This paper presents a comprehensive study of the (2+1) time-fractional nonlinear generalized biological population model (TFNBPM) using the $ J $-transform combined with the optimal homotopy analysis method, a robust semi-analytical technique. The primary focus is to derive analytical solutions for the model and provide a thorough investigation of the convergence properties of these solutions. The proposed method allows for flexibility and accuracy in handling nonlinear fractional differential equations (NFDEs), demonstrating its efficacy through a series of detailed analyses. To validate the results, we present a set of 2D and 3D graphical representations of the solutions, illustrating the dynamic behavior of the biological population over time and space. These visualizations provide insightful perspectives on the population dynamics governed by the model. Additionally, a comparative study is conducted, where our results are juxtaposed with those obtained using other established techniques from the literature. The comparisons underscore the advantages of optimal homotopy analysis $ J $-transform method (optimal HA$ J $-TM), highlighting its consistency and superior convergence in solving complex fractional models.
Citation: Khalid K. Ali, Mohamed S. Mohamed, M. Maneea. Optimal homotopy analysis method for (2+1) time-fractional nonlinear biological population model using $ {{J}} $-transform[J]. AIMS Mathematics, 2024, 9(11): 32757-32781. doi: 10.3934/math.20241567
This paper presents a comprehensive study of the (2+1) time-fractional nonlinear generalized biological population model (TFNBPM) using the $ J $-transform combined with the optimal homotopy analysis method, a robust semi-analytical technique. The primary focus is to derive analytical solutions for the model and provide a thorough investigation of the convergence properties of these solutions. The proposed method allows for flexibility and accuracy in handling nonlinear fractional differential equations (NFDEs), demonstrating its efficacy through a series of detailed analyses. To validate the results, we present a set of 2D and 3D graphical representations of the solutions, illustrating the dynamic behavior of the biological population over time and space. These visualizations provide insightful perspectives on the population dynamics governed by the model. Additionally, a comparative study is conducted, where our results are juxtaposed with those obtained using other established techniques from the literature. The comparisons underscore the advantages of optimal homotopy analysis $ J $-transform method (optimal HA$ J $-TM), highlighting its consistency and superior convergence in solving complex fractional models.
| [1] |
F. Shakeri, M. Dehghan, Numerical solution of a biological population model using He's variational iteration method, Comput. Math. Appl., 54 (2007), 1197–1209. https://doi.org/10.1016/j.camwa.2006.12.076 doi: 10.1016/j.camwa.2006.12.076
|
| [2] |
M. Shakeel, M. Iqbal, S. Mohyud-Din, Closed form solutions for nonlinear biological population model, J. Biol. Syst., 26 (2018), 207–223. https://doi.org/10.1142/S0218339018500109 doi: 10.1142/S0218339018500109
|
| [3] |
M. Uddin, H. Ali, M. Taufiq, On the approximation of a nonlinear biological population model using localized radial basis function method, Math. Comput. Appl., 24 (2019), 54. https://doi.org/10.3390/mca24020054 doi: 10.3390/mca24020054
|
| [4] |
S. Mohyud-Din, A. Ali, B. Bin-Mohsin, On biological population model of fractional order, Int. J. Biomath., 9 (2016), 1650070. https://doi.org/10.1142/S1793524516500704 doi: 10.1142/S1793524516500704
|
| [5] |
Z. Fan, K. Ali, M. Maneea, M. Inc, S. Yao, Solution of time fractional Fitzhugh-Nagumo equation using semi analytical techniques, Results Phys., 51 (2023), 106679. https://doi.org/10.1016/j.rinp.2023.106679 doi: 10.1016/j.rinp.2023.106679
