Numerical investigation of the time-fractional Fisher equation in physics-based diffusion-reaction systems

Mousa J. Huntul; Serkan Alagoz; Berat Karaagac; Mousa J. Huntul; Serkan Alagoz; Berat Karaagac

doi:10.3934/math.2026656

AIMS Mathematics

2026, Volume 11, Issue 6: 15926-15951. doi: 10.3934/math.2026656

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Numerical investigation of the time-fractional Fisher equation in physics-based diffusion-reaction systems

1.
Department of Mathematics, College of Science, Jazan University, P. O. Box 114, Jazan 45142, Saudi Arabia
2.
Department of Physics, Faculty of Science and Arts, Inonu University, Malatya 44000, Turkey
3.
Department of Natural and Mathematical Sciences, Faculty of Engineering, Tarsus University, Mersin 33400, Turkey

Received: 17 April 2026 Revised: 19 May 2026 Accepted: 26 May 2026 Published: 05 June 2026
MSC : 65L60, 65M60, 42A15

In this study, we investigate the numerical solutions of the time-fractional Fisher equation by employing the collocation finite element method (FEM). The fractional derivative is considered in the Caputo sense, which provides a suitable framework for modeling memory-dependent diffusion-reaction processes. The temporal discretization is carried out using the $ L1 $ algorithm, while the spatial discretization is carried out using the unified hyperbolic polynomial B-spline basis within a collocation finite element framework. Two test problems are presented to demonstrate the efficiency of the proposed method, and the obtained numerical results are compared with available exact and reference solutions. The results confirm that the collocation finite element approach is a reliable and effective technique for solving fractional partial differential equations (FPDEs) arising in nonlinear diffusion-reaction models.
- time-fractional Fisher equation,
- unified hyperbolic basis,
- collocation method,
- Caputo derivative,
- stability
Citation: Mousa J. Huntul, Serkan Alagoz, Berat Karaagac. Numerical investigation of the time-fractional Fisher equation in physics-based diffusion-reaction systems[J]. AIMS Mathematics, 2026, 11(6): 15926-15951. doi: 10.3934/math.2026656

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Abstract

In this study, we investigate the numerical solutions of the time-fractional Fisher equation by employing the collocation finite element method (FEM). The fractional derivative is considered in the Caputo sense, which provides a suitable framework for modeling memory-dependent diffusion-reaction processes. The temporal discretization is carried out using the $ L1 $ algorithm, while the spatial discretization is carried out using the unified hyperbolic polynomial B-spline basis within a collocation finite element framework. Two test problems are presented to demonstrate the efficiency of the proposed method, and the obtained numerical results are compared with available exact and reference solutions. The results confirm that the collocation finite element approach is a reliable and effective technique for solving fractional partial differential equations (FPDEs) arising in nonlinear diffusion-reaction models.

