Wavelet$ - $ALA fusion scheme for (2+1)D time$ - $fractional mobile/immobile models

Fengying Zhou; Jiakun Zhang; Fengying Zhou; Jiakun Zhang

doi:10.3934/math.2026567

AIMS Mathematics

2026, Volume 11, Issue 5: 13777-13804. doi: 10.3934/math.2026567

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Wavelet$ - $ALA fusion scheme for (2+1)D time$ - $fractional mobile/immobile models

Fengying Zhou ^,,
Jiakun Zhang

School of Mathematical Sciences, Jiangxi Science and Technology Normal University, Nanchang, Jiangxi, 330038, China

Received: 03 February 2026 Revised: 02 April 2026 Accepted: 08 April 2026 Published: 16 May 2026
MSC : 41A10, 65T60

A numerical scheme is proposed, merging 3D eighth$ - $kind fractional Chebyshev wavelets with the artificial lemming algorithm (ALA) to address (2+1)D time$ - $fractional mobile/immobile models. Initially, we construct the 3D eighth$ - $kind fractional Chebyshev wavelets and investigate their properties, including approximation capability and convergence analysis. Furthermore, by establishing a multivariate Caputo fractional Taylor formula, error bounds for the presented wavelets approximation are derived. Then, we provide a detailed algorithmic procedure for solving the (2+1)D time$ - $fractional mobile/immobile models considered in this paper, incorporating the ALA. Finally, the performance of the proposed method is examined through selected numerical examples. Comparative studies with existing findings provide quantitative evidence of the method's accuracy and the enhanced computational efficiency resulting from ALA incorporation. Moreover, our method remains applicable to 3D space$ - $fractional differential equations, and we validate this claim through a 3D space$ - $fractional Poisson equation.
Citation: Fengying Zhou, Jiakun Zhang. Wavelet$ - $ALA fusion scheme for (2+1)D time$ - $fractional mobile/immobile models[J]. AIMS Mathematics, 2026, 11(5): 13777-13804. doi: 10.3934/math.2026567

