Numerical analysis of fractional heat transfer and porous media equations within Caputo-Fabrizio operator

Yousef Jawarneh; Humaira Yasmin; M. Mossa Al-Sawalha; Rasool Shah; Asfandyar Khan; Yousef Jawarneh; Humaira Yasmin; M. Mossa Al-Sawalha; Rasool Shah; Asfandyar Khan

doi:10.3934/math.20231356

AIMS Mathematics

2023, Volume 8, Issue 11: 26543-26560. doi: 10.3934/math.20231356

Previous Article Next Article

Research article Special Issues

Numerical analysis of fractional heat transfer and porous media equations within Caputo-Fabrizio operator

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Basic Sciences, Preparatory Year Deanship, King Faisal University, Al-Ahsa 31982, Saudi Arabia
3.
Department of Computer Science and Mathematics, Lebanese American University, Beirut Lebanon
4.
Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan

Received: 07 August 2023 Revised: 01 September 2023 Accepted: 05 September 2023 Published: 18 September 2023
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering.
- Yang transform,
- fractional heat transfer equation,
- fractional porous media,
- Adomian decomposition method,
- Homotopy perturbation method
Citation: Yousef Jawarneh, Humaira Yasmin, M. Mossa Al-Sawalha, Rasool Shah, Asfandyar Khan. Numerical analysis of fractional heat transfer and porous media equations within Caputo-Fabrizio operator[J]. AIMS Mathematics, 2023, 8(11): 26543-26560. doi: 10.3934/math.20231356

Related Papers:

Abstract

This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering.

