Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation

M. Mossa Al-Sawalha; Khalil Hadi Hakami; Mohammad Alqudah; Qasem M. Tawhari; Hussain Gissy; M. Mossa Al-Sawalha; Khalil Hadi Hakami; Mohammad Alqudah; Qasem M. Tawhari; Hussain Gissy

doi:10.3934/math.2025324

AIMS Mathematics

2025, Volume 10, Issue 3: 7099-7126. doi: 10.3934/math.2025324

Previous Article Next Article

Research article

Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia
3.
Department of Basic Sciences, School of Electrical Engineering & Information Technology, German Jordanian University, Amman 11180, Jordan

Received: 03 February 2025 Revised: 14 March 2025 Accepted: 21 March 2025 Published: 27 March 2025
MSC : 34A08, 35A15, 35A23

In this paper, we investigate the fractional damped Burgers' equation using two efficient analytical approaches: the Laplace least squares residual power series method and the Laplace least squares variational iteration method. These techniques integrate the Laplace transform with the least squares residual power series and least squares variational iteration methods, providing highly accurate solutions for nonlinear fractional differential equations. The fractional derivatives are considered in the sense of the Caputo operator, allowing for a more realistic description of physical phenomena with memory effects. Comparative studies with exact and numerical solutions demonstrate the reliability and accuracy of the results. The proposed methodologies provide a powerful framework for solving nonlinear fractional models in fluid dynamics, shock wave theory, and applied sciences.
Citation: M. Mossa Al-Sawalha, Khalil Hadi Hakami, Mohammad Alqudah, Qasem M. Tawhari, Hussain Gissy. Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation[J]. AIMS Mathematics, 2025, 10(3): 7099-7126. doi: 10.3934/math.2025324

Related Papers:

Abstract

In this paper, we investigate the fractional damped Burgers' equation using two efficient analytical approaches: the Laplace least squares residual power series method and the Laplace least squares variational iteration method. These techniques integrate the Laplace transform with the least squares residual power series and least squares variational iteration methods, providing highly accurate solutions for nonlinear fractional differential equations. The fractional derivatives are considered in the sense of the Caputo operator, allowing for a more realistic description of physical phenomena with memory effects. Comparative studies with exact and numerical solutions demonstrate the reliability and accuracy of the results. The proposed methodologies provide a powerful framework for solving nonlinear fractional models in fluid dynamics, shock wave theory, and applied sciences.

