Bipolar fuzzy Fourier transform for bipolar fuzzy solution of the bipolar fuzzy heat equation

Akram, Muhammad; Bilal, Muhammad; Shahriari, Mohammadreza; Allahviranloo, Tofigh

doi:10.22111/ijfs.2024.46925.8264

Document Type : Research Paper

Authors

¹ Department of Mathematics, University of the Punjab, Lahore, Pakistan

² Department of Mathematics, University of the Punjab, New Campus, Lahore 4590, Pakistan

³ Faculty of Management, South Tehran Branch, Islamic Azad University, Tehran, Iran

⁴ Faculty of engineering and natural science, Istinye university, Istanbul, Turkey.

https://doi.org/10.22111/ijfs.2024.46925.8264

Abstract

This article presents the exact solution of a bipolar fuzzy heat equation based on bipolar fuzzy Fourier transform under generalized Hukuhara partial (gH-p) differentiability. A bipolar fuzzy Fourier transform is defined, and the related key propositions and fundamental characteristics are discussed. Further, a bipolar fuzzy heat equation model is constructed using gH-differentiability, and the analytical solution of a bipolar fuzzy heat equation with bipolar fuzzy Fourier transform approach is examined. Some illustrative examples are provided to check the suggested methodology’s liability and efficiency. The type of differentiability and the solution of the bipolar fuzzy heat equation are shown graphically, demonstrating the versatility of the proposed
methodology and elucidating the impact of differentiability types on the solution behavior of the bipolar fuzzy heat equation. Additionally, the impact of different parameters on the solution behavior is analyzed, revealing insights into the underlying dynamics.

Keywords

Main Subjects

Fuzzy Math

References

[1] S. Ahmad, A. Ullah, A. Akg¨ul, T. Abdeljawad, Semi-analytical solutions of the 3rd order fuzzy dispersive partial
differential equations under fractional operators, Alexandria Engineering Journal, 60(6) (2021), 5861-5878. https:
//doi.org/10.1016/j.aej.2021.04.065

[2] M. Akram, M. Ali, T. Allahviranloo, A method for solving bipolar fuzzy complex linear systems with real and complex
coefficients, Soft Computing, 26(5) (2022), 2157-2178. https://doi.org/10.1007/s00500-021-06672-7

[3] M. Akram, T. Allahviranloo, W. Pedrycz, M. Ali, Methods for solving LR-bipolar fuzzy linear systems, Soft Computing, 25(1) (2021), 85-108. https://doi.org/10.1007/s00500-020-05460-z

[4] M. Akram, M. Arshad, A novel trapezoidal bipolar fuzzy TOPSIS method for group decision-making, Group Decision
and Negotiation, 28 (2019), 565-584. https://doi.org/10.1007/s10726-018-9606-6

[5] M. Akram, G. Muhammad, Analysis of incommensurate multi-order fuzzy fractional differential equations under
strongly generalized fuzzy Caputo’s differentiability, Granular Computing, 8(4) (2023), 809-825. https://doi.org/
10.1007/s41066-022-00353-y

[6] M. Akram, G. Muhammad, T. Allahviranloo, Bipolar fuzzy linear system of equations, Computational and Applied
Mathematics, 38(69) (2019), 69. https://doi.org/10.1007/s40314-019-0814-8

[7] M. Akram, G. Muhammad, T. Allahviranloo, Explicit analytical solutions of an incommensurate system of fractional
differential equations in a fuzzy environment, Information Sciences, 645 (2023), 119372. https://doi.org/10.
1016/j.ins.2023.119372

[8] M. Akram, G. Muhammad, T. Allahviranloo, W. Pedrycz, Incommensurate non-homogeneous system of fuzzy linear
fractional differential equations using the fuzzy bunch of real functions, Fuzzy Sets and Systems, 473 (2023), 108725.
https://doi.org/10.1016/j.fss.2023.108725

[9] T. Allahviranloo, Difference methods for fuzzy partial differential equtions, Computational Methods in Applied
Mathematics, 2(3) (2006), 233-242. https://doi.org/10.2478/cmam-2002-0014

