Numerical solutions of hybrid fuzzy differential equations in a Hilbert space. (English) Zbl 1451.65102
Summary: The main goal of this work is to study a numerical method for certain hybrid fuzzy differential equations with an application of a reproducing kernel Hilbert space technique for fuzzy differential equations. Meanwhile, we construct a system of orthogonal functions of the space \(W_2^2 [a, b] \oplus W_2^2 [a, b]\) depending on a Gram-Schmidt orthogonalization process to get approximate-analytical solutions of a hybrid fuzzy differential equation. A proof of convergence of this method is also discussed in detail. The exact as well as the approximate solutions are displayed by a series in terms of their \(\alpha \)-cut representation form in the Hilbert space \(W_2^2 [a, b] \oplus W_2^2 [a, b]\). To demonstrate behavior, efficiency, and appropriateness of the present technique, two different numerical experiments are solved numerically in this paper.
MSC:
| 65L99 | Numerical methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A07 | Fuzzy ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34A38 | Hybrid systems of ordinary differential equations |
| 46N20 | Applications of functional analysis to differential and integral equations |
Keywords:
hybrid fuzzy differential equation; fuzzy derivative; Gram-Schmidt process; reproducing kernel function; Hilbert spaceReferences:
| [1] | Abbod, M. F.; von Keyserlingk, D. G.; Linkens, D. A.; Mahfouf, M., Survey of utilisation of fuzzy technology in medicine and healthcare, Fuzzy Sets Syst., 120, 331-349 (2001) |
| [2] | Abu Arqub, O.; Al-Smadi, M., Atangana-Baleanu fractional approach to the solutions of Bagley-Torvik and Painlevé equations in Hilbert space, Chaos Solitons Fractals, 117, 161-167 (2018) · Zbl 1442.65119 |
| [3] | Al-Smadi, M.; Abu Arqub, O., Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of Dirichlet functions type with error estimates, Appl. Math. Comput., 342, 280-294 (2019) · Zbl 1429.65304 |
| [4] | Barro, S.; Marn, R., Fuzzy Logic in Medicine (2002), Physica-Verlag: Physica-Verlag Heidelberg · Zbl 0978.68780 |
| [5] | Bede, B.; Rudas, I. J.; Bencsik, A. L., First order linear fuzzy differential equations under generalized differentiability, Inf. Sci., 177, 1648-1662 (2007) · Zbl 1119.34003 |
| [6] | Buckley, J. J.; Feuring, T., Fuzzy differential equations, Fuzzy Sets Syst., 110, 43-54 (2000) · Zbl 0947.34049 |
| [7] | Caldas, M.; Jafari, M., θ-compact fuzzy topological spaces, Chaos Solitons Fractals, 25, 229-232 (2005) · Zbl 1070.54501 |
| [8] | Casasnovas, J.; Rossell, F., Averaging fuzzy biopolymers, Fuzzy Sets Syst., 152, 139-158 (2005) · Zbl 1060.92032 |
| [9] | Chalco-Can, Y.; Roman-Flores, H., On new solutions of fuzzy differential equations, Chaos Solitons Fractals, 38, 112-119 (2008) · Zbl 1142.34309 |
| [10] | Chang, B. C.; Halgamuge, S. K., Protein motif extraction with neuro-fuzzy optimization, Bioinformatics, 18, 1084-1090 (2002) |
| [11] | Cui, M.; Geng, F., Solving singular two-point boundary value problem in reproducing kernel space, J. Comput. Appl. Math., 205, 6-15 (2007) · Zbl 1149.65057 |
| [12] | El Naschie, M. S., A review of E-infinite theory and the mass spectrum of high energy particle physics, Chaos Solitons Fractals, 19, 209-236 (2004) · Zbl 1071.81501 |
| [13] | El Naschie, M. S., The concepts of E-infinite: an elementary introduction to the Cantorian-fractal theory of quantum physics, Chaos Solitons Fractals, 22, 495-511 (2004) · Zbl 1063.81582 |
