Towards Efficient Solutions of Space-Time Fractional Fuzzy Diffusion Equations: A Methodological Approach

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, P.O. Box 13185/768, Tehran, Iran.

Abstract

This paper aims to introduce a groundbreaking methodology for deriving analytical solutions to the space-time fractional fuzzy diffusion equation. Our approach uniquely incorporates a Caputo generalized Hukuhara fractional derivative (of order $\beta \in (0,2]$) for the second-order spatial derivative, alongside a fuzzy Caputo-Katugampola generalized Hukuhara time-fractional derivative (of order $\alpha \in (0,1)$) for the first-order temporal derivative. The primary objective is to develop explicit and fundamental solutions for both the space-time fractional fuzzy diffusion equation and the time fractional fuzzy diffusion equation, encompassing various forms of fuzzy Caputo-Katugampola generalized Hukuhara time-fractional differentiability. We initiate our study by thoroughly analyzing the fuzzy Fourier and fuzzy $\wp-$Laplace transforms of the equation. To demonstrate the practical utility and effectiveness of our proposed method, we apply it to two specific models: a fuzzy groundwater flow model for computing pressure head, and a fuzzy model for determining the concentration of tumor cells. The results obtained highlight the method's efficiency and precision in addressing the complexities of both the space-time fractional fuzzy diffusion equation and the time fractional fuzzy diffusion equation.

Keywords

Main Subjects

References

[1] T. Allahviranloo, Fuzzy fractional differential operators and equations: Fuzzy fractional differential equations, Germany,
Springer International Publishing, (2020). https://doi.org/10.1007/978-3-030-51272-9
[2] T. Allahviranloo, Z. Gouyandeh, A. Armand, 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, Z. Gouyandeh, A. Armand, A. Hasanoglu, On fuzzy solutions for the heat equation based on
generalized Hukuhara differentiability, Fuzzy Sets and Systems, 265 (2015), 1-23. https://doi.org/10.1016/j.
fss.2014.11.009
[4] A. Alshbeel, A. Azmi, A. K. Alomari, Generalized Caputo-Katugampola for solving fuzzy fractional heat equation,
Results in Nonlinear Analysis, 7(1) (2024), 44-63. https://doi.org/10.31838/rna/2024.07.01.006
[5] M. P. Anderson, W. W. Woessner, R. J. Hunt, Applied groundwater modeling: Simulation of flow and advective
transport, Academic Press, (2015). https://doi.org/10.1016/C2009-0-21563-7
[6] A. Armand, T. Allahviranloo, Z. Gouyandeh, Some fundamental results on fuzzy calculus, Iranian Journal of Fuzzy
Systems, 15(3) (2018), 27-46. https://doi.org/10.22111/IJFS.2018.3948
[7] A. Atangana, Mathematical analysis of groundwater flow models, CRC Press, (2022). https://doi.org/10.1201/
9781003266266
[8] B. Bede, Mathematics of fuzzy sets and fuzzy logic, Springer, London, (2013). https://doi.org/10.1007/
978-3-642-35221-8
[9] J. N. Del Pino, P. D’iaz, Pesticide distribution and movement. Biotherapy, 11 (1998), 69-76. https://doi.org/10.
1023/A:1007961524517
[10] H. Eghlimi, M. S. Asgari, A study of the time-fractional heat equation under the generalized Hukuhara conformable
fractional derivative, Chaos, Solitons and Fractals, 175 (2023), 114007. https://doi.org/10.1016/j.chaos.2023.
114007
[11] M. Friedman, M. Ma, A. Kandel, Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets and
Systems, 106(1) (1999), 35-48. https://doi.org/10.1016/S0165-0114(98)00355-8
[12] 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
[13] Z. Gouyandeh, T. Allahviranloo, S. Abbasbandy, A. Armand, A fuzzy solution of the heat equation under generalized
Hukuhara differentiability by fuzzy Fourier transform, Fuzzy Sets and Systems, 309 (2017), 81-97. https://doi.
org/10.1016/j.fss.2016.04.010
[14] K. M. Hiscock, V. F. Bense, Hydrogeology: Principles and practice, Wiley-Blackwell, (2014). https://doi.org/
10.1144/1470-9236/05-105
[15] N. V. Hoa, H. Vu, T. M. Duc, Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative
approach, Fuzzy Sets and Systems, 375(15) (2019), 70-99. https://doi.org/10.1016/j.fss.2018.08.001
[16] J. Istok, Groundwater modeling by the finite element method (water resources monograph), American Geophysical
Union, (1989). https://doi.org/10.1029/WM013
[17] F. Jarad, T. Abdeljawad, A modified Laplace transform for certain generalized fractional operators, Results in
Nonlinear Analysis, 1(2) (2018), 88-98.
[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] V. Lakshmikantham, T. Bhaskar, J. Devi, Theory of set differential equations in metric spaces, Cambridge Scientific
Publishers, (2006).
[20] J. D. Logan, Applied partial differential equations, Springer Cham, (2015). https://doi.org/10.1007/
978-3-319-12493-3
[21] D. Mohapatra, S. Chakraverty, M. Alshammari, Time fractional heat equation of n + 1-dimension in Type-
1 and Type-2 fuzzy environment, International Journal of Fuzzy Systems, (2023). https://doi.org/10.1007/
s40815-023-01569-z
[22] L. Sajedi, N. Eghbali, H. Aydi, Impulsive coupled system of fractional differential equations with Caputo-
Katugampola fuzzy fractional derivative, Journal of Mathematics, (2021). https://doi.org/10.1155/2021/
7275934
[23] S. Salahshour, T. Allahviranloo, Applications of fuzzy Laplace transforms, Soft Computing, 17 (2013), 145-158.
https://doi.org/10.1007/s00500-012-0907-4
[24] W. Sawangtong, P. Sawangtong, An analytical solution for the Caputo type generalized fractional evolution equation,
Alexandria Engineering Journal, 61 (2022), 5475-5483. https://doi.org/10.1016/j.aej.2021.10.055
[25] 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
[26] W. W. Woessner, E. P. Poeter, Hydrogeologic properties of earth materials and principles
of groundwater flow, The Groundwater Project, (2020). https://books.gw-project.org/
hydrogeologic-properties-of-earth-materials-and-principles-of-groundwater-flow/.
[27] H. C. Wu, The improper fuzzy Riemann integral and its numerical integration, Information Sciences, 111(14)
(1998), 109-137. https://doi.org/10.1016/S0020-0255(98)00016-4
Iranian Journal of Fuzzy Systems
Volume 21, Issue 4
July and August 2024
Pages 179-195

Files

Share

How to cite

Statistics