References

[1]	I. Podlubny, Fractional Differential Equations, San Diego: Academic Press, 1999. http://doi.org/10.1016/S0076-5392(99)80021-6
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006. http://doi.org/10.1016/S0304-0208(06)X8001-5
[3]	F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, London: Imperial College Press, 2010. http://doi.org/10.1142/p614
[4]	R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 355–369. http://doi.org/10.1111/j.1469-1809.1937.tb02153.x doi: 10.1111/j.1469-1809.1937.tb02153.x
[5]	J. D. Murray, Mathematical Biology I: An Introduction, 3 Eds., New York: Springer, 2002. https://doi.org/10.1007/b98868
[6]	L. J. S. Allen, An Introduction to Mathematical Biology, Upper Saddle River: Pearson/Prentice Hall, 2006.
[7]	K. M. Owolabi, S. Jain, Modeling anomalous transport and pattern formation using coupled fractional reaction–diffusion equations, Nonlinear Sci., 7 (2026), 100118. https://doi.org/10.1016/j.nls.2026.100118 doi: 10.1016/j.nls.2026.100118
[8]	K. M. Owolabi, S. Jain, E. Maré, Fractional quantum dynamics in dissipative chemical systems: Memory-driven Schrödinger models for anomalous quantum transport, Comput. Theor. Chem., 1260 (2026), 115759. https://doi.org/10.1016/j.comptc.2026.115759 doi: 10.1016/j.comptc.2026.115759
[9]	K. M. Owolabi, S. Alagoz, Advection–diffusion–reaction modeling of contaminant transport in groundwater: Analysis and simulation, Nonlinear Sci., 000000000 (2025), 100083. https://doi.org/10.1016/j.nls.2025.100083 doi: 10.1016/j.nls.2025.100083
[10]	D. Kumar, S. Chaudhary, V. V. K. S. Kumar, Fractional Crank–Nicolson–Galerkin finite element scheme for the time-fractional nonlinear diffusion equation, Numer. Meth. PDE, 35 (2019), 2056–2075. https://doi.org/10.1002/num.22399 doi: 10.1002/num.22399
[11]	M. A. Yousif, F. K. Hamasalh, Novel simulation of the time fractional Burgers–Fisher equations using a non-polynomial spline fractional continuity method, AIP Adv.., 12 (2022), 115302. http://doi.org/10.1063/5.0128819 doi: 10.1063/5.0128819
[12]	A. Majeed, A. A. Zafar, S. Abbasbandy, An efficient numerical technique for solving time-fractional generalized Fisher's equation, Front. Phys., 8 (2020), 293. http://doi.org/10.3389/fphy.2020.00293 doi: 10.3389/fphy.2020.00293
[13]	A. Majeed, M. Kamran, M. K. Iqbal, D. Baleanu, Solving time fractional Burgers' and Fisher's equations using cubic B-spline approximation method, Adv. Diff. Equ., 2020 (2020), 175. http://doi.org/10.1186/s13662-020-02619-8 doi: 10.1186/s13662-020-02619-8
[14]	J. N. Reddy, An Introduction to the Finite Element Method, 3 Eds., New York: McGraw–Hill, 2006.
[15]	P. Zhu, S. Xie, ADI finite element method for 2D nonlinear time fractional reaction–subdiffusion equation, Amer. J. Comput. Math., 6 (2016), 336–356. http://doi.org/10.4236/ajcm.2016.64034 doi: 10.4236/ajcm.2016.64034
[16]	B. Karaagac, A. Esen, K. M. Owolabi, E. Pindza, A collocation method for solving time fractional nonlinear Korteweg–de Vries–Burgers equation arising in shallow water waves, Int. J. Modern Phys. C, 34 (2023), 2350096. https://doi.org/10.1142/S0129183123500961 doi: 10.1142/S0129183123500961
[17]	B. Karaagac, K. M. Owolabi, A. Esen, Unveiling numerical solutions of Zeldovich model using collocation method via fourth-order uniform hyperbolic polynomial B-spline, Adv. Theory Simul., 9 (2026), e01349. https://doi.org/10.1002/adts.202501349 doi: 10.1002/adts.202501349
[18]	F. Idiz, G. Tangolu, N. Aghazadeh, A. Mohammadi, An effective Legendre wavelet technique for the time-fractional Fisher equation, Comput. Meth. Diff. Equ., 14 (2026), 145–164. http://doi.org/10.22034/cmde.2025.63725.2849 doi: 10.22034/cmde.2025.63725.2849
