The explicit solution to fuzzy differential equation with piecewise constant arguments involving the multi-order fractional derivative in (1,2)

Lai, Van Phut; Ngo, Van Hoa

doi:10.22111/ijfs.2026.54660.9681

Document Type : Research Paper

Authors

Van Phut Lai ¹
Van Hoa Ngo ²

¹ Department of Mathematics, FPT University HCM, Saigon Hi-tech Park, Ho Chi Minh City, Vietnam

² Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, Vietnam

https://doi.org/10.22111/ijfs.2026.54660.9681

Abstract

This paper investigates non-homogeneous linear fuzzy fractional differential equations with piecewise constant arguments involving generalized multi-order Caputo fractional derivatives of order $\mathfrak{q}_{\imath} \in (1,2)$. By employing generalized single-level constrained fuzzy arithmetic, explicit closed-form solutions are obtained. A fuzzy vehicle-following model is included to demonstrate the applicability of the proposed results and to illustrate the effects of fractional orders, kernel functions, and fuzzy disturbances on the solution behavior.

Keywords

Main Subjects

Fuzzy Math

References

[1] K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Local and global existence and uniqueness of solution for
class of fuzzy fractional functional evolution equation, Journal of Function Spaces, 2022 (2022), 7512754. https:
//doi.org/10.1155/2022/7512754

[2] T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equations with uncertainty,
Soft Computing, 16 (2012), 297-302. https://doi.org/10.1007/s00500-011-0743-y

[3] R. Almeida, On the variable-order fractional derivatives with respect to another function, Aequationes Mathematicae,
99 (2025), 805-822. https://doi.org/10.1007/s00010-024-01082-0

[4] T. V. An, V. Lupulescu, N. V. Hoa, Asymptotical stabilization of fuzzy semilinear dynamic systems involving the
generalized Caputo fractional derivative for q ∈ (1, 2), Fractional Calculus and Applied Analysis, 27 (2024), 1186-
1214. https://doi.org/10.1007/s13540-024-00268-2

[5] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, 2013. https://doi.org/10.1007/978-3-642\
-35221-8

[6] S. Busenberg, K. L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, Nonlinear
Phenomena in Mathematical Sciences, (1982), 179-187. https://doi.org/10.1016/B978-0-12-434170-8.
50028-5

[7] M. S. Cecconello, M. T. Mizukoshi, W. Lodwick, Interval nonlinear initial-valued problem using constraint intervals:
Theory and an application to the Sars-Cov-2 outbreak, Information Sciences, 577 (2021), 871-882. https://doi.
org/10.1016/j.ins.2021.08.045

[8] Y. Chalco-Cano, W. A. Lodwick, B. Bede, Single-level constraint interval arithmetic, Fuzzy Sets and Systems, 257
(2014), 146-168. https://doi.org/10.1016/j.fss.2014.06.017

[9] D. Dubois, H. Prade, Fuzzy numbers: An overview, Readings in Fuzzy Sets for Intelligent Systems, (1993), 112-148.
https://doi.org/10.1016/B978-1-4832-1450-4.50015-8

[10] D. Dubois, H. Prade, Gradual elements in a fuzzy set, Soft Computing, 12 (2008), 165-175. https://doi.org/
10.1007/s00500-007-0187-6

[11] Z. Eidinejad, R. Saadati, Hyers-Ulam-Rassias-Kummer stability of the fractional integro-differential equations,
Mathematical Biosciences and Engineering, 19(7) (2022), 6536-6550. https://doi.org/10.3934/mbe.2022308

[12] S. Elaydi, An introduction to difference equations, Third Edition, Springer, 2005. https://doi.org/10.1007/
0-387-27602-5

[13] E. Esmi, F. S. Pedro, L. C. de Barros, W. Lodwick, Fr´echet derivative for linearly correlated fuzzy function,
Information Sciences, 435 (2018), 150-160. https://doi.org/10.1016/j.ins.2017.12.051

[14] N. V. Hoa, Fractional impulsive fuzzy differential systems with multi-order in (1, 2): The solution representation
and asymptotical stabilization, Information Sciences, 719 (2025), 122461. https://doi.org/10.1016/j.ins.2025.
122461

[15] N. V. Hoa, N. D. Phu, Fuzzy discrete fractional calculus and fuzzy fractional discrete equations, Fuzzy Sets and
Systems, 492 (2024), 109073. https://doi.org/10.1016/j.fss.2024.109073

[16] A. Khan, R. Shafqat, A. U. K. Niazi, Existence results of fuzzy delay impulsive fractional differential equation by
fixed point theory approach, Journal of Function Spaces, 2022 (2022), 4123949. https://doi.org/10.1155/2022/
4123949

[17] C. F. Lorenzo, T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynamics, 29
(2002), 57-98 . https://doi.org/10.1023/A:1016586905654

[18] M. Mazandarani, N. Pariz, A. V. Kamyad, Granular differentiability of fuzzy-number-valued functions, IEEE Transactions
and Fuzzy Systems, 26(1) (2018), 310-323. https://doi.org/10.1109/TFUZZ.2017.2659731

[19] N. D. Phu, L. V. Phut, N. V. Hoa, Solving fuzzy non-homogeneous linear differential systems with piecewise
constant arguments involving the short-memory variable-order Caputo fractional derivative, Applied Mathematics
and Computation, 510 (2026), 129672. https://doi.org/10.1016/j.amc.2025.129672

[20] L. V. Phut, Representation of solutions to fuzzy linear fractional differential equations with a piecewise constant
argument, Physica Scripta, 99(6) (2024), 065239. https://doi.org/10.1088/1402-4896/ad4689

[21] A. Piegat, M. Landowski, Horizontal membership function and examples of its applications, International Journal
of Fuzzy Systems, 17 (2015), 22-30. https://doi.org/10.1007/s40815-015-0013-8

[22] S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms and Special
Functions, 1(4) (1993), 277-300. https://doi.org/10.1080/10652469308819027

[23] R. Shafqat, A. U. K. Niazi, M. B. Jeelani, N. H. Alharthi, Existence and uniqueness of mild solution where
α ∈ (1, 2) for fuzzy fractional evolution equations with uncertainty, Fractal and Fractional, 6(2) (2022), 65. https:
//doi.org/10.3390/fractalfract6020065

[24] D. Tavares, R. Almeida, D. F. M. Torres, Caputo derivatives of fractional variable order: Numerical approximations,
Communications in Nonlinear Science and Numerical Simulation, 35 (2016), 69-87. https://doi.org/10.1016/j.
cnsns.2015.10.027

Iranian Journal of Fuzzy Systems

The explicit solution to fuzzy differential equation with piecewise constant arguments involving the multi-order fractional derivative in (1,2)

References

References

Volume 23, Issue 4
July and August 2026
Pages 183-201

The explicit solution to fuzzy differential equation with piecewise constant arguments involving the multi-order fractional derivative in (1,2)

References

References

Volume 23, Issue 4July and August 2026Pages 183-201

Volume 23, Issue 4
July and August 2026
Pages 183-201