Evaluation of regularized long-wave equation via Caputo and Caputo-Fabrizio fractional derivatives

Naveed Iqbal; Saleh Alshammari; Thongchai Botmart; Naveed Iqbal; Saleh Alshammari; Thongchai Botmart

doi:10.3934/math.20221118

AIMS Mathematics

2022, Volume 7, Issue 11: 20401-20419. doi: 10.3934/math.20221118

Previous Article Next Article

Research article

Evaluation of regularized long-wave equation via Caputo and Caputo-Fabrizio fractional derivatives

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 28 June 2022 Revised: 25 August 2022 Accepted: 05 September 2022 Published: 19 September 2022
MSC : 34A34, 35A20, 35A22, 44A10, 33B15

The analytical solution of fractional-order regularized long waves in the context of various operators is presented in this study as a framework for the homotopy perturbation transform technique. To investigate regularized long wave equations, we first establish the Yang transform of the fractional Caputo and Caputo-Fabrizio operators. The fractional order regularized long wave equation is solved using the Yang transform as well. The accuracy of the proposed operators are verified using numerical problems, and the resulting solutions are shown in the figures. The solutions demonstrate how the suggested approach is accurate and suitable for analyzing nonlinear physical and engineering challenges.
- Caputo and Caputo-Fabrizio derivatives,
- Yang transformation,
- approximate-analytical solution,
- regularized long wave equations
Citation: Naveed Iqbal, Saleh Alshammari, Thongchai Botmart. Evaluation of regularized long-wave equation via Caputo and Caputo-Fabrizio fractional derivatives[J]. AIMS Mathematics, 2022, 7(11): 20401-20419. doi: 10.3934/math.20221118

Related Papers:

Abstract

The analytical solution of fractional-order regularized long waves in the context of various operators is presented in this study as a framework for the homotopy perturbation transform technique. To investigate regularized long wave equations, we first establish the Yang transform of the fractional Caputo and Caputo-Fabrizio operators. The fractional order regularized long wave equation is solved using the Yang transform as well. The accuracy of the proposed operators are verified using numerical problems, and the resulting solutions are shown in the figures. The solutions demonstrate how the suggested approach is accurate and suitable for analyzing nonlinear physical and engineering challenges.

