Abstract
In this paper, a new solution for the hybrid fuzzy differential equation under generalized differentiability is proposed. This method is obtained by approximates the solution by (discontinuous) piece-wise polynomial approximations of degree 3. The existence and uniqueness of the solution and convergence of the method are discussed in detail. To illustrate the approach, some examples are finally given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
No data were used to support this study.
References
Abbasbandy S, Allahviranloo T (2002) Numerical solution of fuzzy differential equation by Taylor method. J Comput Method Appl Math 2:113–124
Ahmady N, Allahviranloo T, Ahmady E (2022) An estimation of the solution of first order fuzzy differential equations. In: Allahviranloo T, Salahshour S (eds) Advances in fuzzy integral and differential equations. Studies in fuzziness and soft computing. Springer, Cham
Allahviranloo T (2020) Uncertain information and linear systems. Springer, Cham
Allahviranloo T (2020) Fuzzy fractional differential operators and equations. Springer, Cham
Ahmady N, Allahviranloo T, Ahmady E (2020) A modified Euler method for solving fuzzy differential equations under generalized differentiability. Comput Appl Math 39:104. https://doi.org/10.1007/s40314-020-1112-1
Allahviranloo T, Ahmady N, Ahmady E (2007) Numerical solution of fuzzy differential equations by predictor-corrector method. Inf Sci 177(7):1633–1647
Allahviranloo T, Abbasbandy S, Ahmady N, Ahmady E (2009) Improved predictor-corrector method for solving fuzzy initial value problems. Inf Sci 179:945–955
Allahviranloo T, Gouyandeh Z, Armand A (2015) A full fuzzy method for solving differential equation based on Taylor expansion. J Intell Fuzzy Syst 29:1039–1055
Alur R, Henzinger TA, Sontag ED (1996) Hybrid systems III: Verification and control berlin. Springer, Heidelberg
BaloochShahryari MR, Salashour S (2012) Improved predictor - corrector method for solving fuzzy differential equations under generalized differentiability. J Fuzzy Set Valued Anal 2012:1–16
Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set Syst 151:581–599
Bede B, Stefanini L (2011) Solution of Fuzzy Differential Equations with generalized differentiability using LU-parametric representation. EUSFLAT 1:785–790
Bede B, Stefanini L (2013) Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst 230:119–141
Chalco-Cano Y, Romoman-Flores H (2008) On new solutions of fuzzy differential equations. Chaos Solitons Fractals 38:112–119
Epperson JF (2007) An introduction to numerical methods and analysis. Wiley, Hoboken
Gumah G, Naser MFM, Al-Smadi M, Al-Omari SKQ, Baleanu D (2020) Numerical solutions of hybrid fuzzy differential equations in a Hilbert space. Appl Numer Math 151:402–412
Hukuhara M (1967) Integration des applications mesurables dont la valeur est un compact convex. Funkcial Ekvac 10:205–229
Kandel A, Byatt WJ (1978) Fuzzy differential equations. In Proceedings of International Conference on Cybernetics and Society, Tokyo
Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317
Kim H, Sakthivel R (2012) Numerical solution of hybrid fuzzy differential equations using improved predictor corrector method. Commun Nonlinear Sci Numer Simulat 17:3788–3794
Ma M, Friedman M, Kandel A (1999) Numerical solutions of fuzzy differential equations. Fuzzy Sets Syst 105:133–138
Paripour M, Hajilou E, Hajilou A, Heidari H (2015) Application of Adomian decomposition method to solve hybrid fuzzy differential equations. J Taibah Univ Sci 9:95–103
Prakash P, Kalaiselvi V (2009) Numerical solution of hybrid fuzzy differential equations by predictor-corrector method. Int J Comput Math 86:121–134
Pederson S, Sambandham M (2009) Numerical solution of hybrid fuzzy differential equation IVPs by a characterization theorem. Inf Sci 179:319–328
Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319–330
Stefanini L (2008) A generalization of Hukuhara difference for interval and fuzzy arithmetic. In: Soft Methods for Handling Variability and Imprecision Series on Advances in Soft Computing, vol. 48. Springer, Berlin (An extended version is available online at the RePEc service: https://econpapers.repec.org/paper/urbwpaper/08-5f01.htm)
Funding
This research does not receive any specific funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Communicated by Leonardo Tomazeli Duarte.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ahmady, E., Allahviranloo, T., Ahmady, N. et al. An estimation of the solution of hybrid fuzzy differential equations. Comp. Appl. Math. 42, 133 (2023). https://doi.org/10.1007/s40314-022-02171-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-02171-z