This paper investigates nonlinear fractional chemical-wave models governed by the Atangana–Baleanu–Caputo (ABC) fractional derivative. The motivation of this study arose from the need to model memory-dependent effects in nonlinear chemical-wave propagation, which cannot be adequately described using classical integer-order models. To solve the considered system, an $ \alpha $-Laplace homotopy perturbation method ($ \alpha $-LHPM) was developed by combining the $ \alpha $-Laplace transform with He's homotopy perturbation technique. The proposed approach provides recursive series solutions with rapid convergence and reduced computational complexity. Existence, uniqueness, and Hyers–Ulam stability results were also established under suitable assumptions. As an application, the fractional Belousov–Zhabotinsky dynamical system (BZDS) was analyzed for different fractional orders $ 0 < \sigma\leq1 $. Numerical simulations showed that decreasing the fractional-order parameter slows the wave propagation and produces smoother solution profiles due to stronger memory effects. In the classical case $ \sigma = 1 $, the obtained numerical solutions showed excellent agreement with the exact solutions, with absolute errors of order $ 10^{-5} $ for $ \zeta $ and $ 10^{-4} $ for $ \omega $. These results demonstrate that the proposed $ \alpha $-LHPM is an accurate and efficient semi-analytical tool for solving nonlinear fractional chemical-wave models.
Citation: Faten H. Damag, Amin Saif, Osman Osman, Amel Touati, Khaled Aldwoah. Numerical analysis of fractional nonlinear chemical wave models using the $ \alpha $-Laplace homotopy perturbation method[J]. AIMS Mathematics, 2026, 11(6): 15649-15675. doi: 10.3934/math.2026643
This paper investigates nonlinear fractional chemical-wave models governed by the Atangana–Baleanu–Caputo (ABC) fractional derivative. The motivation of this study arose from the need to model memory-dependent effects in nonlinear chemical-wave propagation, which cannot be adequately described using classical integer-order models. To solve the considered system, an $ \alpha $-Laplace homotopy perturbation method ($ \alpha $-LHPM) was developed by combining the $ \alpha $-Laplace transform with He's homotopy perturbation technique. The proposed approach provides recursive series solutions with rapid convergence and reduced computational complexity. Existence, uniqueness, and Hyers–Ulam stability results were also established under suitable assumptions. As an application, the fractional Belousov–Zhabotinsky dynamical system (BZDS) was analyzed for different fractional orders $ 0 < \sigma\leq1 $. Numerical simulations showed that decreasing the fractional-order parameter slows the wave propagation and produces smoother solution profiles due to stronger memory effects. In the classical case $ \sigma = 1 $, the obtained numerical solutions showed excellent agreement with the exact solutions, with absolute errors of order $ 10^{-5} $ for $ \zeta $ and $ 10^{-4} $ for $ \omega $. These results demonstrate that the proposed $ \alpha $-LHPM is an accurate and efficient semi-analytical tool for solving nonlinear fractional chemical-wave models.
| [1] |
K. Shah, A. R. Seadawy, M. Arfan, Evaluation of one dimensional fuzzy fractional partial differential equations, Alex. Eng. J., 59 (2020), 3347–3353. https://doi.org/10.1016/j.aej.2020.05.003 doi: 10.1016/j.aej.2020.05.003
|
| [2] |
F. H. Damag, F. Alhubairah, M. F. Alshammari, K. M. Saad, A. Kiliçman, A. Saif, et al., Topological analysis with some techniques for solving a fractional tsunami shallow water mathematical model based on Hausdorff locally compact structures and their analytical implications, AIMS Math., 11 (2026), 3957–3985. https://doi.org/10.3934/math.2026159 doi: 10.3934/math.2026159
