Adomian decomposition method, Int. J. Adv. Appl. Math. Mech., 3, 1, 87-91 (2015) · Zbl 1359.35220
[59]	Seikkala, S., On the fuzzy initial value problem, Fuzzy Sets Syst., 24, 319-330 (1987) · Zbl 0643.34005
[60]	Singh, N.; Kumar, M., Adomian decomposition method for solving higher order boundary value problems, Math. Theory Model., 2, 1, 11-22 (2011)
[61]	Abbasbandy, S.; Allahvinloo, T., Numerical solutions of fuzzy differential equation, Math. Comput. Appl., 7, 1, 41-52 (2002) · Zbl 1013.65070
[62]	Abbasbandy, S.; Allahvinloo, T., Numerical solutions of fuzzy differential equations by Taylor method, Comput. Methods Appl. Math., 2, 2, 113-124 (2002)
[63]	Abbasbandy, S.; Allahvinloo, T., Numerical solution of fuzzy differential equation by Runge-Kutta method, Nonlinear Stud., 11, 1, 117-129 (2004) · Zbl 1056.65069
[64]	Hussian, E.; Suhhiem, M., Numerical solution of fuzzy partial differential equations by using modified fuzzy neural networks, Br. J. Math. Comput. Sci., 12, 2, 1-20 (2016)
[65]	Mosleh, M.; Otadi, M., Simulation and evaluation of fuzzy differential equations by fuzzy neural network, Appl. Soft Comput., 12, 9, 2817-2827 (2012)
[66]	Streeter, V.; Wylie, E.; Benjamin, E., Fluid Mechanics (1966), Mc. Graw-Hill Book Company
[67]	Beer, F.; Johnston, E., Mechanics of Materials (1992), McGraw-Hilly
[68]	(Hazewinkel, M., Oscillator Harmonic, Encyclopedia of Mathematics (2001), Springer Science+Business Media B.V. Kluwer Academic Publishers), [1994]
[69]	Pletcher, R.; Tannehill, J.; Anderson, D., Computational Fluid Mechanics and Heat Transfer (2013), Taylor and Francis · Zbl 1285.76003