References

[1]	C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141–159. https://doi.org/10.1016/j.cnsns.2017.04.001 doi: 10.1016/j.cnsns.2017.04.001
[2]	M. Sinan, K. Shah, P. Kumam, I. Mahariq, K. J. Ansari, Z. Ahmad, et al., Fractional order mathematical modeling of typhoid fever disease, Results Phys., 32 (2022), 105044. https://doi.org/10.1016/j.rinp.2021.105044 doi: 10.1016/j.rinp.2021.105044
[3]	S. L. Wu, M. Al-Khaleel, Convergence analysis of the Neumann-Neumann waveform relaxation method for time-fractional RC circuits, Simul. Model. Pract. Theory, 64 (2016), 43–56. https://doi.org/10.1016/j.simpat.2016.01.002 doi: 10.1016/j.simpat.2016.01.002
[4]	Y. Sun, J. Lu, M. Zhu, A. A. Alsolami, Numerical analysis of fractional nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass, J. Low Freq. Noise, Vibrat. Active Control, 44 (2024), 178–189. https://doi.org/10.1177/14613484241285502 doi: 10.1177/14613484241285502
[5]	S. L. Wu, M. Al-Khaleel, Parameter optimization in waveform relaxation for fractional-order $ RC $ circuits, IEEE Trans. Circuits Syst. I: Regular Papers, 64 (2017), 1781–1790. https://doi.org/10.1109/TCSI.2017.2682119 doi: 10.1109/TCSI.2017.2682119
[6]	J. Wang, X. Jiang, X. Yang, H. Zhang, A compact difference scheme for mixed-type time-fractional black-Scholes equation in European option pricing, Math. Meth. Appl. Sci., 48 (2025), 6818–6829. https://doi.org/10.1002/mma.10717 doi: 10.1002/mma.10717
[7]	T. Liu, H. Zhang, X. Yang, The ADI compact difference scheme for three-dimensional integro-partial differential equation with three weakly singular kernels, J. Appl. Math. Comput., 2025. https://doi.org/10.1007/s12190-025-02386-3
[8]	K. Liu, Z. He, H. Zhang, X. Yang, A Crank-Nicolson ADI compact difference scheme for the three-dimensional nonlocal evolution problem with a weakly singular kernel, Comp. Appl. Math., 44 (2025), 164. https://doi.org/10.1007/s40314-025-03125-x doi: 10.1007/s40314-025-03125-x
[9]	Z. Chen, H. Zhang, H. Chen, ADI compact difference scheme for the two-dimensional integro-differential equation with two fractional Riemann-Liouville integral kernels, Fractal Fract., 8 (2024), 707. https://doi.org/10.3390/fractalfract8120707 doi: 10.3390/fractalfract8120707
[10]	X. Yang, W. Wang, Z. Zhou, H. Zhang, An efficient compact difference method for the fourth-order nonlocal subdiffusion problem, Taiwanese J. Math., 29 (2025), 35–66. https://doi.org/10.11650/tjm/240906 doi: 10.11650/tjm/240906
[11]	D. Baleanu, Z. B. Guvenc, J. T. Machado, New trends in nanotechnology and fractional calculus applications, New York: Springer, 2010.
[12]	N. H. Sweilam, M. M. Abou Hasan, D. Baleanu, New studies for general fractional financial models of awareness and trial advertising decisions, Chaos Solit. Fract., 104 (2017), 772–784. https://doi.org/10.1016/j.chaos.2017.09.013 doi: 10.1016/j.chaos.2017.09.013
[13]	D. Baleanu, G. C. Wu, S. D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Solit. Fract., 102 (2017), 99–105. https://doi.org/10.1016/j.chaos.2017.02.007 doi: 10.1016/j.chaos.2017.02.007
[14]	P. Veeresha, D. G. Prakasha, H. Mehmet Baskonus, New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives, Chaos: Interdiscipl. J. Nonlinear Sci., 29 (2019), 013119. https://doi.org/10.1063/1.5074099 doi: 10.1063/1.5074099
[15]	S. Noor, H. A. Alyousef, A. Shafee, R. Shah, S. A. El-Tantawy, A novel analytical technique for analyzing the (3+ 1)-dimensional fractional calogero-bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena, Phys, Scr., 99 (2024), 065257. https://doi.org/10.1088/1402-4896/ad49d9 doi: 10.1088/1402-4896/ad49d9
[16]	E. F. D. Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers's equation, Math. Model. Anal., 21 (2016), 188–198. https://doi.org/10.3846/13926292.2016.1145607 doi: 10.3846/13926292.2016.1145607
[17]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[18]	X. Yang, Z. Zhang, Superconvergence analysis of a robust orthogonal Gauss collocation method for 2D fourth-order subdiffusion equations, J. Sci. Comput., 100 (2024), 62. https://doi.org/10.1007/s10915-024-02616-z doi: 10.1007/s10915-024-02616-z
[19]	X. Shen, X. Yang, H. Zhang, The high-order ADI difference method and extrapolation method for solving the two-dimensional nonlinear parabolic evolution equations, Mathematics, 12 (2024), 3469. https://doi.org/10.3390/math12223469 doi: 10.3390/math12223469