[10] T. Allahviranloo, S. Abbasbandy, H. Rouhparvar, The exact solutions of fuzzy wave like equations with variable
coefficients by a variational iteration method, Applied Soft Computing, 11(2) (2011), 2186-2192. https://doi.org/
10.1016/j.asoc.2010.07.018

[11] T. Allahviranloo, K. M. Afshar, Numerical methods for fuzzy linear partial differential equations under new definition for derivative, Iranian Journal of Fuzzy Systems, 7(3) (2010), 33-50. https://doi.org/10.22111/IJFS.2010.
187

[12] T. Allahviranloo, Z. Gouyandeh, A. Armand, A. Hasanoglu, On fuzzy solution for 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

[13] T. Allahviranloo, N. Taheri, An analytic approximation to the solution of fuzzy heat equation by adomian decompostion method, International Journal of Contemporary Mathematical Sciences, 4(3) (2009), 105-114.

[14] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to
fuzzy ordinary differential equations, Fuzzy Sets and Systems, 151(3) (2005), 581-599. https://doi.org/10.1016/
j.fss.2004.08.001

[15] B. Bede, G. Lovane, I. Esposito, Fuzzy Fourier transforms and their application to fingerprint identification, Journal
of Discrete Mathematical Sciences and Cryptography, 8(1) (2005), 59-79. https://doi.org/10.1080/09720529.
2005.10698021

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

[17] A. M. Bertone, R. M. Jafelice, L. C. de Barros, R. C. Bassanezi, On fuzzy solution for partial differential equations,
Fuzzy Sets and Systems, 219 (2013), 68–80. https://doi.org/10.1016/j.fss.2012.12.002

[18] J. J. Buckley, Fuzzy complex number, Fuzzy Sets and Systems, 33(3) (1989), 333-345. https://doi.org/10.1016/
0165-0114(89)90122-X

[19] J. J. Buckley, T. Feuring, Introduction to fuzzy partial differential equtions, Fuzzy Sets and Systems, 105(2) (1999),
241-248. https://doi.org/10.1016/S0165-0114(98)00323-6

[20] S. S. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Transaction on Systems, 2(1) (1972), 30-34.
https://doi.org/10.1109/TSMC.1972.5408553

[21] Y. Y. Chen, Y. T. Chang, B. S. Chen, Fuzzy solution to partial differential equations: Adoptive approach, IEEE
Transactions on Fuzzy Systems, 17(1) (2009), 116-127. https://doi.org/10.1109/TFUZZ.2008.2005010

[22] F. Dayan, N. Ahmed, M. Rafiq, A. Akg¨ul, A. Raza, M. O. Ahmad, F. Jarad, Construction and numerical analysis
of a fuzzy non-standard computational method for the solution of an SEIQR model of COVID-19 dynamics, AIMS
Mathematics, 7(5) (2022), 8449-8470. http://hdl.handle.net/20.500.12416/7409

[23] D. Dubois, H. Prade, Operations on fuzzy numbers, International Journal of Systems Sciences, 9(6) (1978), 613-626.
https://doi.org/10.1080/00207727808941724

[24] D. Dubois, H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Sets and
Systems, 8(1) (1982), 1-17. https://doi.org/10.1016/0165-0114(82)90025-2

[25] D. Dubois, H. Prade, Towards fuzzy differential calculus part 2: Integration on fuzzy intervals, Fuzzy Sets and
Systems, 8(2) (1982), 105-116. https://doi.org/10.1016/0165-0114(82)90001-X

[26] D. Dubois, H. Prade, Towards fuzzy differential calculus part 3: Differentiation, Fuzzy Sets and Systems, 8(3)
(1982), 225-233. https://doi.org/10.1016/S0165-0114(82)80001-8

[27] Z. Gouyandeh, T. Allahviranloo, S. Abbasbandy, A. Armand, A fuzzy solution of heat equation under generalized
Hukuhara differentiability by fuzzy Fourier transform, In Fuzzy Sets and Systems, 309 (2017), 81-97. https://doi.
org/10.1016/j.fss.2016.04.010

[28] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24(3) (1987), 301-317. https://doi.org/10.
1016/0165-0114(87)90029-7

[29] A. Khan, M. Farman, A. Akg¨ul, Decision making under Pythagorean fuzzy soft environment, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 31(5) (2023), 773-793. https://doi.org/10.1142/
S0218488523500368

[30] S. Nagalakshmi, G. S. Kumar, Solution for one dimensional fuzzy heat equation under generalized Hukuhara differentiability, AIP Conference Proceedings, 2375(1) (2021), 020031. https://doi.org/10.1063/5.0066523

[31] M. Oberguggenbberger, Fuzzy and weak solution to differential equations, in: Proceedings of the 10th International
IPMU Conference, (2004), 517-524.

