Fuzzy Fractional Calculus: A Comprehensive Overview with a Focus on weighted Caputo-type generalized Hukuhara Differentiability and Analytical Solutions for Fuzzy Fractional Differential Equations

Ezzati, Reza; Zabihi, Samaneh

doi:10.22111/ijfs.2024.45607.8045

Document Type : Research Paper

Authors

Reza Ezzati ¹
Samaneh Zabihi ²

¹ Islamic Azad University, karaj branch

² Department of Mathematics, North Tehran Branch, Islamic Azad University, Tehran, Iran

https://doi.org/10.22111/ijfs.2024.45607.8045

Abstract

This paper introduces a novel approach to obtaining analytical solutions for
fuzzy fractional differential equations in the context of weighted Caputo-type generalized
Hukuhara derivatives. The paper proposes the use of non-singular kernels to improve the
accuracy of fractional calculus in fuzzy space and establishes the uniqueness of solutions for
fuzzy fractional differential equations. The paper also introduces the concept of fuzzy Laplace
transforms to facilitate the solution of these equations. Practical examples, such as the fuzzy
fractional Newton’s law of heating and cooling, are provided to demonstrate the effectiveness
of the proposed method. Overall, this paper contributes to the development of practical solutions
for real-world problems in fuzzy space and enhances the accuracy of fractional calculus
in this context.

Keywords

Main Subjects

Fuzzy approximation

References

[1] R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with
uncertainty, Nonlinear Analysis: Theory, Methods and Applications, 72 (2010), 59-62. https://doi.org/10.1016/
j.na.2009.11.029

[2] T. Allahviranloo, A. Armand, Z. Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo
derivative, Journal of Intelligent and Fuzzy Systems, 26 (2014), 1481-1490. https://doi.org/10.3233/IFS-130831

[3] T. Allahviranloo, B. Ghanbari, On the fuzzy fractional differential equation with interval Atangana-Baleanu fractional
derivative approach, Chaos, Solitons and Fractals, 130 (2020), 109397. https://doi.org/10.1016/j.chaos.
2019.109397

[4] 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

[5] M. Al-Refai, On weighted Atangana-Baleanu fractional operators, Advances in Difference Equations, 3 (2020).
https://doi.org/10.1186/s13662-019-2471-z

[6] M. Al-Smadi, O. A. Arqub, D. Zeidan, Fuzzy fractional differential equations under the Mittag-Leffler kernel differential
operator of the ABC approach: Theorems and applications, Chaos, Solitons and Fractals, 146 (2021), 110891.
https://doi.org/10.1016/j.chaos.2021.110891

[7] A. Armand, T. Allahviranloo, S. Abbasbandy, Z. Gouyandeh, The fuzzy generalized Taylor’s expansion with
application in fractional differential equations, Iranian Journal of Fuzzy Systems, 16(2) (2019), 57-72. https:
//doi.org/10.22111/IJFS.2019.4542

[8] S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty, Nonlinear Analysis: Theory,
Methods and Applications, 74 (2011), 85-93. https://doi.org/10.1016/j.na.2011.02.048

[9] S. Arshad, V. Luplescu, Fractional differential equation with fuzzy initial condition, Electronic Journal of Differential
Equations, 34 (2011), 1-8.

[10] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013),
119-141. https://doi.org/10.1016/j.fss.2012.10.003

[11] M. Friedman, M. Ma, A. Kandel, Numerical solutions of fuzzy differential and integral equations, in: Fuzzy Modeling
and Dynamics. Fuzzy Sets and Systems, 106(1) (1999), 35-48. https://doi.org/10.1016/S0165-0114(98)
00355-8

[12] S. G. Gal, Approximation theory in fuzzy setting, In: Anastassiou, G. A. (ed.) Handbook of Analytic-Computational
Methods in Applied Mathematics. ch13, 617-666. Chapman and Hall/CRC, Boca Raton, 2000.

