Solutions of Fuzzy Fractional Abel Differential Equations using Residual Power Series Method

P. Prakash; N. Nithyadevi

doi:https://doi.org/10.14445/22315373/IJMTT-V69I3P506

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 69 | Issue 3 | Year 2023 | Article Id. IJMTT-V69I3P506 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I3P506

Solutions of Fuzzy Fractional Abel Differential Equations using Residual Power Series Method

P. Prakash, N. Nithyadevi

Received	Revised	Accepted	Published
06 Feb 2023	07 Mar 2023	20 Mar 2023	31 Mar 2023

Citation :

P. Prakash, N. Nithyadevi, "Solutions of Fuzzy Fractional Abel Differential Equations using Residual Power Series Method," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 3, pp. 39-49, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I3P506

Abstract

This paper describes the computed approach of such fuzzy fractional Abel differential equation (FFADE) according to a specific make using the extended power series (PS) formula in case of nonlinear. The methodology is based on applying representational processing to create a fractional power series solution in the form of a residual power series (RPS) with the smallest number of computations possible. The suggested approach is consistent with the initial problem difficulty, and the results are promising. The effective computational examples offered to ensure the method and explain the numerical expressions of the analytic solution to its potency, flexibility, and efficiency towards answering similar fractional equations. To demonstrate answer, visual and numerical data were provided and statistically evaluated.

Keywords

Abel initial value problems, Fuzzy, Fractional differential equations, Strongly generalized differentiability, Residual power series.