|
| [3] |
R. Ashraf, R. Nawaz, O. Alabdali, N. Fewster-Young, A. H. Ali, F. Ghanim, et al., A new hybrid optimal auxiliary function method for approximate solutions of non-linear fractional partial differential equations, Fractal Fract., 7 (2023), 673. https://doi.org/10.3390/fractalfract7090673 doi: 10.3390/fractalfract7090673
|
| [4] |
A. Iqbal, R. Nawaz, H. Hina, A. G. Ahmad, H. Emadifar, Utilizing the optimal auxiliary function method for the approximation of a nonlinear long wave system considering Caputo fractional order, Complexity, 2024 (2024), 8357221. https://doi.org/10.1155/2024/8357221 doi: 10.1155/2024/8357221
|
| [5] |
A. E. Hamza, O. Osman, A. Ali, A. Alsulami, K. Aldwoah, A. Mustafa, et al., Fractal-fractional-order modeling of liver fibrosis disease and its mathematical results with subinterval transitions, Fractal Fract., 8 (2024), 638. https://doi.org/10.3390/fractalfract8110638 doi: 10.3390/fractalfract8110638
|
| [6] |
M. Alhazmi, S. Saber, Glucose-insulin regulatory system: cahaos control and stability analysis via Atangana–Baleanu fractal-fractional derivatives, Alex. Eng. J., 122 (2025), 77–90. https://doi.org/10.1016/j.aej.2025.02.066 doi: 10.1016/j.aej.2025.02.066
|
| [7] |
N. Almutairi, S. Saber, Application of a time-fractal fractional derivative with a power-law kernel to the Burke-Shaw system based on Newton's interpolation polynomials, MethodsX, 12 (2024), 102510. https://doi.org/10.1016/j.mex.2023.102510 doi: 10.1016/j.mex.2023.102510
|
| [8] |
M. Adel, K. Aldwoah, F. Alahmadi, M. S. Osman, The asymptotic behavior for a binary alloy in energy and material science: The unified method and its applications, J. Ocean Eng. Sci., 9 (2024), 373–378. https://doi.org/10.1016/j.joes.2022.03.006 doi: 10.1016/j.joes.2022.03.006
|
| [9] |
S. T. M. Thabet, I. Kedim, M. E. Samei, T. Abdeljawad, Analysis study of hybrid Caputo–Atangana–Baleanu fractional pantograph system under integral boundary conditions, Math. Model. Anal., 30 (2025), 386–404. https://doi.org/10.3846/mma.2025.22328 doi: 10.3846/mma.2025.22328
|
| [10] |
S. T. M. Thabet, A. Boutiara, M. E. Samei, I. Kedim, M. Vivas-Cortez, Efficient results on fractional Langevin–Sturm–Liouville problem via generalized Caputo–Atangana–Baleanu derivatives, PLoS One, 19 (2024), e0311141. https://doi.org/10.1371/journal.pone.0311141 doi: 10.1371/journal.pone.0311141
|
| [11] |
S. T. M. Thabet, I. Kedim, T. Abdeljawad, Exploring the solutions of Hilfer delayed Duffing problem on the positive real line, Bound. Value Probl., 2024 (2024), 95. https://doi.org/10.1186/s13661-024-01903-w doi: 10.1186/s13661-024-01903-w
|
| [12] |
Y. Li, F. Liu, I. W. Turner, T. Li, Time-fractional diffusion equation for signal smoothing, Appl. Math. Comput., 326 (2018), 108–116. https://doi.org/10.1016/j.amc.2018.01.007 doi: 10.1016/j.amc.2018.01.007
|
| [13] |
K. A. Aldwoah, M. A. Almalahi, K. Shah, M. Awadalla, R. H. Egami, Dynamics analysis of dengue fever model with harmonic mean type under fractal-fractional derivative, AIMS Math., 9 (2024), 13894–13926. https://doi.org/10.3934/math.2024676 doi: 10.3934/math.2024676
|
| [14] |
F. H. Damag, On comparing analytical and numerical solutions of time Caputo fractional Kawahara equations via some techniques, Mathematics, 13 (2025), 2995. https://doi.org/10.3390/math13182995 doi: 10.3390/math13182995
|
| [15] |
J. Duan, Z. Wei, Y. Chen, L. Wang, W. Zhang, Global dynamics and mixed-mode oscillations in nonlinear ship rolling with implications for capsizing, Ocean Eng., 341 (2025), 122805. https://doi.org/10.1016/j.oceaneng.2025.122805 doi: 10.1016/j.oceaneng.2025.122805
