[1] J. Shin, S-K Park and J. Kim, A hybrid FEM for solving the Allen–Cahn equation, Appl. Math. Comput. 244 (2014) 606–612, https://doi.org/10.1016/j.amc.2014.07.040.
[2] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik 12 (1972) 30–39, https://doi.org/10.1007/BF00289234.
[3] M. Alber, T. Glimm, H. G. E. Hentschel, B. Kazmierczak, Y-T Zhang, J. Zhu, and S. A. Newman, The morphostatic limit for a model of skeletal pattern formation in the vertebrate limb, Bull. Math. Biol. 70 (2) (2008) 460–483, https://doi.org/10.1007/s11538-007-9264-3.
[4] I. A. Pavel’chak, A numerical method for determining the localized initial condition in the FitzHugh-Nagumo and Aliev-Panfilov models, MoscowUniv. Comput. Math. Cybern. 35 (2011) 105–112, https://doi.org/10.3103/S0278641911030071.
[5] M. W. Yasin, M. S. Iqbal, N. Ahmed, A. Akgül, A. Raza, M. Rafiq and M. B. Riaz, Numerical scheme and stability analysis of stochastic FitzHugh-Nagumo model, Results Phys. 32 (2022) p. 105023, https://doi.org/10.1016/j.rinp.2021.105023.
[6] R. Chertovskih, E. L. Rempel and E. V. Chimanski, Magnetic field generation by intermittent convection, Phys. Lett. A 381 (38) (2017) 3300–3306, https://doi.org/10.1016/j.physleta.2017.08.025.
[7] M. Dehghan and V. Mohammadi, The boundary knot method for the solution of two-dimensional advection reaction-diffusion and Brusselator equations, Int. J. Numer. Methods Heat Fluid Flow 31 (1) (2020) 106–133,
https://doi.org/10.1108/HFF-10-2019-0731.
[8] Z. Lin, R. Ruiz-Baier and C. Tian, Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion, J. Comput. Phys. 256 (2014) 806–823, https://doi.org/10.1016/j.jcp.2013.09.009.
[9] E. H. Twizell, A. B. Gumel and Q Cao, A second-order scheme for the “Brusselator” reaction–diffusion system, J. Math. Chem. 26 (1999) 297–316, https://doi.org/10.1023/A:1019158500612.
[10] M. Dehghan and M. Abbaszadeh, Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction–diffusion system with and without cross-diffusion, Comput. Methods Appl. Mech. Eng. 300 (2016) 770–797, https://doi.org/10.1016/j.cma.2015.11.033.
[11] R. Jiwari and J. Yuan, A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes, J. Math. Chem. 52 (2014) 1535–1551, https://doi.org/10.1007/s10910-014-0333-1.
[12] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer New York, 2003.
[13] R. C. Sarker and S. K. Sahani, Turing pattern dynamics in an SI epidemic model with superdiffusion, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 32 (11) (2022) p. 2230025, https://doi.org/10.1142/S0218127422300257.
[14] R. M. Jena, S. Chakraverty, H. Rezazadeh and D. Domiri Ganji, On the solution of time-fractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions, Math. Methods Appl. Sci. 43 (7) (2020) 3903–3913,
https://doi.org/10.1002/mma.6141.
[15] F. Muhammad Khan, A. Ali, K. Shah, A. Khan and I. Mahariq, Analytical approximation of brusselator model via ladm, Math. Probl. Eng. 2022 (2022) Article ID 8778805, https://doi.org/10.1155/2022/8778805.
[16] S. Alshammari, M. M. Al-Sawalha and J. R. Humaidi, Fractional view study of the Brusselator reaction–diffusion model occurring in chemical reactions, Fractal Fract. 7 (2023) p. 108, https://doi.org/10.3390/ fractalfract7020108.
[17] K. M. Saad, Fractal-fractional Brusselator chemical reaction, Chaos Solit. Fractals 150 (2021) p. 111087, https://doi.org/10.1016/j.chaos.2021.111087.