References

[1] Rania Saadeh et al., “Application of Fractional Residual Power Series Algorithm to Solve Newell–Whitehead–Segel Equation of Fractional Order,” Symmetry, vol. 11, no. 12, 2019.
[CrossRef] [Goggle Scholar] [Publisher link]
[2] Iryna Komashynska et al., “Analytical Approximate Solutions of Systems of Multipantograph Delay Differential Equations Using Residual Power-series Method,” Australian Journal of Basic and Applied Sciences, vol. 8, no.10, pp. 664-675, 2014.
[CrossRef] [Google Scholar]
[3] Ahmad El-Ajou et al., “New Results on Fractional Power Series: Theories and Applications,” Entropy, vol. 15, no. 12, pp. 5305–5323, 2013.
[CrossRef] [Google Scholar] [Publisher link]
[4] Iryana Komashynska et al., “Approximate Analytical Solution by Residual Power Series Method for System of Fredholm Integral Equations,” Applied Mathematics Information Sciences, vol. 10, no. 3, pp. 975–985 2016.
[Google Scholar]
[5] Asad Freihet et al., “Construction of Fractional Power Series Solutions to Fractional Stiff System using Residual Functions Algorithm,” Advances in Difference Equations, 2019.
[CrossRef] [Google Scholar] [Publisher link]
[6] A.A. Alderremy et al., “A Fuzzy Fractional Model of Coronavirus (COVID-19) and Its Study with Legendre Spectral Method,” Results in Physics, vol. 21, 2021.
[CrossRef] [Google Scholar] [Publisher link]
[7] F. Karimi et al., “Solving Riccati Fuzzy Differential Equations,” New Mathematics and Natural Computation, vol. 17, no. 1, pp. 29-43, 2021.
[CrossRef] [Google Scholar] [Publisher link]
[8] S. Ahmad et al., “Fuzzy Fractional-order Model of the Novel Coronavirus,” Advances in Difference Equations, 2020.
[CrossRef] [Google Scholar] [Publisher link]
[9] M. Z. Ahmad, M. K. Hasan, and S. Abbasbandy, “Solving Fuzzy Fractional Differential Equations Using Zadeh’s Extension Principle,” Hindawi Publishing Corporation The Scientific World Journal, 2013.
[CrossRef] [Google Scholar] [Publisher link]
[10] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, "Theory and Applications of Fractional Differential Equations,” Elsevier, 2006.
[Google Scholar] [Publisher link]
[11] Heman Dutta, Ahmat Ocak Akdemir, and Abdon Atangana, Fractional Order Analysis: Theory, Methods and Applications, John Wiley and Sons Ltd, Hoboken, United States, 2020.
[Google Scholar] [Publisher link]
[12] Dumitru Baleanu, Alireza K. Golmankhaneh, and Ali K. Golmankhaneh, “Solving of the Fractional Non-linear and Linear Schrodinger Equations by Homotopy Perturbation Method,” Romanian Journal of Physics, vol. 54, pp. 823–832, 2009.
[CrossRef]
[13] Dumitru Baleanu, Octavian G. Mustafa, and Ravi P. Agarwal, “On the Solution Set for a Class of Sequential Fractional Differential Equations,” Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 38, 2010.
[CrossRef] [Google Scholar] [Publisher link]
[14] Mohammad Al-Smadi et al., “Analytical Approximations of Partial Differential Equations of Fractional Order with Multistep Approach,” Journal of Computational and Theoretical Nanoscience, vol. 13, no. 11, 2016.
[CrossRef] [Google Scholar] [Publisher link]
[15] Antoni Ferragut, Johanna D. García-Saldaña, and Claudia Valls, “Phase Portraits of Abel Quadratic Differential Systems of Second Kind with Symmetries,” Dynamical Systems, vol. 34, no. 2, pp. 301-333, 2019.
[CrossRef] [Google Scholar] [Publisher link]
[16] K.I. Al-Dosary, N.K. Al-Jubouri, and H.K. Abdullah, “On the Solution of Abel Differential Equation by Adomian Decomposition Method,” Applied Mathematical Sciences, vol. 2, no. 43, pp. 2105 - 2118, 2008.
[Google Scholar]
[17] Barnabas Bede, and Sorin G. Gal, “Generalizations of the Differentiability of Fuzzy-number-valued Functions with Applications to Fuzzy Differential Equations,” Fuzzy Sets and Systems, vol. 151, pp. 581-599, 2005.
[CrossRef] [Google Scholar] [Publisher link]
[18] Y. Chalco-Cano, and H. Roman-Flores, “On New Solutions of Fuzzy Differential Equations,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 112-119, 2008.
[CrossRef] [Google Scholar]
[19] Asad Freihet et al., “Construction of Fractional Power Series Solutions to Fractional Stiff System using Residual Functions Algorithm,” Advances in Difference Equations, 2019.
[CrossRef] [Google Scholar] [Publisher link]
[20] Roy Goetschel, and William Voxman, “Elementary Fuzzy Calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31-43, 1986.
[CrossRef] [Publisher link]
[21] M. P. Markakis, “Closed-form Solutions of Certain Abel Equations of the First Kind,” Applied Mathematics Letters, vol. 22, no. 9, pp. 1401–1405, 2009.[CrossRef] [Google Scholar] [Publisher link]
[22] S. Salahshour, T. Allahviranloo, and S. Abbasbandy, “Solving Fuzzy Fractional Differential Equations by Fuzzy Laplace Transform,” Communications in Nonlinear science and Numberical simulation, vol. 17, no. 3, pp. 1372-1381, 2012.
[CrossRef] [Google Scholar] [Publisher link]
[23] Seppo Seikkala, “On the Fuzzy Initial Value Problem,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 319-330, 1987.
[CrossRef] [Google Scholar] [Publisher link]
[24] Fritz Schwarz, “Symmetry Analysis of Abel’s Equation,” Studies in Applied Mathematics, vol. 100, no. 3, pp. 269-294, 1998.
[CrossRef] [Google Scholar] [Publisher link]
[25] Mohammed Shqair et al., “Adaptation of Conformable Residual Power Series Scheme in Solving Nonlinear Fractional Quantum Mechanics Problems,” Applied Sciences, vol. 10, no. 3, 2020.
[CrossRef] [Google Scholar] [Publisher link]
[26] S. G. Samko, A. A. Kilbas, and O. I. Marichev, “Fractional Integrals and Derivatives,” Theory and Applications, Gordon and Breach Science Publishers, India, 1993.