|
| [6] |
K. Ali, M. Maneea, Optical solitons using optimal homotopy analysis method for time-fractional (1+1)-dimensional coupled nonlinear Schrodinger equations, Optik, 283 (2023), 170907. https://doi.org/10.1016/j.ijleo.2023.170907 doi: 10.1016/j.ijleo.2023.170907
|
| [7] |
K. Ali, M. Maneea, M. Mohamed, Solving nonlinear fractional models in superconductivity using the q-Homotopy analysis transform method, J. Math., 2023 (2023), 6647375. https://doi.org/10.1155/2023/6647375 doi: 10.1155/2023/6647375
|
| [8] |
S. Alsallami, M. Maneea, E. Khalil, S. Abdel-Khalek, K. Ali, Insights into time fractional dynamics in the Belousov-Zhabotinsky system through singular and non-singular kernels, Sci. Rep., 13 (2023), 22347. https://doi.org/10.1038/s41598-023-49577-1 doi: 10.1038/s41598-023-49577-1
|
| [9] |
M. Du, Z. Wang, H. Hu, Measuring memory with the order of fractional derivative, Sci. Rep., 3 (2013), 3431. https://doi.org/10.1038/srep03431 doi: 10.1038/srep03431
|
| [10] |
A. Ivanov, Fractional activation functions in feedforward artificial neural networks, Proceedings of 20th International Symposium on Electrical Apparatus and Technologies (SIELA), 2018, 1–4. https://doi.org/10.1109/SIELA.2018.8447139 doi: 10.1109/SIELA.2018.8447139
|
| [11] |
K. Ali, A. Wazwaz, M. Maneea, Efficient solutions for fractional Tsunami shallow-water mathematical model: a comparative study via semi analytical techniques, Chaos Soliton. Fract., 178 (2024), 114347. https://doi.org/10.1016/j.chaos.2023.114347 doi: 10.1016/j.chaos.2023.114347
|
| [12] |
K. Raslan, K. Ali, Numerical study of MHD-duct flow using the two-dimensional finite difference method, Appl. Math. Inf. Sci., 14 (2020), 693–697. https://doi.org/10.18576/amis/140417 doi: 10.18576/amis/140417
|
| [13] |
A. Ali, M. Asjad, M. Usman, M. Inc, Numerical solutions of a heat transfer for fractional maxwell fluid flow with water based clay nanoparticles; a finite difference approach, Fractal Fract., 5 (2021), 242. https://doi.org/10.3390/fractalfract5040242 doi: 10.3390/fractalfract5040242
|
| [14] |
Z. Zhang, G. Li, Lie symmetry analysis and exact solutions of the time-fractional biological population model, Physica A, 540 (2020), 123134. https://doi.org/10.1016/j.physa.2019.123134 doi: 10.1016/j.physa.2019.123134
|
| [15] |
Z. Zhang, Z. Lin, Local symmetry structure and potential symmetries of time-fractional partial differential equations, Stud. Appl. Math., 147 (2021), 363–389. https://doi.org/10.1111/sapm.12374 doi: 10.1111/sapm.12374
|
| [16] |
H. Zhu, Z. Zhang, J. Zheng, The time-fractional (2+1)-dimensional Hirota-Satsuma-Ito equations: Lie symmetries, power series solutions and conservation laws, Commun. Nonlinear Sci., 115 (2022), 106724. https://doi.org/10.1016/j.cnsns.2022.106724 doi: 10.1016/j.cnsns.2022.106724
|
| [17] |
M. Akram, T. Ihsan, T. Allahviranloo, Solving Pythagorean fuzzy fractional differential equations using Laplace transform, Granul. Comput., 8 (2023), 551–575. https://doi.org/10.1007/s41066-022-00344-z doi: 10.1007/s41066-022-00344-z
|
| [18] |
S. Rashid, K. Kubra, K. Abualnaja, Fractional view of heat-like equations via the Elzaki transform in the settings of the Mittag-Leffler function, Math. Method. Appl. Sci., 46 (2023), 11420–11441. https://doi.org/10.1002/mma.7793 doi: 10.1002/mma.7793
|
| [19] |
K. Ali, M. Mohamed, M. Maneea, Exploring optical soliton solutions of the time fractional q-deformed Sinh-Gordon equation using a semi-analytic method, AIMS Mathematics, 8 (2023), 27947–27968. https://doi.org/10.3934/math.20231429 doi: 10.3934/math.20231429
|
| [20] |
D. Albogami, D. Maturi, H. Alshehri, Adomian decomposition method for solving fractional Time-Klein-Gordon equations using Maple, Applied Mathematics, 14 (2023), 411–418. https://doi.org/10.4236/am.2023.146024 doi: 10.4236/am.2023.146024