References

[1]	S. Pandit, Local radial basis functions and scale-3 Haar wavelets operational matrices based numerical algorithms for generalized regularized long wave model, Wave Motion, 109 (2022), 102846. https://doi.org/10.1016/j.wavemoti.2021.102846 doi: 10.1016/j.wavemoti.2021.102846
[2]	A. Saad Alshehry, M. Imran, A. Khan, W. Weera, Fractional view analysis of Kuramoto-Sivashinsky equations with Non-Singular kernel operators, Symmetry, 14 (2022), 1463. https://doi.org/10.3390/sym14071463 doi: 10.3390/sym14071463
[3]	R. Mittal, S. Pandit, Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems, Eng. Comput., 35 (2018), 1907–1931. https://doi.org/10.1108/ec-09-2017-0347 doi: 10.1108/ec-09-2017-0347
[4]	S. Kumar, R. Jiwari, R. Mittal, Radial basis functions based meshfree schemes for the simulation of non-linear extended Fisher-Kolmogorov model, Wave Motion, 109 (2022), 102863. https://doi.org/10.1016/j.wavemoti.2021.102863 doi: 10.1016/j.wavemoti.2021.102863
[5]	M. Caputo, Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969. (In Italian)
[6]	Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 21 (2008), 194–199. https://doi.org/10.1016/j.aml.2007.02.022 doi: 10.1016/j.aml.2007.02.022
[7]	O. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal., 15 (2012), 700–711. https://doi.org/10.2478/s13540-012-0047-7 doi: 10.2478/s13540-012-0047-7
[8]	K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[9]	Y. Rossikhin, M. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results, Appl. Mech. Rev., 63 (2009). https://doi.org/10.1115/1.4000563 doi: 10.1115/1.4000563
[10]	A. Akdemir, A. Karaoglan, M. Ragusa, E. Set, Fractional integral inequalities via Atangana-Baleanu operators for convex and concave functions, J. Funct. Space., 2021 (2021), 1–10. https://doi.org/10.1155/2021/1055434 doi: 10.1155/2021/1055434
[11]	M. Beddani, B. Hedia, An existence results for a fractional differential equation with $\phi$-fractional derivative, Filomat, 36 (2022), 753–762. https://doi.org/10.2298/fil2203753b doi: 10.2298/fil2203753b
[12]	E. Ilhan, Analysis of the spread of Hookworm infection with Caputo-Fabrizio fractional derivative, Turk. J. Sci., 7 (2022), 43–52. https://doi.org/10.28919/jmcs/5995 doi: 10.28919/jmcs/5995
[13]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[14]	Y. Zhang, H. Sun, H. Stowell, M. Zayernouri, S. Hansen, A review of applications of fractional calculus in Earth system dynamics, Chaos, Soliton. Fract., 102 (2017), 29–46. https://doi.org/10.1016/j.chaos.2017.03.051 doi: 10.1016/j.chaos.2017.03.051
[15]	Y. Rossikhin, Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids, Appl. Mech. Rev., 63 (2009). https://doi.org/10.1115/1.4000246 doi: 10.1115/1.4000246
[16]	A. Carpinteri, F. Mainardi, Eds., 2014. Fractals and fractional calculus in continuum mechanics, (Vol. 378). Springer. https://doi.org/
[17]	A. S. Alshehry, M. Imran, W. Weera, Fractional-View analysis of Fokker-Planck equations by ZZ transform with Mittag-Leffler kernel, Symmetry, 14 (2022), 1513. https://doi.org/10.3390/sym14081513 doi: 10.3390/sym14081513
[18]	C. Lederman, J. Roquejoffre, N. Wolanski, Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames, CR Math., 334 (2002), 569–574. https://doi.org/10.1016/s1631-073x(02)02299-9 doi: 10.1016/s1631-073x(02)02299-9
[19]	V. Kulish, J. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng., 124 (2002), 803–806. https://doi.org/10.1115/1.1478062 doi: 10.1115/1.1478062
[20]	F. Meral, T. Royston, R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlinear Sci., 15 (2010), 939–945. https://doi.org/10.1016/j.cnsns.2009.05.004 doi: 10.1016/j.cnsns.2009.05.004
[21]	R. Bagley, P. Torvik, Fractional calculus in the transient analysis of viscoelastically damped structures, AIAA J., 23 (1985), 918–925. https://doi.org/10.2514/3.9007 doi: 10.2514/3.9007
[22]	K. Nonlaopon, A. Alsharif, A. Zidan, A. Khan, Y. Hamed, Numerical investigation of fractional-order Swift-Hohenberg Equations via a novel transform, Symmetry, 13 (2021), 1263. https://doi.org/10.3390/sym13071263 doi: 10.3390/sym13071263
[23]	E. Elsayed, R. Shah, K. Nonlaopon, The analysis of the fractional-order Navier-Stokes equations by a novel approach, J. Funct. Space., 2022 (2022), 1–18. https://doi.org/10.1155/2022/8979447 doi: 10.1155/2022/8979447
[24]	R. Shah, H. Khan, D. Baleanu, P. Kumam, M. Arif, A novel method for the analytical solution of fractional Zakharov-Kuznetsov equations, Adv. Differ. Equ-Ny., 2019 (2019). https://doi.org/10.1186/s13662-019-2441-5 doi: 10.1186/s13662-019-2441-5
