Numerical solutions of fuzzy equal width models via generalized fuzzy fractional derivative operators

Rehana Ashraf; Saima Rashid; Fahd Jarad; Ali Althobaiti; Rehana Ashraf; Saima Rashid; Fahd Jarad; Ali Althobaiti

doi:10.3934/math.2022152

AIMS Mathematics

2022, Volume 7, Issue 2: 2695-2728. doi: 10.3934/math.2022152

Previous Article Next Article

Research article

Numerical solutions of fuzzy equal width models via generalized fuzzy fractional derivative operators

1.
Department of Mathematics, Lahore College Women University, 54000, Lahore, Pakistan
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Çankaya University, Ankara, Turkey
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
Department of Mathematics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 20 August 2021 Accepted: 12 November 2021 Published: 18 November 2021
MSC : 46S40, 47H10, 54H25

The Shehu homotopy perturbation transform method (SHPTM) via fuzziness, which combines the homotopy perturbation method and the Shehu transform, is the subject of this article. With the assistance of fuzzy fractional Caputo and Atangana-Baleanu derivatives operators, the proposed methodology is designed to illustrate the reliability by finding fuzzy fractional equal width (EW), modified equal width (MEW) and variants of modified equal width (VMEW) models with fuzzy initial conditions (ICs). In cold plasma, the proposed model is vital for generating hydro-magnetic waves. We investigated SHPTM's potential to investigate fractional nonlinear systems and demonstrated its superiority over other numerical approaches that are accessible. Another significant aspect of this research is to look at two significant fuzzy fractional models with differing nonlinearities considering fuzzy set theory. Evaluating various implementations verifies the method's impact, capabilities, and practicality. The level impacts of the parameter $ \hbar $ and fractional order are graphically and quantitatively presented, demonstrating good agreement between the fuzzy approximate upper and lower bound solutions. The findings are numerically examined to crisp solutions and those produced by other approaches, demonstrating that the proposed method is a handy and astonishingly efficient instrument for solving a wide range of physics and engineering problems.
- Shehu transform,
- Caputo fractional derivative,
- AB-fractional operator,
- homotopy perturbation method,
- equal width equation,
- modified equal width equation
Citation: Rehana Ashraf, Saima Rashid, Fahd Jarad, Ali Althobaiti. Numerical solutions of fuzzy equal width models via generalized fuzzy fractional derivative operators[J]. AIMS Mathematics, 2022, 7(2): 2695-2728. doi: 10.3934/math.2022152

Related Papers:

Abstract

The Shehu homotopy perturbation transform method (SHPTM) via fuzziness, which combines the homotopy perturbation method and the Shehu transform, is the subject of this article. With the assistance of fuzzy fractional Caputo and Atangana-Baleanu derivatives operators, the proposed methodology is designed to illustrate the reliability by finding fuzzy fractional equal width (EW), modified equal width (MEW) and variants of modified equal width (VMEW) models with fuzzy initial conditions (ICs). In cold plasma, the proposed model is vital for generating hydro-magnetic waves. We investigated SHPTM's potential to investigate fractional nonlinear systems and demonstrated its superiority over other numerical approaches that are accessible. Another significant aspect of this research is to look at two significant fuzzy fractional models with differing nonlinearities considering fuzzy set theory. Evaluating various implementations verifies the method's impact, capabilities, and practicality. The level impacts of the parameter $ \hbar $ and fractional order are graphically and quantitatively presented, demonstrating good agreement between the fuzzy approximate upper and lower bound solutions. The findings are numerically examined to crisp solutions and those produced by other approaches, demonstrating that the proposed method is a handy and astonishingly efficient instrument for solving a wide range of physics and engineering problems.

