[1] Allen, S.M. and Cahn, J.W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall 27 (1979) 1085–1095.
[2] Benes, M., Chalupecky, V. and Mikula, K. Geometrical image segmenta-tion by the Allen–Cahn equation, Appl. Numer. Math. 51 (2004) 187–205.
[3] Boettinger, W.J., Warren, J.A., Beckermann, C. and Karma, A. Phase-field simulation of solidification, Annu. Rev. Mater. Sci. 32 (2002) 163–194.
[4] De Marchi, S., Martinez, A. and Perracchione, E. Fast and stable rational RBF-based partition of unity interpolation, J. Comput. Appl. Math. 349 (2019) 331–343.
[5] Driscoll, A. and Heryudono, R.H. Adaptive residual subsampling methods for radial basis function interpolation and collocation problems, Comput. Math. Appl. 53 (2007) 927–939.
[6] Elliott, C.M. and Stinner, B. Computation of two-phase biomembranes with phase dependent material parameters using surface finite element, Commun. Comput. Phys. 13 (2013) 325–360.
[7] Elliott, C.M. and Stuart, A.M. The global dynamics of discrete semilin-ear parabolic equations, SIAM J. Numer. Anal. 30 (1993) 1622–1663.
[8] Esedoglu, S. and Tsai, Y.H.R. Threshold dynamics for the piecewise constant Mumford-Shan functional, J. Comput. Phys. 211 (2006) 367–384.
[9] Fan, D. and Chen, L.Q. Computer simulation of grain growth using a continuum field model, Acta Mater. 45 (1997) 611–622.
[10] Fasshauer, G.E. Meshfree Approximation Methods with Matlab, World Scientific, 2007.
[11] Feng, X., Li, Y. and Zhang, Y. Finite element methods for the stochastics Allen–Cahn equation with gradient-type multiplicative noise, SIAM J. Numer. Anal. 55(1) (2017) 194–216.
[12] Feng, X., Song, H., Tang, T. and Yang, J. Nonlinear stability of the implicit-explicit methods for the Allen–Cahn equation, Inverse Probl. Imaging 7 (2013) 679–695.
[13] Golubovic, L., Levandovsky, A. and Moldovan, D. Interface dynamics and far-from-equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: Continuum theory insights, East Asian J. Appl. Math. 1 (2011) 297–371.
[14] Heidari, M., Mohammadi, M. and De Marchi, S. A shape preserv-ing quasi-interpolation operator based on a new transcendental RBF, Dolomites Research Notes on Approximation 14 (1) (2021) 56–73.
[15] Heydari, M.H. and Hosseininia, M. A new variable-order fractional derivative with non-singular Mittag-Leffler kernel: application to variable-order fractional version of the 2D Richard equation, Engineering with Computers 38 (2) (2022) 1759–1770.
[16] Hosseininia, M., Heydari, M.H., Avazzadeh, Z. and Maalek Ghaini, F.M. A hybrid method based on the orthogonal Bernoulli polynomials and radial basis functions for variable order fractional reaction-advection-diffusion equation, Engineering Analysis with Boundary Elements 127(2021) 18–28.
[17] Jakobsson, S., Andersson, B. and Edelvik, F. Rational radial basis func-tion interpolation with applications to antenna design, J. Comput. Appl. Math. 233 (2009) 889–904.
[18] Jafari-Varzaneh, H.A. and Hosseini, S.M. A new map for the Chebyshev pseudospectral solution of differential equations with large gradients, Nu-mer. Algorithms 69 (2015) 95–108.
[19] Jeong, D. and Kim, J. An explicit hybrid finite difference scheme for the Allen–Cahn equation, J. Comput. Appl. Math. 340 (2018) 247–255.
[20] Kay, D.A. and Tomasi, A. Color image segmentation by the vector valued Allen–Cahn phase-field model: A multigrid solution, IEEE Trans. Image Process. 18 (2009) 2330–2339.
[21] Kim, J. Phase-field models for multi-component fluid flows, Commun. Comput. Phys. 12 (2012) 613–661.
[22] Kim, J., Jeong, D., Yang, S. and Choi, Y. A finite difference method for a conservative Allen–Cahn equation on non-flat surfaces, J. Comput. Phys. 334 (2017) 170–181.
[23] Kobayashi, R., Warren, J.A. and Carter, W.C. A continuum model of grain boundaries, Phys. D 140 (2000) 141-150.
[24] Krill, C.E. and Chen, L.Q. Computer simulation of 3-D grain growth using a phase-field model, Acta Mater. 50 (2002) 3057–3073.
[25] Lee, H. and Lee, J. A semi-analytical Fourier spectral method for the Allen–Cahn equation, Comput. Math. Appl. 68 (3) (2014) 174–184.
[26] Lee, H.G., Shin, J. and Lee, J.Y. First and second order operator splitting method for phase-field crystal equation, J. Comput. Phys. 299 (2015) 82–91.
[27] Li, Y. and Kim, J. Multiphase image segmentation using a phase-field model, Comput. Math. Appl. 62 (2011) 737–745.
[28] Li, Y., Lee, H.G. and Kim, J. A fast, robust and accurate operator splitting method for Phase-field simulation of crystal growth, J. Cryst. Growth 321 (2011) 176–182.
[29] Mohammadi, V., Mirzaei, D. and Dehghan, M. Numerical simulation and error estimation of the time-dependent Allen–Cahn equation on sur-faces with radial basis functions, J. Sci. Comput. 79 (2019) 493–516.
[30] Mohammadi, M., Mokhtari, R. and Schaback, R. A meshless method for solving the 2d brusselator reaction-diffusion system, Comput. Model. Eng. Sci. 101 (2014) 113–138.
[31] Naqvi, S.L., Levesley, J. and Ali, S. Adaptive radial basis function for time dependent partial differential equations, J. Prime Res. Math. 13(2017) 90–106.
[32] Niu, J., Xu, M. and Yao, G. An eﬀicient reproducing kernel method for solving the Allen–Cahn equation, Appl. Math. Lett. 89 (2019) 78–84.
[33] Perracchione, E. Rational RBF-based partition of unity method for eﬀi-ciently and accurately approximating 3D objects, Comp. Appl. Math. 37(2018) 4633-4648.
[34] Saberi Zafarghandi, F. and Mohammadi, M. Numerical approximations for the Riesz space fractional advection-dispersion equations via radial basis functions, Appl. Numer. Math. 144 (2019) 59–82.
[35] Sarra, S.A. and Bai, Y. A rational radial basis function method for accu-rately resolving discontinuities and steep gradients, Appl. Numer. Math. 130 (2018) 131–142.
[36] Schaback, R. Kernel-Based Meshless Methods, Gottingen, 2011.
[37] Shiralizadeh, M., Alipanah, A. and Mohammadi, M. Numerical solu-tion of one-dimensional Sine-Gordon equation using rational radial basis functions, J. Math. Model. 10 (3)(2022) 387–405.
[38] Wendland, H. Scattered data approximation, Cambridge University Press, 2004.
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