|
| [16] |
J. Duan, Z. Wei, G. Li, D. Li, C. Grebogi, Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems, Nonlinear Dyn., 112 (2024), 12565–12577. https://doi.org/10.1007/s11071-024-09678-6 doi: 10.1007/s11071-024-09678-6
|
| [17] |
T. Yazgan, E. Ilhan, E. Çelik, H. Bulut, On the new hyperbolic wave solutions to Wu–Zhang system models, Opt. Quant. Electron., 54 (2022), 298. https://doi.org/10.1007/s11082-022-03683-y doi: 10.1007/s11082-022-03683-y
|
| [18] |
A. R. Seadawy, M. Iqbal, D. Lu, Propagation of kink and anti-kink wave solitons for the nonlinear damped modified Korteweg–de Vries equation arising in ion-acoustic waves in an unmagnetized collisional dusty plasma, Phys. A, 544 (2020), 123560. https://doi.org/10.1016/j.physa.2019.123560 doi: 10.1016/j.physa.2019.123560
|
| [19] |
A. Akgul, A. Cordero, J. R. Torregrosa, A fractional Newton method with $2\alpha$-order of convergence and its stability, Appl. Math. Lett., 98 (2019), 344–351. https://doi.org/10.1016/j.aml.2019.06.028 doi: 10.1016/j.aml.2019.06.028
|
| [20] |
M. Nadeem, S. H. Alotaibi, F. M. Alotaibi, Y. Alsayaad, Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable, PLoS One, 19 (2024), e0304871. https://doi.org/10.1371/journal.pone.0304871 doi: 10.1371/journal.pone.0304871
|
| [21] |
M. U. Rahman, M. Arfan, Z. Shah, E. Alzahrani, Evolution of a fractional mathematical model for drinking under Atangana–Baleanu Caputo derivatives, Phys. Scr., 96 (2021), 115203. https://doi.org/10.1088/1402-4896/ac1218 doi: 10.1088/1402-4896/ac1218
|
| [22] |
J. H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng., 178 (1999), 257–262. https://doi.org/10.1016/S0045-7825(99)00018-3 doi: 10.1016/S0045-7825(99)00018-3
|
| [23] |
S. Ş. Ş. Kiliç, E. Çelik, Complex solutions to the higher-order nonlinear Boussinesq type wave equation transform, Ricerche Mat., 73 (2024), 1793–1800. https://doi.org/10.1007/s11587-022-00698-1 doi: 10.1007/s11587-022-00698-1
|
| [24] |
B. R. Sontakke, A. S. Shelke, A. S. Shaikh, Solution of non-linear fractional differential equations by variational iteration method and applications, Far East J. Math. Sci., 110 (2019), 113–129. https://doi.org/10.17654/MS110010113 doi: 10.17654/MS110010113
|
| [25] |
S. A. Ahmed, M. Elbadri, A. A. Hassan, W. Hdidi, Numerical solutions of time-fractional Whitham–Broer–Kaup equations via Sumudu decomposition method, J. Math., 2023 (2023), 4664866. https://doi.org/10.1155/2023/4664866 doi: 10.1155/2023/4664866
|
| [26] |
S. M. El-Sayed, D. Kaya, Exact and numerical traveling wave solutions of Whitham–Broer–Kaup equations, Appl. Math. Comput., 167 (2005), 1339–1349. https://doi.org/10.1016/j.amc.2004.08.012 doi: 10.1016/j.amc.2004.08.012
|
| [27] |
F. H. Damag, A. Saif, On solving modified time Caputo fractional Kawahara equations in the framework of Hilbert algebras using the Laplace residual power series method, Fractal Fract., 9 (2025), 301. https://doi.org/10.3390/fractalfract9050301 doi: 10.3390/fractalfract9050301
|
| [28] |
B. Boutarfa, A. Akgül, M. Inc, New approach for the Fornberg–Whitham type equations, J. Comput. Appl. Math., 312 (2017), 13–26. https://doi.org/10.1016/j.cam.2015.09.016. doi: 10.1016/j.cam.2015.09.016
|
| [29] |
H. Xu, W. Cheng, J. Cui, Multiple-soliton and periodic solutions to space-time fractional Whitham–Broer–Kaup equations, Eur. Phys. J. Spec. Top., 231 (2022), 2353–2357. https://doi.org/10.1140/epjs/s11734-021-00374-9 doi: 10.1140/epjs/s11734-021-00374-9
|
| [30] |