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Abstract

A numerical scheme is proposed, merging 3D eighth$ - $kind fractional Chebyshev wavelets with the artificial lemming algorithm (ALA) to address (2+1)D time$ - $fractional mobile/immobile models. Initially, we construct the 3D eighth$ - $kind fractional Chebyshev wavelets and investigate their properties, including approximation capability and convergence analysis. Furthermore, by establishing a multivariate Caputo fractional Taylor formula, error bounds for the presented wavelets approximation are derived. Then, we provide a detailed algorithmic procedure for solving the (2+1)D time$ - $fractional mobile/immobile models considered in this paper, incorporating the ALA. Finally, the performance of the proposed method is examined through selected numerical examples. Comparative studies with existing findings provide quantitative evidence of the method's accuracy and the enhanced computational efficiency resulting from ALA incorporation. Moreover, our method remains applicable to 3D space$ - $fractional differential equations, and we validate this claim through a 3D space$ - $fractional Poisson equation.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[2]	Y. C. Keluskar, N. G. Singhaniya, V. A. Vyawahare, C. S. Jage, P. Patil, G. Espinosa-Paredes, Solution of nonlinear fractional-order models of nuclear reactor with parallel computing: Implementation on GPU platform, Ann. Nucl. Energy, 195 (2024), 110134. https://doi.org/10.1016/j.anucene.2023.110134 doi: 10.1016/j.anucene.2023.110134
[3]	D. Amilo, K. Sadri, B. Kaymakamzade, E. Hincal, A mathematical model with fractional-order dynamics for the combined treatment of metastatic colorectal cancer, Commun. Nonlinear Sci. Numer. Simul., 130 (2024), 107756. https://doi.org/10.1016/j.cnsns.2023.107756 doi: 10.1016/j.cnsns.2023.107756
[4]	W. J. Li, Y. N. Pang, Application of Adomian decomposition method to nonlinear systems, Adv. Difference Equ., 2020 (2020), 67. https://doi.org/10.1186/s13662-020-2529-y doi: 10.1186/s13662-020-2529-y
[5]	X. Zhang, H. Wang, Z. Luo, L. Wei, A high-accuracy compact finite difference scheme for time-fractional diffusion equations, Revista de la Unión Matemática Argentina, 24 (2025), 1952–1968. https://doi.org/10.33044/revuma.4665 doi: 10.33044/revuma.4665
[6]	J. H. He, M. L. Jiao, K. A. Gepreel, Y. Khan, Homotopy perturbation method for strongly nonlinear oscillators, Math. Comput. Simul., 204 (2023), 243–258. https://doi.org/10.1016/j.matcom.2022.08.005 doi: 10.1016/j.matcom.2022.08.005
[7]	M. Adel, M. M. Khader, S. Algelany, High-dimensional Chaotic Lorenz system: Numerical treatment using Changhee polynomials of the appell type, Fractal Fract., 7 (2023), 398. https://doi.org/10.3390/fractalfract7050398 doi: 10.3390/fractalfract7050398
[8]	Y. Chen, X. Zhang, L. Wei, High precision interpolation method for mixed 2D time-space fractional convection-diffusion equation, Int. J. Dynam. Control, 13 (2025), 333. https://doi.org/10.1007/s40435-025-01842-z doi: 10.1007/s40435-025-01842-z
[9]	J. C. Mason, D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002.
[10]	E. H. Doha, W. M. Abd-Elhameed, M. A. Bassuony, New algorithms for solving high even-order differential equations using third and fourth Chebyshev–Galerkin methods, J. Comput. Phys., 236 (2013), 563–579. https://doi.org/10.1016/j.jcp.2012.11.009 doi: 10.1016/j.jcp.2012.11.009
[11]	W. M. Abd-Elhameed, Y. H. Youssri, A. K. Amin, A. G. Atta, Eighth-kind Chebyshev polynomials collocation algorithm for the nonlinear time-fractional generalized Kawahara equation, Fractal Fract., 7 (2023), 652. https://doi.org/10.3390/fractalfract7090652 doi: 10.3390/fractalfract7090652
[12]	M. Ahmadinia, H. Afshariarjmand, M. Salehi, Numerical solution of multi-dimensional It{\^o} Volterra integral equations by the second kind Chebyshev wavelets and parallel computing process, Appl. Math. Comput., 450 (2023), 127988. https://doi.org/10.1016/j.amc.2023.127988 doi: 10.1016/j.amc.2023.127988
[13]	J. Shahni, R. Singh, A novel collocation approach using Chebyshev wavelets for solving fourth-order Emden-Fowler type equations, J. Comput. Sci., 77 (2024), 102243. https://doi.org/10.1016/j.jocs.2024.102243 doi: 10.1016/j.jocs.2024.102243
[14]	F. Mohammadi, C. Cattani, A generalized fractional-order Legendre wavelet Tau method for solving fractional differential equations, J. Comput. Appl. Math., 339 (2018), 306–316. https://doi.org/10.1016/j.cam.2017.09.031 doi: 10.1016/j.cam.2017.09.031
[15]	S. Behera, S. S. Ray, An efficient numerical method based on Euler wavelets for solving fractional order pantograph Volterra delay-integro-differential equations, J. Comput. Appl. Math., 406 (2022), 113825. https://doi.org/10.1016/j.cam.2021.113825 doi: 10.1016/j.cam.2021.113825
[16]	B. Hazarika, G. Methi, R. Aggarwal, Application of generalized Haar wavelet technique on simultaneous delay differential equations, J. Comput. Appl. Math., 449 (2024), 115977. https://doi.org/10.1016/j.cam.2024.115977 doi: 10.1016/j.cam.2024.115977
[17]	H. L. Qiao, A. J. Cheng, A fast finite difference method for 2D time fractional mobile/immobile equation with weakly singular solution, Fractal Fract., 9 (2025), 204. https://doi.org/10.3390/fractalfract9040204 doi: 10.3390/fractalfract9040204
[18]	Z. H. Sheng, Y. Liu, Y. H. Li, A double-parameter shifted convolution quadrature formula and its application to fractional mobile/immobile transport equations, Appl. Math. Lett., 162 (2025), 109388. https://doi.org/10.1016/j.aml.2024.109388 doi: 10.1016/j.aml.2024.109388
[19]	H. F. Jiang, D. Xu, W. L. Qiu, J. Zhou, An ADI compact difference scheme for the two-dimensional semilinear time-fractional mobile–immobile equation, Comput. Appl. Math., 39 (2020), 287. https://doi.org/10.1007/s40314-020-01345-x doi: 10.1007/s40314-020-01345-x
[20]	Y. N. Xiao, H. Cui, R. A. Khurma, P. A. Castillo, Artificial lemming algorithm: A novel bionic meta-heuristic technique for solving real-world engineering optimization problems, Artif. Intell. Rev., 58 (2025), 84. https://doi.org/10.1007/s10462-024-11023-7 doi: 10.1007/s10462-024-11023-7
[21]	I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
[22]	J. K. Zhang, F. Y. Zhou, N. W. Mao, Numerical optimization algorithm for solving time-fractional telegraph equations, Phys. Scr., 100 (2025), 045237. https://doi.org/10.1088/1402-4896/adbf6f doi: 10.1088/1402-4896/adbf6f
[23]	Z. M. Odibat, N. T. Shawagfeh, Generalized Taylor's formula, Appl. Math. Comput., 186 (2007), 286–293. https://doi.org/10.1016/j.amc.2006.07.102
[24]	J. Q. Xie, Z. B. Yao, R. R. Wu, X. F. Ding, J. Zhang, Block-pulse functions method for solving three-dimensional fractional Poisson type equations with Neumann boundary conditions, Bound. Value Probl., 2018 (2018), 26. https://doi.org/10.1186/s13661-018-0945-7 doi: 10.1186/s13661-018-0945-7

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