[20]	Y. Shi, X. Yang, Z. Zhang, Construction of a new time-space two-grid method and its solution for the generalized Burgers' equation, Appl. Math. Lett., 158 (2024), 109244. https://doi.org/10.1016/j.aml.2024.109244 doi: 10.1016/j.aml.2024.109244
[21]	X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7 doi: 10.1007/s10915-024-02511-7
[22]	X. Yang, Z. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
[23]	Q. Wang, Homotopy perturbation method for fractional KdV-Burgers's equation, Chaos Solit. Fract., 35 (2008), 843850. https://doi.org/10.1016/j.chaos.2006.05.074 doi: 10.1016/j.chaos.2006.05.074
[24]	S. Anil Sezer, A. Yildırım, S. Tauseef Mohyud-Din, He's homotopy perturbation method for solving the fractional KdV-Burgers's-Kuramoto equation, Int. J. Numer. Meth. Heat Fluid Flow, 21 (2011), 448–458.
[25]	G. C. Wu, D. Baleanu, Variational iteration method for the Burgers's' flow with fractional derivatives-new Lagrange multipliers, Appl. Math. Model., 37 (2013), 6183–6190. https://doi.org/10.1016/j.apm.2012.12.018 doi: 10.1016/j.apm.2012.12.018
[26]	M. Inc, The approximate and exact solutions of the space-and time-fractional Burgers's equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476–484. https://doi.org/10.1016/j.jmaa.2008.04.007 doi: 10.1016/j.jmaa.2008.04.007
[27]	A. Esen, N. M. Yagmurlu, O. Tasbozan, Approximate analytical solution to time-fractional damped Burgers' and Cahn-Allen equations, Appl. Math. Inf. Sci., 7 (2013), 1951–1956. http://doi.org/10.12785/amis/070533 doi: 10.12785/amis/070533
[28]	H. Jafari, V. Daftardar-Gejji, Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition, Appl. Math. Comput., 180 (2006), 488–497. https://doi.org/10.1016/j.amc.2005.12.031 doi: 10.1016/j.amc.2005.12.031
[29]	J. Lu, Y. Sun, Numerical approaches to time fractional boussinesq-burgers equations, Fractals, 29 (2021), 2150244. https://doi.org/10.1142/S0218348X21502443 doi: 10.1142/S0218348X21502443
[30]	J. Singh, D. Kumar, M. A. Qurashi, Baleanu, Analysis of a new fractional model for damped Bergers' equation, Open Phys., 15 (2017), 35–41. https://doi.org/10.1515/phys-2017-0005 doi: 10.1515/phys-2017-0005
[31]	B. M. Vaganan, M. S. Kumaran, Kummer function solutions of damped Burgers's equations with time-dependent viscosity by exact linearization, Nonlinear Anal.: Real World Appl., 9 (2008), 2222–2233. https://doi.org/10.1016/j.nonrwa.2007.08.001 doi: 10.1016/j.nonrwa.2007.08.001
[32]	W. Malfliet, Approximate solution of the damped Burgers's equation, J. Phys. A: Math. Gen., 26 (1993), L723. https://doi.org/10.1088/0305-4470/26/16/003 doi: 10.1088/0305-4470/26/16/003
[33]	F. Yilmaz, B. Karasozen, Solving optimal control problems for the unsteady Burgers's equation in COMSOL Multiphysics, J. Comput. Appl. Math., 235 (2011), 4839–4850. https://doi.org/10.1016/j.cam.2011.01.002 doi: 10.1016/j.cam.2011.01.002
[34]	P. Rosenau, J. M. Hyman, Compactons: solitons with finite wavelength, Phys. Rev. Lett., 70 (1993), 564. https://doi.org/10.1103/PhysRevLett.70.564 doi: 10.1103/PhysRevLett.70.564
[35]	B. Mihaila, A. Cardenas, F. Cooper, A. Saxena, Stability and dynamical properties of Rosenau-Hyman compactons using Pade approximants, Phys. Rev. E-Stat. Nonlinear Soft Matter Phys., 81 (2010), 056708. https://doi.org/10.1103/PhysRevE.81.056708 doi: 10.1103/PhysRevE.81.056708
[36]	F. Rus, F. R. Villatoro, Self-similar radiation from numerical Rosenau-Hyman compactons, J. Comput. Phys., 227 (2007), 440–454. https://doi.org/10.1016/j.jcp.2007.07.024 doi: 10.1016/j.jcp.2007.07.024
[37]	F. Rus, F. R. Villatoro, Numerical methods based on modified equations for nonlinear evolution equations with compactons, Appl. Math. Comput., 204 (2008), 416–422. https://doi.org/10.1016/j.amc.2008.06.056 doi: 10.1016/j.amc.2008.06.056
[38]	O. S. Iyiola, G. O. Ojo, O. Mmaduabuchi, The fractional Rosenau-Hyman model and its approximate solution, Alex. Eng. J., 55 (2016), 1655–1659. https://doi.org/10.1016/j.aej.2016.02.014 doi: 10.1016/j.aej.2016.02.014
[39]	J. Singh, D. Kumar, R. Swroop, S. Kumar, An efficient computational approach for time-fractional Rosenau-Hyman equation, Neural Comput. Applic., 30 (2018), 3063–3070. https://doi.org/10.1007/s00521-017-2909-8 doi: 10.1007/s00521-017-2909-8