R. Shah, H. Khan, D. Baleanu, Fractional Whitham–Broer–Kaup equations within modified analytical approaches, Axioms, 8 (2019), 125. https://doi.org/10.3390/axioms8040125 doi: 10.3390/axioms8040125
|
| [31] |
L. Wang, X. Chen, Approximate analytical solutions of time fractional Whitham–Broer–Kaup equations by a residual power series method, Entropy, 17 (2015), 6519–6533. https://doi.org/10.3390/e17096519 doi: 10.3390/e17096519
|
| [32] | D. B. Dhaigude, S. B. Kiwne, R. M. Dhaigude, Monotone iterative scheme for weakly coupled system of finite difference reaction–diffusion equations, Commun. Appl. Anal., 12 (2008), 161. |
| [33] |
A. M. Zhabotinsky, Belousov–Zhabotinsky reaction, Scholarpedia, 2 (2007), 1435. https://doi.org/10.4249/scholarpedia.1435 doi: 10.4249/scholarpedia.1435
|
| [34] |
A. Ali, K. Shah, R. A. Khan, Numerical treatment for traveling wave solutions of fractional Whitham–Broer–Kaup equations, Alex. Eng. J., 57 (2018), 1991–1998. https://doi.org/10.1016/j.aej.2017.04.012 doi: 10.1016/j.aej.2017.04.012
|
| [35] |
S. A. M. Alsallami, M. Maneea, E. M. Khalil, S. Abdel-Khalek, K. 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
|
| [36] |
K. K. Ali, M. S. Mohamed, M. Maneea, Exploring optical soliton solutions of the time fractional q-deformed Sinh–Gordon equation using a semi-analytic method, AIMS Math., 8 (2023), 27947–27968. https://doi.org/10.3934/math.20231429 doi: 10.3934/math.20231429
|
| [37] |
B. Karaagac, K. M. Owolabi, E. Pindza, Analysis and new simulations of fractional Noyes–Field model using Mittag-Leffler kernel, Sci. Afr., 17 (2022), e01384. https://doi.org/10.1016/j.sciaf.2022.e01384 doi: 10.1016/j.sciaf.2022.e01384
|
| [38] |
C. Baishya, P. Veeresha, Fractional approach for Belousov–Zhabotinsky reactions model with unified technique, Progr. Fract. Differ. Appl., 10 (2024), 295–311. https://doi.org/10.18576/pfda/100210 doi: 10.18576/pfda/100210
|
| [39] |
M. Naeem, N. H. Aljahdaly, R. Shah, W. Weera, The study of fractional-order convection-reaction-diffusion equation via an Elzake Atangana–Baleanu operator, AIMS Math., 7 (2022), 18080–18098. https://doi.org/10.3934/math.2022995 doi: 10.3934/math.2022995
|
| [40] |
K. M. Owolabi, A. Atangana, Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction–diffusion systems, Comput. Appl. Math., 37 (2018), 2166–2189. https://doi.org/10.1007/s40314-017-0445-x doi: 10.1007/s40314-017-0445-x
|
| [41] |
L. Akinyemi, A fractional analysis of Noyes–Field model for the nonlinear Belousov–Zhabotinsky reaction, Comput. Appl. Math., 39 (2020), 175. https://doi.org/10.1007/s40314-020-01212-9 doi: 10.1007/s40314-020-01212-9
|
| [42] |
W. Alfwzan, S. W. Yao, F. M. Allehiany, S. Ahmad, S. Saifullah, M. Inc, Analysis of fractional non-linear tsunami shallow water mathematical model with singular and non singular kernels, Results Phys., 52 (2023), 106707. https://doi.org/10.1016/j.rinp.2023.106707 doi: 10.1016/j.rinp.2023.106707
|
| [43] |
L. Debnath, The double Laplace transforms and their properties with applications to functional, integral and partial differential equations, Int. J. Appl. Comput. Math., 2 (2016), 223–241. https://doi.org/10.1007/s40819-015-0057-3 doi: 10.1007/s40819-015-0057-3
|
| [44] | I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space. Carpath. J. Math., 26 (2010), 103–107. |
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Faten H. Damag, Amin Saif, Osman Osman, Amel Touati, Khaled Aldwoah. Numerical analysis of fractional nonlinear chemical wave models using the $ \alpha $-Laplace homotopy perturbation method[J]. AIMS Mathematics, 2026, 11(6): 15649-15675. doi: 10.3934/math.2026643
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