References

[1]	I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
[2]	R. Hilfer, Applications of fractional calculus in physics, Word Scientific, Singapore, 2000.
[3]	A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and application of fractional differential equations, North Holland Math. Stud., 204, 2006.
[4]	R. L. Magin, Fractional Calculus in bioengineering, Begell House Publishers, 2006.
[5]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, London-New York: Gordon and Breach, Yverdon, 1993.
[6]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 73 (2015). doi: 10.12785/pfda/010201. doi: 10.12785/pfda/010201
[7]	A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel, theory and application to heat transfer model, Therm. Sci., 20 (2016). doi: 10.2298/TSCI160111018A. doi: 10.2298/TSCI160111018A
[8]	M. H. 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, Soliton. Fract., 146 (2021). doi: 10.1016/j.chaos.2021.110891. doi: 10.1016/j.chaos.2021.110891
[9]	M. H. Al-Smadi, O. A. Arqub, S. Momani, Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense, Phys. Scripta, 95 (2020). doi: 10.1088/1402-4896/ab96e0. doi: 10.1088/1402-4896/ab96e0
[10]	O. A. Arqub, M. H. Al-Smadi, An adaptive numerical approach for the solutions of fractional advection-diffusion and dispersion equations in singular case under Riesz's derivative operator, Physica A, 540 (2019). doi: 10.1016/j.physa.2019.123257. doi: 10.1016/j.physa.2019.123257
[11]	M. H. Al-Smadi, O. A. Arqub, S. Hadid, An attractive analytical technique for coupled system of fractional partial differential equations in shallow water waves with conformable derivative, Commun. Theor. Phy., 72 (2020). doi: 10.1088/1572-9494/ab8a29. doi: 10.1088/1572-9494/ab8a29
[12]	S. Kumar, D. Zeidan, An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation, Appl. Numer. Math., 170 (2021), 190–207. doi: 10.1016/j.apnum.2021.07.025. doi: 10.1016/j.apnum.2021.07.025
[13]	D. Zeidan, S. Govekar, M. Pandey, Discontinuity wave interactions in generalized magnetogasdynamics, Acta Astronaut., 180 (2021), 110–114. doi: 10.1016/j.actaastro.2020.12.025. doi: 10.1016/j.actaastro.2020.12.025
[14]	D. Zeidan, C. K. Chau, T. T. Lu, On the characteristic Adomian decomposition method for the Riemann problem, Math. Method. Appl. Sci., 44 (2021), 8097–8112. doi: 10.1002/mma.5798. doi: 10.1002/mma.5798
[15]	S. Sil, T. R. Sekhar, D. Zeidan, Nonlocal conservation laws, nonlocal symmetries and exact solutions of an integrable soliton equation, Chaos, Solit. Fract., 139 (2020). doi: 10.1016/j.chaos.2020.110010. doi: 10.1016/j.chaos.2020.110010
[16]	E. Babolian, J. Saeidian, M. Paripour, Application of the homotopy analysis method for solving equal-width wave and modified equal-width wave equations, Z. Nat. forsch., 64a (2009), 685–690. doi: 10.1515/zna-2009-1103. doi: 10.1515/zna-2009-1103
[17]	R. Arora, M. J. Siddiqui, V. P. Singh, Solution of the modified equal width equation, its variant and non-homogeneous Burgers' equation by RDT method, Am. J. Comput. Appl. Math., 1 (2011), 53–56. doi: 10.5923/j.ajcam.20110102.10. doi: 10.5923/j.ajcam.20110102.10
[18]	L. R. T. Gardner, G. A. Gardner, Solitary waves of the equal width wave equation, J. Comput. Phys., 101 (1992), 218–223. doi: 10.1016/0021-9991(92)90054-3. doi: 10.1016/0021-9991(92)90054-3
[19]	E. Yusufoglu, A. Bekir, Numerical simulation of equal-width wave equation, Comput. Math. Appl., 54 (2007), 1147–1153.
[20]	K. R. Raslan, Collocation method using quartic B-spline for the equal width (EW) equation, Appl. Math. Comput., 168 (2005), 795–805. doi: 10.1016/j.camwa.2006.12.080. doi: 10.1016/j.camwa.2006.12.080
[21]	A. Dogan, Application of Galerkin's method to equal width wave equation, Appl. Math. Comput., 160 (2005), 65–76. doi: 10.1016/j.amc.2003.08.105. doi: 10.1016/j.amc.2003.08.105
[22]	S. I. Zaki, A least square finite element scheme for the EW equation, Comput. Method. Appl. M., 189 (2000), 587–594. doi: 10.1002/(SICI)1099-0887. doi: 10.1002/(SICI)1099-0887
[23]	S. I. Zaki, Solitary waves induced by the boundary forced EW equation, Comput. Method. Appl. M., 190 (2001), 4881–4887. doi: 10.1016/S0045-7825(99)00462-4. doi: 10.1016/S0045-7825(99)00462-4