References

[1]	D. Kumar, J. Singh, K. Tanwar, D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler laws, Int. J. Heat Mass Tran., 138 (2019), 1222–1227. https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.094 doi: 10.1016/j.ijheatmasstransfer.2019.04.094
[2]	L. M. Yan, Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations, Therm. Sci., 17 (2013), 1409–1414. https://doi.org/10.2298/TSCI1305409Y doi: 10.2298/TSCI1305409Y
[3]	Q. Liu, H. Peng, Z. Wang, Convergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesis, J. Differ. Equations, 314 (2022), 251–286. https://doi.org/10.1016/j.jde.2022.01.021 doi: 10.1016/j.jde.2022.01.021
[4]	H. Jin, Z. Wang, L. Wu, Global dynamics of a three-species spatial food chain model, J. Differ. Equations, 333 (2022), 144–183. https://doi.org/10.1016/j.jde.2022.06.007 doi: 10.1016/j.jde.2022.06.007
[5]	H. Y. Jin, Z. A. Wang, Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Cont. Dyn-A, 40 (2020), 3509–3527. https://doi.org/10.3934/dcds.2020027 doi: 10.3934/dcds.2020027
[6]	M. Pan, L. Zheng, F. Liu, C. Liu, X. Chen, A spatial-fractional thermal transport model for nanofluid in porous media, Appl. Math. Model., 53 (2018), 622–634. https://doi.org/10.1016/j.apm.2017.08.026 doi: 10.1016/j.apm.2017.08.026
[7]	H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah, Investigating families of soliton solutions for the complex structured coupled fractional Biswas-Arshed model in birefringent fibers using a novel analytical technique, Fractal Fract., 7 (2023), 491. https://doi.org/10.3390/fractalfract7070491 doi: 10.3390/fractalfract7070491
[8]	F. Liu, Z. Sun, H. Bian, M. Ding, X. Meng, Identification and classification of the flow pattern of hydrogen-air-steam mixture gas under steam condensation, Int. J. Therm. Sci., 183 (2023), 107854. https://doi.org/10.1016/j.ijthermalsci.2022.107854 doi: 10.1016/j.ijthermalsci.2022.107854
[9]	B. Bai, D. Rao, T. Chang, Z. Guo, A nonlinear attachment-detachment model with adsorption hysteresis for suspension-colloidal transport in porous media, J. Hydrol., 578 (2019), 124080. https://doi.org/10.1016/j.jhydrol.2019.124080 doi: 10.1016/j.jhydrol.2019.124080
[10]	M. Pan, L. Zheng, F. Liu, X. Zhang, Modeling heat transport in nanofluids with stagnation point flow using fractional calculus, Appl. Math. Model., 40 (2016), 8974–8984. https://doi.org/10.1016/j.apm.2016.05.044 doi: 10.1016/j.apm.2016.05.044
[11]	M. K. Alaoui, K. Nonlaopon, A. M. Zidan, A. Khan, R. Shah, Analytical investigation of fractional-order Cahn-Hilliard and gardner equations using two novel techniques, Mathematics, 10 (2022), 1643. https://doi.org/10.3390/math10101643 doi: 10.3390/math10101643
[12]	P. Sunthrayuth, N. H. Aljahdaly, A. Ali, R. Shah, I. Mahariq, A. M. J. Tchalla, $\psi$-Haar wavelet operational matrix method for fractional relaxation-oscillation equations containing $\psi$-Caputo fractional derivative, J. Funct. Space., 2021 (2021), 7117064. https://doi.org/10.1155/2021/7117064 doi: 10.1155/2021/7117064
[13]	C. Guo, J. Hu, J. Hao, S. Celikovsky, X. Hu, Fixed-time safe tracking control of uncertain high-order nonlinear pure-feedback systems via unified transformation functions, Kybernetika, 59 (2023), 342–364. https://doi.org/10.14736/kyb-2023-3-0342 doi: 10.14736/kyb-2023-3-0342
[14]	C. Guo, J. Hu, Y. Wu, S. Celikovsky, Non-singular fixed-time tracking control of uncertain nonlinear pure-feedback systems with practical state constraints, IEEE T. Circuits-I, 70 (2023), 3746–3758. https://doi.org/10.1109/TCSI.2023.3291700 doi: 10.1109/TCSI.2023.3291700
[15]	Q. Meng, Q. Ma, Y. Shi, Adaptive fixed-time stabilization for a class of uncertain nonlinear systems, IEEE T. Automat. Contr., 2023, 1–8. https://doi.org/10.1109/TAC.2023.3244151
[16]	A. S. Alshehry, H. Yasmin, F. Ghani, R. Shah, K. Nonlaopon, Comparative analysis of Advection- Dispersion equations with Atangana-Baleanu fractional derivative, Symmetry, 15 (2023), 819. https://doi.org/10.3390/sym15040819 doi: 10.3390/sym15040819
[17]	M. A. Shah, H. Yasmin, F. Ghani, S. Abdullah, I. Khan, R. Shah, Fuzzy fractional gardner and Cahn-Hilliard equations with the Atangana-Baleanu operator, Front. Phys., 11 (2023), 1169548. https://doi.org/10.3389/fphy.2023.1169548 doi: 10.3389/fphy.2023.1169548
[18]	H. Yasmin, A. S. Alshehry, A. Khan, R. Shah, K. Nonlaopon, Numerical analysis of the fractional-order belousov zhabotinsky system, Symmetry, 15 (2023), 834. https://doi.org/10.3390/sym15040834 doi: 10.3390/sym15040834
[19]	N. Iqbal, A. M. Albalahi, M. S. Abdo, W. W. Mohammed, Analytical analysis of fractional-order Newell-Whitehead-Segel equation: A modified homotopy perturbation transform method, J. Funct. Space., 2022 (2022), 3298472. https://doi.org/10.1155/2022/3298472 doi: 10.1155/2022/3298472
[20]	H. K. Jassim, M. G. Mohammed, Natural homotopy perturbation method for solving nonlinear fractional gas dynamics equations, Int. J. Nonlinear Appl., 12 (2021), 812–820. https://doi.org/10.22075/IJNAA.2021.4936 doi: 10.22075/IJNAA.2021.4936
[21]	Y. J. Yang, S. Q. Wang, Fractional residual method coupled with Adomian decomposition method for solving local fractional differential equations, Therm. Sci., 26 (2022), 2667–2675. https://doi.org/10.2298/TSCI2203667Y doi: 10.2298/TSCI2203667Y
[22]	M. Alesemi, N. Iqbal, A. A. Hamoud, The analysis of fractional-order proportional delay physical models via a novel transform, Complexity, 2022 (2022), 2431533. https://doi.org/10.1155/2022/2431533 doi: 10.1155/2022/2431533
[23]	F. Chen, Q. Q. Liu, Adomian decomposition method combined with padé approximation and laplace transform for solving a model of HIV infection of CD4$^+$ T cells, Discrete Dyn. Nat. Soc., 2015 (2015), 584787. https://doi.org/10.1155/2015/584787 doi: 10.1155/2015/584787
[24]	D. Chen, Q. Wang, Y. Li, H. Zhou, Y. Fan, A general linear free energy relationship for predicting partition coefficients of neutral organic compounds, Chemosphere, 247 (2020), 125869. https://doi.org/10.1016/j.chemosphere.2020.125869 doi: 10.1016/j.chemosphere.2020.125869
[25]	S. Ahmad, A. Ullah, A. Akgül, M. De la Sen, A novel homotopy perturbation method with applications to nonlinear fractional order KdV and burger equation with exponential-decay kernel, J. Funct. Space., 2021 (2021), 8770488. https://doi.org/10.1155/2021/8770488 doi: 10.1155/2021/8770488
[26]	S. Pamuk, Solution of the porous media equation by Adomian's decomposition method, Phys. Lett. A, 344 (2005), 184–188. https://doi.org/10.1016/j.physleta.2005.06.068 doi: 10.1016/j.physleta.2005.06.068
[27]	A. D. Polyanin, V. F. Zaitsev, Handbook of nonlinear partial differential equations: Exact solutions, methods, and problems, New York: CRC, 2003. https://doi.org/10.1201/9780203489659
[28]	D. D. Ganji, A. Sadighi, Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations, J. Comput. Appl. Math., 207 (2007), 24–34. https://doi.org/10.1016/j.cam.2006.07.030 doi: 10.1016/j.cam.2006.07.030

Reader Comments

Your name:*

Email:*
© 2023 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)