[18] M. Izadi and H. M. Srivastava, Fractional clique collocation technique for numerical simulations of fractional-order Brusselator chemical model, Axioms 11 (11) (2022) p. 654, https://doi.org/10.3390/axioms11110654.
[19] J. Singh, M. M. Rashidi, D. Kumar and R. Swroop, A fractional model of a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions, Nonlinear Eng. 5 (4) (2016) 277–285, https://doi.org/10.1515/nleng-2016-0041.
[20] H. Fallahgoul, S. Focardi and F. Fabozzi, Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application, Academic Press, 2016.
[21] F. Mainardi, Fractional calculus: theory and applications, Mathematics 6 (9) (2018) p. 145,
https://doi.org/10.3390/math6090145.
[22] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
[23] Z. Jackiewicz, General Linear Methods for Ordinary Differential Equations, John Wiley & Sons, 2009.
[24] J Li, F. Liu, L. Feng and I. Turner, A novel finite volume method for the Riesz space distributed-order diffusion equation, Comput. Math. Appl. 74 (4) (2017) 772–783, https://doi.org/10.1016/j.camwa.2017.05.017.
[25] M. Chen and W. Deng, A second-order accurate numerical method for the space– time tempered fractional diffusion-wave equation, Appl. Math. Lett. 68 (2017) 87–93, https://doi.org/10.1016/j.aml.2016.12.010.
[26] A. Mohebbi, Finite difference and spectral collocation methods for the solution of semilinear time fractional convection-reaction-diffusion equations with time delay, J. Appl. Math. Comput. 61 (2019) 635–656, https://doi.org/10.1007/s12190-019-01267-w.
[27] A. Saadatmandi and M. Dehghan, A new operational matrix for solving fractionalorder differential equations, Comput. Math. Appl. 59 (3) (2010) 1326–1336, https://doi.org/10.1016/j.camwa.2009.07.006.
[28] A. Saadatmandi and M. Dehghan, A tau approach for solution of the space fractional diffusion equation, Comput. Math. Appl. 62 (3) (2011) 1135–1142, https://doi.org/10.1016/j.camwa.2011.04.014.
[29] M. Pourbabaee and A. Saadatmandi, A new operational matrix based on Müntz–Legendre polynomials for solving distributed order fractional differential equations, Math. Comput. Simulation 194 (2022) 210–235, https://doi.org/10.1016/j.matcom.2021.11.023.
[30] M. Dehghan and M. Abbaszadeh, An efficient technique based on finite difference/ finite element method for solution of two-dimensional space/multi-time fractional Bloch–Torrey equations, Appl. Numer. Math. 131 (2018) 190–206,
https://doi.org/10.1016/j.apnum.2018.04.009.
[31] M. R. Ahmadi Darani and A. Saadatmandi, The operational matrix of fractional derivative of the fractional-order Chebyshev functions and its applications, Comput. Methods Differ. Equ. 5 (1) (2017) 67–87.
[32] M. Pourbabaee and A. Saadatmandi, Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations, Comput. Methods Differ. Equ. 9 (3) (2021) 858–873, https://doi.org/10.22034/CMDE.2020.38506.1695.
[33] A. Saadatmandi, A. Khani and M. R. Azizi, Numerical calculation of fractional derivatives for the sinc functions via Legendre polynomials, Math. Interdisc. Res. 5 (2) (2020) 71–86, https://doi.org/10.22052/MIR.2018.96632.1074.
[34] M. Abbaszade and A. Mohebbi, Fourth-order numerical solution of a fractional PDE with the nonlinear source term in the electroanalytical chemistry, Iranian J. Math. Chem. 3 (2) (2012) 195–220, https://doi.org/ 10.22052/IJMC.2012.5147.
[35] M. Abbaszadeh, Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation, Appl. Math. Lett. 88 (2019) 179–185,
https://doi.org/10.1016/j.aml.2018.08.024.
[36] H. Ding, C. Li and Y. Chen, High-order algorithms for Riesz derivative and their applications (ii), J. Comput. Phys. 293 (2015) 218–237, https://doi.org/10.1016/j.jcp.2014.06.007.