[13] M. Ghaffari, T. Allahviranloo, S. Abbasbandy, M. Azhini, On the fuzzy solutions of time-fractional problems,
Iranian Journal of Fuzzy Systems, 18(3) (2021), 51-66. https://doi.org/10.22111/IJFS.2021.6081

[14] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43. https://doi.
org/10.1016/0165-0114(86)90026-6

[15] K. Hattaf, New generalized definition of fractional derivative with non-singular kernel, Computation, 8 (2020), 49.
https://doi.org/10.3390/computation8020049

[16] K. Hattaf, On some properties of the new generalized fractional derivative with non-singular kernel, Mathematical
Problems in Engineering, 2021 (2021). https://doi.org/10.1155/2021/1580396

[17] N. V. Hoa, H. Vu, T. Minh Duc, Fuzzy fractional differential equations under Caputo-Katugampola fractional
derivative approach, fuzzy Sets and Systems, 375 (2019), 70-99. https://doi.org/10.1016/j.fss.2018.08.001

[18] M. Keshavarz, T. Allahviranloo, Fuzzy fractional diffusion processes and drug release, Fuzzy Sets and Systems,
436 (2022), 82-101. https://doi.org/10.1016/j.fss.2021.04.001

[19] M. Keshavarz, E. Qahremani, T. Allahviranloo, Solving a fuzzy fractional diffusion model for cancer tumor by using
fuzzy transforms, Fuzzy Sets and Systems, 443 (2022), 198-220. https://doi.org/10.1016/j.fss.2021.10.009

[20] V. Lakshmikantham, T. Bhaskar, J. Devi, Theory of set differential equations in metric spaces, Cambridge Scientific
Publishers, 2006.

[21] F. Mainardi, On some properties of the Mittag-Leffler function E℘(−t℘), completely monotone for t > 0 with
0 < ℘ < 1, Discrete and Continuous Dynamical Systems-B, 19(7) (2014), 2267-2278. https://doi.org/10.48550/
arXiv.1305.0161

[22] H. E. Mamouni, M. E. Younoussi, Z. Hajhouji, K. Hattaf, N. Yousfi, Existence and uniqueness results of solutions
for Hattaf-type fractional differential equations with application to epidemiology, Communications in Mathematical
Biology and Neuroscience, 2022 (2022), 39. https://doi.org/10.28919/cmbn/7329

[23] M. Mortezaee, M. Ghovatmand, A. Nazemi, An application of generalized fuzzy hyperbolic model for solving fractional
optimal control problems with Caputo-Fabrizio derivative, Neural Processing Letters, 52 (2020), 1997-2020.
https://doi.org/10.1007/s11063-020-10334-4

[24] S. Salahshour, T. Allahviranloo, Applications of fuzzy Laplace transforms, Soft Computing, 17 (2013), 145-158.
https://doi.org/10.1007/s00500-012-0907-4

[25] S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace
transforms, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 1372-1381. https://doi.
org/10.1016/j.cnsns.2011.07.005

[26] S. Salahshour, M. Khezerloo, S. Hajighasemi, M. Khorasany, Solving fuzzy integral equations of the second kind by
fuzzy Laplace transform method, International Journal of Industrial Mathematics, 4(1) (2012), 21-29.

[27] V. M. Tam, On existence and stability results to fuzzy Caputo fractional differential inclusions driven by fuzzy
mixed quasivariational inequalities, Iranian Journal of Fuzzy Systems, 20(3) (2023), 43-59. https://doi.org/10.
22111/IJFS.2023.7638

[28] H. Viet Long, N. Thi Kim Son, H. Thi Thanh Tam, The solvability of fuzzy fractional partial differential equations
under Caputo gH-differentiability, Fuzzy Sets and Systems, 309 (2017), 35-63. https://doi.org/10.1016/j.fss.
2016.06.018

Fuzzy Fractional Calculus: A Comprehensive Overview with a Focus on weighted Caputo-type generalized Hukuhara Differentiability and Analytical Solutions for Fuzzy Fractional Differential Equations

References

References

Volume 21, Issue 2March and April 2024Pages 19-34

Volume 21, Issue 2
March and April 2024
Pages 19-34