|
| [21] |
S. Maitama, W. Zhao, Beyond Sumudu transform and natural transform: J-transform properties and applications, J. Appl. Anal. Comput., 10 (2020), 1223–1241. https://doi.org/10.11948/20180258 doi: 10.11948/20180258
|
| [22] |
B. Singh, A. Kumar, M. Gupta, Efficient new approximations for space-time fractional multi-dimensional telegraph equation, Int. J. Appl. Comput. Math., 8 (2022), 218. https://doi.org/10.1007/s40819-022-01343-z doi: 10.1007/s40819-022-01343-z
|
| [23] |
B. Singh, A. Kumar, S. Rai, D. Prakasha, Study of nonlinear time-fractional hyperbolic-like equations with variable coefficients via semi-analytical technique: differential J-transform method, Int. J. Mod. Phys. B, 38 (2024), 2450001. https://doi.org/10.1142/S0217979224500012 doi: 10.1142/S0217979224500012
|
| [24] |
V. Srivastava, S. Kumar, M. Awasthi, B. Singh, Two-dimensional time fractional-order biological population model and its analytical solution, Egyptian Journal of Basic and Applied Sciences, 1 (2014), 71–76. https://doi.org/10.1016/j.ejbas.2014.03.001 doi: 10.1016/j.ejbas.2014.03.001
|
| [25] |
O. Acana, M. Al Qurashi, D. Baleanu, New exact solution of generalized biological population model, J. Nonlinear Sci. Appl., 10 (2017), 3916–3929. https://doi.org/10.22436/jnsa.010.07.44 doi: 10.22436/jnsa.010.07.44
|
| [26] |
P. Veeresha, D. Prakasha, An efficient technique for two-dimensional fractional order biological population model, Int. J. Model. Simul. Sci., 2020 (2020), 2050005. https://doi.org/10.1142/S1793962320500051 doi: 10.1142/S1793962320500051
|
| [27] |
D. Ziane, M. Hamdi Cherif, D. Baleanu, K. Belghaba, Non-differentiable solution of nonlinear biological population model on cantor sets, Fractal Fract., 4 (2020), 5. https://doi.org/10.3390/fractalfract4010005 doi: 10.3390/fractalfract4010005
|
| [28] |
M. Alaroud, A. Alomari, N. Tahat, A. Ishak, Analytical computational scheme for multivariate nonlinear time-fractional generalized biological population model, Fractal Fract., 7 (2023), 176. https://doi.org/10.3390/fractalfract7020176 doi: 10.3390/fractalfract7020176
|
| [29] | I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 1999. |
| [30] |
R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
|
| [31] |
A. Elsaid, M. Latif, M. Maneea, Similarity solutions for solving Riesz fractional partial differential equations, Progr. Fract. Differ. Appl., 2 (2016), 293–298. https://doi.org/10.18576/pfda/020407 doi: 10.18576/pfda/020407
|
| [32] | S. Liao, Beyond perturbation: introduction to the homotopy analysis method, New York: Chapman and Hall/CRC, 2003. https://doi.org/10.1201/9780203491164 |
| [33] |
V. Marinca, N. Herisanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, Int. Commun. Heat Mass, 35 (2008), 710–715. https://doi.org/10.1016/j.icheatmasstransfer.2008.02.010 doi: 10.1016/j.icheatmasstransfer.2008.02.010
|
| [34] |
S. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci., 15 (2010), 2003–2016. https://doi.org/10.1016/j.cnsns.2009.09.002 doi: 10.1016/j.cnsns.2009.09.002
|
| [35] |
A. El-Ajou, O. Arqub, Z. Al Zhour, S. Momani, New results on fractional power series: theories and applications, Entropy, 15 (2013), 5305–5323. https://doi.org/10.3390/e15125305 doi: 10.3390/e15125305
|
| [36] |
Y. Liu, Z. Li, Y. Zhang, Homotopy perturbation method to fractional biological population equation, Fractional Differential Calculus, 1 (2011), 117–124. https://doi.org/10.7153/fdc-01-07 doi: 10.7153/fdc-01-07
|
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Khalid K. Ali, Mohamed S. Mohamed, M. Maneea. Optimal homotopy analysis method for (2+1) time-fractional nonlinear biological population model using $ {{J}} $-transform[J]. AIMS Mathematics, 2024, 9(11): 32757-32781. doi: 10.3934/math.20241567
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