| [14] | El Naschie, M. S., From experimental quantum optics to quantum gravity via a fuzzy Kahler manifold, Chaos Solitons Fractals, 25, 969-977 (2005) · Zbl 1070.81118 |
| [15] | Feng, G.; Chen, G., Adaptative control of discrete-time chaotic systems: a fuzzy control approach, Chaos Solitons Fractals, 23, 459-467 (2005) · Zbl 1061.93501 |
| [16] | Geng, F., Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. Math. Model., 215, 2095-2102 (2009) · Zbl 1178.65085 |
| [17] | Goetschel, R.; Voxman, W., Elementary fuzzy calculus, Fuzzy Sets Syst., 18, 31-43 (1986) · Zbl 0626.26014 |
| [18] | Gumah, G.; Moaddy, K.; Al-Smadi, M.; Hashim, I., Solutions of uncertain Volterra integral equations by fitted reproducing kernel Hilbert space method, J. Funct. Spaces, 2016, Article 2920463 pp. (2016) · Zbl 1347.65194 |
| [19] | Gumah, G.; Naser, M.; Al-Smadi, M.; Al-Omari, S., Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations, Adv. Differ. Equ., 2018, Article 475 pp. (2018) · Zbl 1451.65235 |
| [20] | Guo, M.; Xue, X.; Li, R., Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets Syst., 138, 601-615 (2003) · Zbl 1084.34072 |
| [21] | Hasan, S.; Al-Smadi, M.; Freihet, A.; Momani, S., Two computational approaches for solving a fractional obstacle system in Hilbert space, Adv. Differ. Equ., 2019, Article 55 pp. (2019) · Zbl 1458.65093 |
| [22] | Jiang, W.; Guo-Dong, Q.; Bin, D., \( H_\infty\) variable universe adaptative fuzzy control for chaotic systems, Chaos Solitons Fractals, 24, 1075-1086 (2005) · Zbl 1083.93013 |
| [23] | Jie, L.; Yuesheng, L., A new method for computing reproducing kernels, Northeast. Math. J., 14, 467-473 (1998) · Zbl 0939.65080 |
| [24] | Kaleva, O., Fuzzy differential equations, Fuzzy Sets Syst., 24, 301-317 (1987) · Zbl 0646.34019 |
| [25] | Kaleva, O., A note on fuzzy differential equations, Nonlinear Anal., 64, 895-900 (2006) · Zbl 1100.34500 |
| [26] | Lakshmikantham, V.; Mohapatra, R. N., Theory of Fuzzy Differential Equations and Inclusions (2003), Taylor & Francis: Taylor & Francis London · Zbl 1072.34001 |
| [27] | Li, C.; Cui, M., The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Appl. Math. Comput., 143, 393-399 (2003) · Zbl 1034.47030 |
| [28] | Okutmustur, B., Reproducing Kernel Hilbert Spaces (2005), Bilkent University |
| [29] | Park, J. H., Intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 22, 1039-1046 (2004) · Zbl 1060.54010 |
| [30] | Pederson, S.; Sambandham, M., Numerical solution to hybrid fuzzy systems, Math. Comput. Model., 45, 1133-1144 (2007) · Zbl 1123.65069 |
| [31] | Pederson, S.; Sambandham, M., Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem, Inf. Sci., 179, 3, 319-328 (2009) · Zbl 1165.65041 |
| [32] | Prakash, P.; Kalaiselvi, V., Numerical solution of hybrid fuzzy differential equations by predictor-corrector method, Int. J. Comput. Math., 86, 1, 121-134 (2009) · Zbl 1158.65049 |
| [33] | Saadeh, R.; Al-Smadi, M.; Gumah, G.; Khalil, H.; Ali Khan, R., Numerical investigation for solving two-point fuzzy boundary value problems by reproducing kernel approach, Appl. Math. Inf. Sci., 10, 6, 2117-2129 (2016) |
| [34] | Zadeh, L. A., Fuzzy sets, Inf. Control, 8, 3, 338-353 (1965) · Zbl 0139.24606 |
| [35] | Zhang, S.; Liu, L.; Diao, L., Reproducing kernel functions represented by form of polynomials, (Proceedings of the Second Symposium International Computer Science and Computational Technology, vol. 26 (2009)), 353-358 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