[37] X. Zhao, Z. Sun and Z. Hao, A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional schrödinger equation, SIAM J. Sci. Comput. 36 (6) (2014) A2865– A2886, https://doi.org/10.1137/140961560.
[38] C. Li, Q. Yi and A. Chen, Finite difference methods with non-uniform meshes for nonlinear fractional differential equations, J. Comput. Phys. 316 (2016) 614–631, https://doi.org/10.1016/j.jcp.2016.04.039.
[39] A. H. Bhrawy and M. A. Zaky, An improved collocation method for multi-dimensional space–time variable-order fractional schrödinger equations, Appl. Numer. Math. 111 (2017) 197–218, https://doi.org/10.1016/j.apnum.2016.09.009.
[40] A. H. Bhrawy, M. A. Zaky and R. A. Van Gorder, A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation, Numer. Algor. 71 (1) (2016) 151–180, https://doi.org/10.1007/s11075-015-9990-9.
[41] M. Abbaszadeh and M. Dehghan, An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate, Numer. Algor. 75 (2017) 173–211, https://doi.org/10.1007/s11075-016-0201-0.
[42] A. S. Hendy and M. A. Zaky, Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations, Eng. Comput. 38 (2022) 1351–1363, https://doi.org/10.1007/s00366-020-01095-8.
[43] A. S. Hendy, M. A. Zaky and J. E. Macías-Díaz, On the dissipativity of some Caputo time-fractional subdiffusion models in multiple dimensions: theoretical and numerical investigations, J. Comput. Appl. Math. 400 (2022) p. 113748,
https://doi.org/10.1016/j.cam.2021.113748.
[44] M. A. Zaky, A. S. Hendy and D. Suragan, Logarithmic Jacobi collocation method for Caputo–Hadamard fractional differential equations, Appl. Numer. Math. 181 (2022) 326– 346, https://doi.org/10.1016/j.apnum.2022.06.013.
[45] A. Bhrawy and M. Zaky, A fractional-order Jacobi tau method for a class of time-fractional pdes with variable coefficients, Math. Methods Appl. Sci. 39 (7) (2016) 1765–1779, https://doi.org/10.1002/mma.3600.
[46] H. Liao, P. Lyu and S. Vong, Second-order BDF time approximation for Riesz space-fractional diffusion equations, Int. J. Comput. Math. 95 (1) (2018) 144–158, https://doi.org/10.1080/00207160.2017.1366461.
[47] M. Hamid, M. Usman, Y. Yan and Z. Tian, An efficient numerical scheme for fractional characterization of MHD fluid model, Chaos Solitons Fractals 162 (2022) p. 112475, https://doi.org/10.1016/j.chaos.2022.112475.
[48] M. Abbaszadeh, M. Golmohammadi and M. Dehghan, Simulation of activator–inhibitor dynamics based on cross-diffusion Brusselator reaction–diffusion system via a differential quadrature-radial point interpolation method (DQ-RPIM) technique, Eur. Phys. J. Plus 136 (1) (2021) 1–26,
https://doi.org/10.1140/epjp/s13360-020-00872-0.
[49] M. Abbaszadeh and M. Dehghan, A reduced order finite difference method for solving space-fractional reaction-diffusion systems: the Gray-Scott model, Eur. Phys. J. Plus 134 (2019) 1–15, https://doi.org/10.1140/epjp/i2019-12951-0.
[50] Z. Mao and G. E. Karniadakis, A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative, SIAM J. Numer. Anal. 56 (1) (2018) 24–49, https://doi.org/10.1137/16M1103622.
[51] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations 22 (3) (2006) 558–576, https://doi.org/10.1002/num.20112.
[52] W. Bu, Y. Tang, Y. Wu and J. Yang, Finite difference/finite element method for two-dimensional space and time fractional Bloch–Torrey equations, J. Comput. Phys. 293 (2015) 264–279, https://doi.org/10.1016/j.jcp.2014.06.031.
)