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Open Access Research Article

A semi-analytical approach for solving time-fractional nonlinear KdV equations using fractional multiple transformed iterative method

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* Corresponding author · click or hover a name for details

pp. 1031–1041Vol. 28Issue 3-BApril 2025DOI: 10.47974/JIM-2013XML
Received:
15 Oct 2024
Published Online:
08 May 2025
Article type:
Research Article
Language:
EN
Article no.:
JIM-2013
Pages:
1031–1041

Abstract

This paper introduces a novel successful hybrid method of solving different kinds for time-fractional nonlinear partial differential equation (TFr-NLPDEs). A technique combine Shehu, & Sumudu integral transform of fractional order derivative, that described in the Caputo sense, with the variational iteration method (VIM). Time-fractional equations such as the Kortwe-De Veries (KdV) and the Kortweg-De Veries Burgers (KdV-B) equations are solved by the suggested method with results occurred as a convergent series. Some numerical examples confirm the effectiveness of the suggested method under appropriate initial conditions. The resulting series solutions converge to the exact solutions of the equations. The results are compared with the exact solutions and to some existing approaches in the literature to show the efficiency, accuracy, and simplicity of the method.

Keywords

Subject Classifications

47B4944A30

References

[1] S. S. Ray, A. Atangana, S. C. O. Noutchie, M. Kurulay, N. Bildik, and A. Kilicman, “Fractional calculus and its applications in applied mathematics and other sciences,” Math. Probl. Eng., vol. 2014, Art. ID 849395 (2014). [Online]. Available: http://dx.doi.org/10.1155/2014/849395
[2] T. U. Khan, M. A. Khan, and Y. M. Chu, “A new generalized Hilfer-type fractional derivative with applications to space-time diffusion equation,” Results Phys., vol. 22, p. 103953 (2021). [Online]. Available: https://doi.org/10.1016/j.rinp.2021.103953
[3] M. F. El Amin, “Derivation of fractional-derivative models of multiphase fluid flows in porous media,” J. King Saud Univ. Sci., vol. 33, p. 101346 (2021).
[4] B. Ross, “A brief history and exposition of the fundamental theory of fractional calculus,” in Fractional Calculus, Springer Lecture Notes in Mathematics, vol. 57, pp. 1–36 (1975).
[5] I. Podlubny, Fractional Differential Equations, New York, NY, USA: Academic Press (2009).
[6] M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel,” Prog. Fract. Differ. Appl., vol. 1, pp. 73–85 (2015).
[7] V. Lakshmikantham and A. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Anal. Theory Methods Appl., vol. 69, pp. 2677–2682 (2008).
[8] R. S. Saha and A. K. Gupta, “An approach with Haar wavelet collocation method for numerical simulations of modified KdV and modified Burgers equations,” Comput. Model. Eng. Sci., vol. 103, no. 5, pp. 315–341 (2014).
[9] D. Kaya, S. Gulbahar, A. Yokus, and M. Gulbahar, “Solutions of the fractional combined KdV, mKdV equation with collocation method using radial basis function and their geometrical obstructions,” Adv. Differ. Equ., vol. 2018, Art. no. 1–16.
[10] N. Parumasur, R. A. Adetona, and P. Singh, “Efficient solution of Burgers’, modified Burgers’ and KdV–Burgers’ equations using B-spline approximation functions,” Mathematics, vol. 11, no. 8, p. 1847 (2023). [Online]. Available: https://doi.org/10.3390/math11081847
[11] C. Li and Y. Wang, “Numerical algorithm based on Adomian decomposition for fractional differential equations,” Comput. Math. Appl., vol. 57, pp. 1672–1681 (2009).
[12] G. C. Wu and E. Lee, “Fractional variational iteration method and its application,” Phys. Lett. A, vol. 374, pp. 2506–2509 (2010).