[24]	K. L. Wang, A novel approach for fractal Burgers-BBM equation and its variational principle, Fractals, 29 (2021). doi: 10.1142/S0218348X21500596. doi: 10.1142/S0218348X21500596
[25]	J. H. He, A tutorial review on fractal spacetime and fractional calculus, Int. J. Theor. Phys., 53 (2014), 3698–3718. doi: 10.1007/s10773-014-2123-8. doi: 10.1007/s10773-014-2123-8
[26]	J. F. Lu, An analytical approach to fractional Bousinesq-Burgers equations, Therm. Sci., 24 (2020), 2581–2588. doi: 10.2298/TSCI2004581L. doi: 10.2298/TSCI2004581L
[27]	J. Lu, Y. Sun, Numerical approaches to time fractional Boussinesq-Burges equations, Fractals, (2021). doi: 10.1142/S0218348X21502443. doi: 10.1142/S0218348X21502443
[28]	A. M. Wazwaz, The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants, Commun. Nonlin. Sci., 11 (2006), 148–160. doi: 10.1016/j.cnsns.2004.07.001. doi: 10.1016/j.cnsns.2004.07.001
[29]	J. Lu, He's variational iteration method for the modified equal width equation. Chaos Solit. Fract., 39 (2009). doi: 10.1016/j.chaos.2007.06.104. doi: 10.1016/j.chaos.2007.06.104
[30]	A. Esen, S. Kutluay, Solitary wave solutions of the modified equal width wave equation, Commun. Nonlin. Sci., 13 (2008), 1538–1546. doi: 10.1016/j.cnsns.2006.09.018. doi: 10.1016/j.cnsns.2006.09.018
[31]	A. A. Esen, A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic Bsplines, Int. J. Comput. Math., 83 (2009), 449–459. doi: 10.1080/00207160600909918. doi: 10.1080/00207160600909918
[32]	W. Rui, S. Xie, B. He, Y. Long, Integral bifurcation method and its application for solving the modified equal width wave equation and its variants, Rostocker Math. Kolloq., 62 (2007), 87–106.
[33]	W. M. Taha, M. S. M. Noorani, Application of the $G/G^{'}$-expansion method for the generalized Fisher's equation and modified equal width equation, J. Assoc. Arab Univ. Basic, 15 (2014). doi: 10.1016/j.jaubas.2013.05.006. doi: 10.1016/j.jaubas.2013.05.006
[34]	A. Goswami, J. Singh, D. Kumar, Sushila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A, 524 (2019), 563–575. doi: 10.1016/j.physa.2019.04.058. doi: 10.1016/j.physa.2019.04.058
[35]	H. N. Hassan, An accurate numerical solution for the modified equal width wave equation using the Fourier pseudo-spectral method, J. Appl. Math. Phys., 4 (2016), 1054–1067. doi: 10.4236/jamp.2016.46110. doi: 10.4236/jamp.2016.46110
[36]	J. Biazar, Z. Ayati, H. Ebrahimi, New solitonary solutions for modified equal-width wave equations using exp-function method, Int. J. Nonlin. Dyn. Eng. Sci., 1 (2009), 109–114.
[37]	H. Wang, L. Chen, H. Wang, Exact travelling wave solutions of the modified equal width equation via the dynamical system method, Nonlin. Anal. Differ. Equ., 4 (2016), 9–15. doi: 10.12988/nade.2016.5824. doi: 10.12988/nade.2016.5824
[38]	J. H. He, Homotopy perturbation technique, Comput. Method. Appl. M., 178 (1999), 257–262. doi: 10.1016/S0045-7825(99)00018-3. doi: 10.1016/S0045-7825(99)00018-3
[39]	J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solit. Fract., 26 (2005), 695–700. doi: 10.1016/j.chaos.2005.03.006. doi: 10.1016/j.chaos.2005.03.006
[40]	J. H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solit. Fract., 26 (2005), 827–833. doi: 10.1016/j.chaos.2005.03.007. doi: 10.1016/j.chaos.2005.03.007
[41]	T. Allahviranloo, W. Pedrycz, Soft numerical computing in uncertain dynamic systems, Elsevier, Academic Press, 2020.
[42]	L. C. Barros, R. C. Bassanezi, W. A. Lodwick, A first course in fuzzy logic, fuzzy dynamical systems, and biomathematics, Stud. Fuzziness Soft Comput., 2017.
[43]	B. Bede, Mathematics of fuzzy sets and fuzzy logic, Stud. Fuzziness Soft Comput., 2013. doi: 10.1007/978-3-642-35221-8. doi: 10.1007/978-3-642-35221-8
[44]	V. F. Wasques, B. Laiate, F. Santo Pedro, E. Esmi, L. C. Barros, Interactive fuzzy fractional differential equation: Application on HIV dynamics, IPMU (2020), 198–211. doi: 10.1007/978-3-030-50153-2_15. doi: 10.1007/978-3-030-50153-2_15
[45]	L. A. Zadeh, Linguistic variables, approximate reasoning and disposition, Med. Inform., (1983), 173–186. doi: 10.3109/14639238309016081. doi: 10.3109/14639238309016081
[46]	C. V. Negoita, D. A. Ralescu, Applications of fuzzy sets to systems analysis, Wiley, New York, 1975.
[47]	S. S. L. Chang, L. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern., 2 (1972), 30–34. doi: 10.1109/TSMC.1972.5408553. doi: 10.1109/TSMC.1972.5408553