[19]	H. M. Srivastava, M. A. Yousif, P. O. Mohammed, T. Abdeljawad, D. Baleanu, N. Chorfi, Solving time-fractional Fisher models by non-polynomial splines in terms of logarithmic derivatives, Fractals, 33 (2025), 2540142. http://doi.org/https://doi.org/10.1142/S0218348X25401425 doi: 10.1142/S0218348X25401425
[20]	P. Roul, An accurate numerical method and its analysis for time-fractional Fisher's equation, Soft Comput., 28 (2024), 11495–11514. https://doi.org/10.1007/s00500-024-09885-8 doi: 10.1007/s00500-024-09885-8
[21]	L. Chai, L. Wu, X. Yang, A fast parallel difference method for solving the time-fractional generalized Fisher equation, J. Appl. Anal. Comput., 15 (2025), 1216–1240. http://doi.org/10.11948/20240159 doi: 10.11948/20240159
[22]	A. G. Atta, Y. H. Youssri, Enhanced spectral collocation Gegenbauer approach for the time-fractional Fisher equation, Math. Meth. Appl. Sci., 47 (2024), 14173–14187. https://doi.org/10.1002/mma.10263 doi: 10.1002/mma.10263
[23]	M. Kashif, A Vieta–Lucas collocation and non-standard finite difference technique for solving space-time fractional-order Fisher equation, Math. Model. Anal., 30 (2025), 1–16. https://doi.org/10.3846/mma.2025.19839 doi: 10.3846/mma.2025.19839
[24]	R. Choudhary, S. Singh, D. Kumar, A high-order numerical technique for generalized time-fractional Fisher's equation, Math. Meth. Appl. Sci., 46 (2023), 16050–16071. https://doi.org/10.1002/mma.9435 doi: 10.1002/mma.9435
[25]	A. S. Rahby, Z. Yang, Theoretical and numerical investigation of long-time behaviors for time-fractional Fisher equations, J. Appl. Math. Comput., 71 (2025), 6527–6547. https://doi.org/10.1007/s12190-025-02534-9 doi: 10.1007/s12190-025-02534-9
[26]	S. Kwak, Y. Nam, S. Kang, J. Kim, Computational analysis of a normalized time-fractional Fisher equation, Appl. Math. Lett., 166 (2025), 109542. https://doi.org/10.1016/j.aml.2025.109542 doi: 10.1016/j.aml.2025.109542
[27]	A. Almuneef, A. Hagag, A fractional semi-analytical iterative method for the approximate treatment of Fisher's equations, Rev. Int. Métodos Numér. Cálc. Diseño Ing., 40 (2024), 50. http://doi.org/10.23967/j.rimni.2024.10.56315 doi: 10.23967/j.rimni.2024.10.56315
[28]	A. El-Sayed, A. Arafa, I. Hanafy, A. Hagag, An approximate study of Fisher's equation by using a semi-analytical iterative method, Progr. Fract. Diff. Appl., 9 (2023), 397–407. http://doi.org/10.18576/pfda/090305 doi: 10.18576/pfda/090305
[29]	M. J. Huntul, Bell polynomial-based semi-discretization approach for the extended Fisher-Kolmogorov equations, AIMS Math., 10 (2025), 29263–29284. http://doi.org/10.3934/math.20251286 doi: 10.3934/math.20251286
[30]	M. J. Huntul, M. Modanli, Numerical approach for solving the inverse problem: A two-dimensional time-fractional boundary value problem, AIMS Math., 11 (2026), 7078–7097. http://doi.org/10.3934/math.2026291 doi: 10.3934/math.2026291
[31]	M. U. Manzoor, M. Yaseen, M. Awadalla, H. Zaway, A uniform hyperbolic polynomial B-spline approach for solving the fractional diffusion-wave equations in the Caputo-Fabrizio sense, AIMS Math., 10 (2025), 17049–17081 http://doi.org/10.3934/math.2025765 doi: 10.3934/math.2025765
[32]	Y. Lu, G. Wang, X. Yang, Uniform hyperbolic polynomial B-spline curves, Comput. Aided Geomet. Design, 19 (2002), 379–393. https://doi.org/10.1016/S0167-8396(02)00092-4 doi: 10.1016/S0167-8396(02)00092-4
[33]	K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Amsterdam: Elsevier, 1974.
[34]	J. Rashidinia, M. N. Rasoulizadeh, Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation, Wave Motion, 90 (2019), 152–167. http://doi.org/10.1016/j.wavemoti.2019.05.006 doi: 10.1016/j.wavemoti.2019.05.006
[35]	C. Li, F. Zeng, Numerical Methods for Fractional Calculus, Boca Raton: Chapman & Hall/CRC, 2015. http://doi.org/10.1201/b18503

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