[13] A. A. Hemeda, “Modified homotopy perturbation method for solving fractional differential equations,” J. Appl. Math., vol. 2014, Art. ID 594245, 9 pages.
[14] B. Ghazanfari and F. Veisi, “Homotopy analysis method for the fractional nonlinear equations,” J. King Saud Univ. Sci., vol. 23, pp. 389–393 (2011).
[15] H. Honggang and Z. Yamin, “Conformable double Laplace–Sumudu transform decomposition method of fractional partial differential equations,” Complexity, vol. 2022, Art. ID 7602254, 8 pages.
[16] F. K. Mohanned and A. Al-Fayadh, “Solving Fredholm integro-differential equation of fractional order by using Sawi homotopy perturbation method,” J. Phys.: Conf. Ser., vol. 2322, p. 012056 (2022).
[17] A. A. Shams, S. Rania, Q. Ahmad, and M. E. Tarig, “Modified conformable double Laplace–Sumudu approach with applications,” Heliyon, May 2023, Art. ID e15891. [Online]. Available: https://doi.org/10.1016/j.heliyon.2023.e15891
[18] G. O. Ojo and N. I. Mahmudov, “Aboodh transform iterative method for spatial diffusion of a biological population with fractional-order,” Mathematics, vol. 9, p. 155 (2021).
[19] M. G. Al-Safi, R. M. Fawzi, and W. R. Abd Al-Hussein, “Numerical solutions for the nonlinear PDEs of fractional orders by using new double integral transform with variational iteration method,” Baghdad Sci. J., vol. 20, no. 3suppl., pp. 1087–1098 (2023).
[20] H. Anac, “A local fractional Elzaki transform decomposition method for the nonlinear system of local fractional partial differential equations,” Fractal Fract., vol. 6, p. 167 (2022).
[21] B. Sikora, “Remarks on the Caputo fractional derivative,” Faculty of Applied Mathematics, Silesian Univ. Technol., vol. 5, pp. 76–84 (2023).
[22] H. Eltayeb and A. Kilicman, “A note on the Sumudu transforms and differential equations,” Appl. Math. Sci., vol. 4, no. 22, pp. 1089–1098 (2010).
[23] S. Maitama and W. Zhao, “New integral transform: Shehu transform—a generalization of Sumudu and Laplace transform for solving differential equations,” Int. J. Anal. Appl., vol. 17, no. 2, pp. 167–190 (2019).
[24] V. G. Gupta, B. Sharma, and A. Kilicman, “A note on fractional Sumudu transform,” J. Appl. Math., vol. 2010, Art. ID 154189, 9 pages.
[25] T. G. Thange and A. R. Gade, “Fractional Shehu transform and its applications,” South East Asian J. Math. Math. Sci., vol. 17, no. 2, pp. 1–14 (2021).
[26] A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler function and its properties,” J. Math. Anal. Appl., vol. 336, pp. 797–811 (2007).
[27] A. Tassadiq and A. Alruban, “On modification of the Gamma function by using Mittag-Leffler function,” J. Math., vol. 2021, Art. ID 9991762, 12 pages.
[28] Q. D. Katatbeh and F. B. M. Belgacem, “Applications of the Sumudu transform to fractional differential equations,” Nonlinear Stud., vol. 18, no. 1, pp. 99–112 (2011).
[29] G. C. Wu, “A fractional variational iteration method for solving fractional nonlinear differential equations,” Comput. Math. Appl., vol. 61, pp. 2186–219 (2011).
[30] H. M. Ali, “An efficient approximate-analytical method to solve time-fractional KdV and KdVB equations,” Inf. Sci. Lett., vol. 9, no. 3, pp. 189–198 (2020).
[31] M. M. Abbood, H. H. Ebrahim, A. Al-Fayadh, and A. J. Obaid, “Measure defined on Γ–algebra and some of their generalizations,” J. Interdiscip. Math., vol. 26, no. 5, pp. 821–827 (2023), doi: 10.47974/JIM-1502.
[32] A. S. Mohammed, S. H. Asaad, and I. S. Ahmed, “New properties of strongly regular left almost semi-group,” J. Discret. Math. Sci. Cryptogr., vol. 25, no. 2, pp. 615–622 (2022), doi: 10.1080/09720529.2021.1982491.
[33] I. S. Ahmed, H. H. Ebrahim, and A. Al-Fayadh, “Γ–algebra of sets and some of its properties,” Iraqi J. Sci., vol. 63, no. 11, pp. 4918–4927 (2022), doi: 10.24996/ijs.2022.63.11.28.

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