[25]	N. Iqbal, A. Akgul, R. Shah, A. Bariq, M. Mossa Al-Sawalha, A. Ali, On solutions of fractional-order gas dynamics equation by effective techniques, J. Funct. Space., 2022 (2022), 1–14. https://doi.org/10.1155/2022/3341754 doi: 10.1155/2022/3341754
[26]	Y. Qin, A. Khan, I. Ali, M. Al Qurashi, H. Khan, D. Baleanu, An efficient analytical approach for the solution of certain fractional-order dynamical systems, Energies, 13 (2020), 2725. https://doi.org/10.3390/en13112725 doi: 10.3390/en13112725
[27]	M. Alaoui, R. Fayyaz, A. Khan, M. Abdo, Analytical investigation of Noyes-Field model for time-fractional Belousov-Zhabotinsky reaction, Complexity, 2021 (2021), 1–21. https://doi.org/10.1155/2021/3248376 doi: 10.1155/2021/3248376
[28]	M. Areshi, A. Khan, K. Nonlaopon, Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform, AIMS Math., 7 (2022), 6936–6958. https://doi.org/10.3934/math.2022385 doi: 10.3934/math.2022385
[29]	T. Botmart, R. Agarwal, M. Naeem, A. Khan, R. Shah, On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators, AIMS Math., 7 (2022), 12483–12513. https://doi.org/10.3934/math.2022693 doi: 10.3934/math.2022693
[30]	M. Alqhtani, K. Saad, R. Shah, W. Weera, W. Hamanah, Analysis of the Fractional-order local Poisson Equation in fractal porous media, Symmetry, 14 (2022), 1323. https://doi.org/10.3390/sym14071323 doi: 10.3390/sym14071323
[31]	M. Mohamed, M. Yousif, A. Hamza, Solving nonlinear fractional partial differential equations using the Elzaki transform method and the homotopy perturbation method, Abstr. Appl. Anal., 2022 (2022), 1–9. https://doi.org/10.1155/2022/4743234 doi: 10.1155/2022/4743234
[32]	M. Alaoui, R. Fayyaz, A. Khan, R. Shah, M. Abdo, Analytical investigation of Noyes-Field model for time-fractional Belousov-Zhabotinsky reaction, Complexity, 2021 (2021), 1–21. https://doi.org/10.1155/2021/3248376 doi: 10.1155/2021/3248376
[33]	M. Yavuz, Characterizations of two different fractional operators without singular kernel, Math. Model. Nat. Pheno., 14 (2019), 302. https://doi.org/10.1051/mmnp/2018070 doi: 10.1051/mmnp/2018070
[34]	P. Sunthrayuth, F. Ali, A. Alderremy, R. Shah, S. Aly, Y. Hamed, et al., The numerical investigation of fractional-order Zakharov-Kuznetsov equations, Complexity, 2021 (2021), 1–13. https://doi.org/10.1155/2021/4570605 doi: 10.1155/2021/4570605
[35]	M. Naeem, O. Azhar, A. Zidan, K. Nonlaopon, R. Shah, Numerical analysis of fractional-order parabolic equations via Elzaki Transform, J. Funct. Space., 2021 (2021), 1–10. https://doi.org/10.1155/2021/3484482 doi: 10.1155/2021/3484482
[36]	R. Ali, K. Pan, The solution of the absolute value equations using two generalized accelerated overrelaxation methods, Asian-Eur. J. Math., 15 (2021). https://doi.org/10.1142/s1793557122501546 doi: 10.1142/s1793557122501546
[37]	R. Ali, A. Ali, S. Iqbal, Iterative methods for solving absolute value equations, J. Math. Comput. Sci., 26 (2021), 322–329. https://doi.org/10.22436/jmcs.026.04.01 doi: 10.22436/jmcs.026.04.01
[38]	T. B. Benjamin, J. L. Bona, J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. A, 272 (1972) 47–78. https://doi.org/10.1098/rsta.1972.0032 doi: 10.1098/rsta.1972.0032
[39]	M. Yavuz, T. Abdeljawad, Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel, Adv. Differ. Equ-Ny., 2020 (2020). https://doi.org/10.1186/s13662-020-02828-1 doi: 10.1186/s13662-020-02828-1
[40]	T. Achouri, K. Omrani, Numerical solutions for the damped generalized regularized long-wave equation with a variable coefficient by Adomian decomposition method, Commun. Nonlinear Sci., 14 (2009), 2025–2033. https://doi.org/10.1016/j.cnsns.2008.07.011 doi: 10.1016/j.cnsns.2008.07.011
[41]	A. Goswami, J. Singh, D. Kumar, S. Gupta, Sushila, An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma, J. Ocean Eng. Sci., 4 (2019), 85–99. https://doi.org/10.1016/j.joes.2019.01.003 doi: 10.1016/j.joes.2019.01.003
[42]	Y. Khan, R. Taghipour, M. Falahian, A. Nikkar, A new approach to modified regularized long wave equation, Neural Comput. Appl., 23 (2012), 1335–1341. https://doi.org/10.1007/s00521-012-1077-0 doi: 10.1007/s00521-012-1077-0
[43]	X. Yang, A new integral transform method for solving steady heat-transfer problem, Therm. Sci., 20 (2016), 639–642. https://doi.org/10.2298/tsci16s3639y doi: 10.2298/tsci16s3639y
[44]	M. Caputo, M. Fabrizio, On the singular kernels for fractional derivatives, some applications to partial differential equations, Progr. Fract. Differ. Appl., 7 (2021), 1–4. https://doi.org/10.18576/pfda/070201 doi: 10.18576/pfda/070201

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)