[48]	L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci., 8 (1975), 199–249. doi: 10.1016/0020-0255(75)90036-5. doi: 10.1016/0020-0255(75)90036-5
[49]	M. Hukuhara, Intégration des applications mesurables dont la valeur est un compact convex, Funkc. Ekvacioj, 10 (1967), 205–229.
[50]	R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlin. Anal., 72 (2010), 59–62. doi: 10.1016/j.na.2009.11.029. doi: 10.1016/j.na.2009.11.029
[51]	S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty, Nonlin. Anal., 74 (2011), 85–93. doi: 10.1016/j.na.2011.02.048. doi: 10.1016/j.na.2011.02.048
[52]	S. Arshad, V. Luplescu, Fractional differential equation with fuzzy initial conditon, Electron. J. Differ. Eq., 34 (2011), 1–8. doi: 10.1016/j.dam.2010.11.006. doi: 10.1016/j.dam.2010.11.006
[53]	T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equations with uncertainty, Soft. Comput. Fus. Found. Meth. Appl., 16 (2012), 297–302. doi: 10.1007/s00500-011-0743-y. doi: 10.1007/s00500-011-0743-y
[54]	S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlin. Sci Numer Simu., 17 (2012), 1372–1381. doi: 10.1016/j.cnsns.2011.07.005. doi: 10.1016/j.cnsns.2011.07.005
[55]	T. Allahviranloo, S. Abbasbandy, S. Salahshour, Fuzzy fractional differential equations with Nagumo and Krasnoselskii-Krein condition, Conference of the European Society for Fuzzy Logic, 2011. doi: 10.2991/eusflat.2011.39.
[56]	S. Bushnaq, Z. Ullah, A. Ullah, K. Shah, Solution of fuzzy singular integral equation with Abel's type kernel using a novel hybrid method, Adv. Differ. Equ., 2020 (2020). doi: 10.1186/s13662-020-02623-y. doi: 10.1186/s13662-020-02623-y
[57]	Z. Ullah, A. Ullah, K. Shah, D. Baleanu, Computation of semi-analytical solutions of fuzzy nonlinear integral equations, Adv. Differ. Equ., 2020 (2020), doi: 10.1186/s13662-020-02989-z. doi: 10.1186/s13662-020-02989-z
[58]	S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, P. Agarwal, On analytical aolutions of the fractional differential equation with uncertainty: application to the Basset problem, Entropy, 17 (2015), 885–902. doi: 10.3390/e17020885. doi: 10.3390/e17020885
[59]	S. Ahmad, A. Ullah, A. Akgül, T. Abdeljawad, Semi-analytical solutions of the 3rd order fuzzy dispersive partial differential equations under fractional operators, Alexandria Eng. J., 60 (2021), 5861–5878. doi: 10.1016/j.aej.2021.04.065. doi: 10.1016/j.aej.2021.04.065
[60]	K. Shah, Aly R. Seadawy, M. Arfan, Evaluation of one dimensional fuzzy fractional partial differential equations, Alexandria Eng. J., 59 (2020), 3347–3353. doi: 10.1016/j.aej.2020.05.003. doi: 10.1016/j.aej.2020.05.003
[61]	T. Allahviranloo, Fuzzy fractional differential operators and equation studies in fuzziness and soft computing, Berlin: Springer, 2021.
[62]	H. J. Zimmermann, Fuzzy set theory and its applications, Dordrecht: Kluwer Academic Publishers, 1991.
[63]	L. A. Zadeh, Fuzzy sets, Infor. Cont., 8 (1965), 338–353. doi: 10.1016/S0019-9958(65)90241-X. doi: 10.1016/S0019-9958(65)90241-X
[64]	T. Allahviranloo, M. B. Ahmadi, Fuzzy Lapalce transform, Soft comput., 14 (2010), 235–243. doi: 10.1007/s00500-008-0397-6. doi: 10.1007/s00500-008-0397-6
[65]	S. Maitama, W. Zhao, Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives, Comput. Appl. Math., 40 (2021). doi: 10.1007/s40314-021-01476-9. doi: 10.1007/s40314-021-01476-9
[66]	S. Maitama, W. Zhao. New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations, Int. J. Anal. Appl., 17 (2019), 167–190. doi: 10.28924/2291-8639-17-2019-167. doi: 10.28924/2291-8639-17-2019-167
[67]	A. Bokhari, D. Baleanu, R. Belgacema, Application of Shehu transform to Atangana-Baleanu derivatives, J. Math. Comp. Sci., 20 (2020), 101–107. doi: 10.22436/jmcs.020.02.03. doi: 10.22436/jmcs.020.02.03
[68]	A. Ghorbani, Beyond Adomian polynomials: He polynomials, Chaos Solit. Fract., 39 (2009), 1486–1492. doi: 10.1016/j.chaos.2007.06.034. doi: 10.1016/j.chaos.2007.06.034
[69]	A. Ghorbani, J. Saberi-Nadjafi, He's homotopy perturbation method for calculating adomian polynomials, Int. J. Nonlin. Sci. Numer. Simul., 8 (2007), 229–232. doi: 10.1515/IJNSNS.2007.8.2.229. doi: 10.1515/IJNSNS.2007.8.2.229

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)