[32] U. M. Pirzada, D. C. Vakaskar, Fuzzy solution of homogeneous heat equation having solution in Fourier series
form, SeMA, 76 (2019), 181-194. https://doi.org/10.1007/s40324-018-0169-x

[33] M. L. Puri, D. A. Ralescu, Differentials of fuzzy functions, Journal of Mathematical Analysis and Application,
91(12) (1983), 552-558. https://doi.org/10.1016/0022-247X(83)90169-5

[34] M. Qayyum, E. Ahmad, A. Akg¨ul, S. M. El Din, Fuzzy-fractional modeling of Korteweg-de Vries equations in
Gaussian-Caputo sense: New solutions via extended He-Mahgoub algorithm, Ain Shams Engineering Journal, 15(4)
(2024), 102623. https://doi.org/10.1016/j.asej.2023.102623

[35] M. Qayyum, A. Tahir, A. Bariq, A. Akg¨ul, S. T. Saeed, Modeling and analysis of thin film flow of fuzzified
Johnson Segalman nanofluid using fuzzy extension of He-Laplace scheme, Mathematical and Computer Modelling
of Dynamical Systems, 29(1) (2023), 286-314. https://doi.org/10.1080/13873954.2023.2276440

[36] M. Saqib, M. Akram, S. Bashir, Certain efficient iterative methods for bipolar fuzzy system of linear equations,
Journal of Intelligent and Fuzzy Systems, 39(3) (2020), 3971-3985. https://doi.org/10.3233/JIFS-200084

[37] M. Saqib, M. Akram, S. Bashir, T. Allahviranloo, Numerical solution of bipolar fuzzy initial value problem, Journal
of Intelligent and Fuzzy Systems, 40(1) (2021), 1309-1341. https://doi.org/10.3233/JIFS-201619

[38] M. Saqib, M. Akram, S. Bashir, T. Allahviranloo, A Runge-Kutta numerical method to approximate the solution
of bipolar fuzzy initial value problems, Computational and Applied Mathematics, 40(4) (2021), 151. https://doi.
org/10.1007/s40314-021-01535-1

[39] S. Seikkala, On the fuzzy initial value problm, Fuzzy Sets and Systems, 24(3) (1987), 319-330. https://doi.org/
10.1016/0165-0114(87)90030-3

[40] L. Stefanini, A generalization of hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and
Systems, 161(11) (2010), 1564-1584. https://doi.org/10.1016/j.fss.2009.06.009

[41] L. Stefanini, B. Bede, Generalized hukuhara differentiability of fuzzy-number-valued functions with interval differential equations, Nonlinear Analysis, 71 (2009), 1311-1328. https://doi.org/10.1016/j.na.2008.12.005

[42] A. Ullah, A. Ullah, S. Ahmad, I. Ahmad, A. Akg¨ul,On solutions of fuzzy fractional order complex population
dynamical model, Numerical Methods for Partial Differential Equations, 39(6) (2023), 4595-4615. https://doi.
org/10.1002/num.22654

[43] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353. https://doi.org/10.1016/
S0019-9958(65)90241-X

[44] W. R. Zhang, Bipolar fuzzy sets and relations: A computational framework for cognitive modeling and multiagent
decision analysis, In Proceedings of IEEE Conference, (1994), 305-309. https://doi.org/10.1109/IJCF.1994.
375115

Bipolar fuzzy Fourier transform for bipolar fuzzy solution of the bipolar fuzzy heat equation

References

References

Volume 21, Issue 3May and June 2024Pages 19-36

Volume 21, Issue 3
May and June 2024
Pages 19-36