[40]	R. Yulita Molliq, M. S. M. Noorani, Solving the fractional Rosenau-Hyman equation via variational iteration method and homotopy perturbation method, Int. J. Differ. Equ., 2012 (2012), 472030. https://doi.org/10.1155/2012/472030 doi: 10.1155/2012/472030
[41]	M. Senol, O. Tasbozan, A. Kurt, Comparison of two reliable methods to solve fractional Rosenau-Hyman equation, Math. Meth. Appl. Sci., 44 (2021), 7904–7914. https://doi.org/10.1002/mma.5497 doi: 10.1002/mma.5497
[42]	M. Cinar, A. Secer, M. Bayram, An application of Genocchi wavelets for solving the fractional Rosenau-Hyman equation, Alex. Eng. J., 60 (2021), 5331–5340. https://doi.org/10.1016/j.aej.2021.04.037 doi: 10.1016/j.aej.2021.04.037
[43]	S. O. Ajibola, A. S. Oke, W. N. Mutuku, LHAM approach to fractional order Rosenau-Hyman and Burgers' equations, Asian Res. J. Math., 16 (2020), 1–14. https://doi.org/10.9734/ARJOM/2020/v16i630192 doi: 10.9734/ARJOM/2020/v16i630192
[44]	M. Alaroud, Application of Laplace residual power series method for approximate solutions of fractional IVP's, Alex. Eng. J., 61 (2022), 1585–1595. https://doi.org/10.1016/j.aej.2021.06.065 doi: 10.1016/j.aej.2021.06.065
[45]	M. Alquran, M. Ali, M. Alsukhour, I. Jaradat, Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics, Results Phys., 19 (2020), 103667. https://doi.org/10.1016/j.rinp.2020.103667 doi: 10.1016/j.rinp.2020.103667
[46]	H. Aljarrah, M. Alaroud, A. Ishak, M. Darus, Approximate solution of nonlinear time-fractional PDEs by Laplace residual power series method, Mathematics, 10 (2022), 1980. https://doi.org/10.3390/math10121980 doi: 10.3390/math10121980
[47]	A. Shafee, Y. Alkhezi, R. Shah, Efficient solution of fractional system partial differential equations using Laplace residual power series method, Fractal Fract., 7 (2023), 429. https://doi.org/10.3390/fractalfract7060429 doi: 10.3390/fractalfract7060429
[48]	M. N. Oqielat, T. Eriqat, O. Ogilat, A. El-Ajou, S. E. Alhazmi, S. Al-Omari, Laplace-residual power series method for solving time-fractional reaction-diffusion model, Fractal Fract., 7 (2023), 309. https://doi.org/10.3390/fractalfract7040309 doi: 10.3390/fractalfract7040309
[49]	N. Anjum, J. H. He, Laplace transform: making the variational iteration method easier, Appl. Math. Lett., 92 (2019), 134–138. https://doi.org/10.1016/j.aml.2019.01.016 doi: 10.1016/j.aml.2019.01.016
[50]	N. Bildik, A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 65–70. https://doi.org/10.1515/IJNSNS.2006.7.1.65 doi: 10.1515/IJNSNS.2006.7.1.65
[51]	N. A. Shah, I. Dassios, E. R. El-Zahar, J. D. Chung, S. Taherifar, The variational iteration transform method for solving the time-fractional Fornberg-Whitham equation and comparison with decomposition transform method, Mathematics, 9 (2021), 141. https://doi.org/10.3390/math9020141 doi: 10.3390/math9020141
[52]	S. M. Kenneth, B. Ross, An introduction to the fractional calculus and fractional differential equations, Hoboken: Wiley, 1993.
[53]	R. Almeida, D. F. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett., 22 (2009), 1816–1820. https://doi.org/10.1016/j.aml.2009.07.002 doi: 10.1016/j.aml.2009.07.002
[54]	R. Almeida, D. F. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1490–1500. https://doi.org/10.1016/j.cnsns.2010.07.016 doi: 10.1016/j.cnsns.2010.07.016
[55]	O. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A: Math. Theor., 40 (2007), 6287. https://doi.org/10.1088/1751-8113/40/24/003 doi: 10.1088/1751-8113/40/24/003
[56]	R. Kumar, R. Koundal, Generalized least square homotopy perturbations for system of fractional partial differential equations, preprint paper, 2018. https://doi.org/10.48550/arXiv.1805.06650
[57]	A. El-Ajou, Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach, Eur. Phys. J. Plus, 136 (2021), 229. https://doi.org/10.1140/epjp/s13360-020-01061-9 doi: 10.1140/epjp/s13360-020-01061-9
[58]	Z. Korpinar, M. Inc, E. Hınçal, D. Baleanu, Residual power series algorithm for fractional cancer tumor models, Alex. Eng. J., 59 (2020), 1405–1412. https://doi.org/10.1016/j.aej.2020.03.044 doi: 10.1016/j.aej.2020.03.044
[59]	J. Zhang, Z. Wei, L. Li, C. Zhou, Least-squares residual power series method for the time-fractional differential equations, Complexity, 2019 (2019), 6159024. https://doi.org/10.1155/2019/6159024 doi: 10.1155/2019/6159024

Reader Comments

Your name:*

Email:*
© 2025 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)