Mô phỏng số dựa trên phương pháp collocation cho phương trình Allen-Cahn bậc phân số

B. Ahmad, S.K. Ntouyas, A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals 83, 234–241 (2016) G. Akagi, G. Schimperna, A. Segatti, Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations. J. Differ. Equ. 261, 2935–2985 (2016) S.M. Allen, J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979) A. Atangana, A. Akgül, Can transfer function and Bode diagram be obtained from Sumudu transform. Alex. Eng. J. 59, 1971–1984 (2020) A. Bekir, O. Guner, A.C. Cevikel, Fractional complex transform and exp-function methods for fractional differential equations. Abstr. Appl. Anal. 8, 426–462 (2013) A. Bekir, O. Guner, O. Unsal, The first integral method for exact solutions of nonlinear fractional differential equations. J. Comput. Nonlinear Dyn. 10, 210–221 (2015) T.A. Biala, S.N. Jator, Block implicit Adams methods for fractional differential equations. Chaos Solitons Fractals 81, 365–377 (2015) A. Esen, N.M. Yagmurlu, O. Tasbozan, Approximate analytical solution to time-fractional damped Burger and Cahn-Allen equations. Appl. Math. Inf. Sci. 7, 1951–1956 (2013) R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashcuk, Symmetry properties of fractional diffusion equations. Phys. Scr. T136, 014016 (2009) C.A. Hall, On error bounds for spline interpolation. J. Approx. Theory 1, 209–218 (1968) T. Hou, T. Tang, J. Yang, Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations. J. Sci. Comput. 72, 1214–1231 (2017) C. Huang, M. Stynes, Optimal \(H^1\) spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation. Adv. Comput. Math. 46(2020). https://doi.org/10.1007/s10444-020-09805-y M. Inc, A. Yusuf, A.I. Aliyu, D. Baleanu, Time-fractional Cahn-Allen and time-fractional Klein-Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis. Physica A 493, 94–106 (2018) H. Jafari, H. Tajadodi, D. Baleanu, Application of a homogeneous balance method to exact solutions of nonlinear fractional evolution equations. J. Comput. Nonlinear Dyn. 9(2), 021019 (2014). https://doi.org/10.1115/1.4025770 R. Jiwari, Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions. Comput. Phys. Commun. 193, 55–65 (2015) R. Jiwari, S. Pandit, M.E. Koksal, A class of numerical algorithms based on cubic trigonometric B-spline functions for numerical Simulation of nonlinear parabolic problems. Comput. Appl. Math. 38, 140 (2019). https://doi.org/10.1007/s40314-019-0918-1 B. Ji, H.L. Liao, L. Zhang, Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation. Adv. Comput. Math. 46, 37 (2020). https://doi.org/10.1007/s10444-020-09782-2 M.K. Kadalbajoo, P. Arora, B-spline collocation method for the singular-perturbation problem using artificial viscosity. Comput. Math. Appl. 57, 650–663 (2009) N. Khalid, M. Abbas, M.K. Iqbal, D. Baleanu, A numerical investigation of Caputo time fractional Allen-Cahn equation using redefined cubic B-spline functions. Adv. Differ. Equ. 2020, 158 (2020). https://doi.org/10.1186/s13662-020-02616-x A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204 (Elsevier Science B.V, Amsterdam, 2006) V. Kiryakova, Generalised Fractional Calculus and Applications, Pitman Research Notes in Mathematics 301 (Longman, London, 1994) C.P. Li, F. Zeng, Numerical Methods for Fractional Calculus (CRC Press, New York, 2015) C. Liu, J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D: Nonlinear Phenom. 179, 211–228 (2003) H. Liu, A. Cheng, H. Wang, A fast Galerkin finite element method for a space-time fractional Allen-Cahn equation. J. Comput. Appl. Math. 368, 112482 (2020) H. Liu, A. Cheng, H. Wang, J. Zhao, Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation. Comput. Math. Appl. 76, 1876–1892 (2018) Z. Liu, X. Li, J. Huang, Accurate and efficient algorithms with unconditional energy stability for the time fractional Cahn-Hilliard and Allen-Cahn equations, Numer. Methods Partial. Differ. Equ. 37, 2613–2633 (2021) K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley & Sons, New York, 1993) R.C. Mittal, S. Dahiya, Numerical Simulation of three-dimensional telegraphic equation using cubic B-spline differential quadrature method. Appl. Math. Comput. 313, 442–452 (2017) R.C. Mittal, R.K. Jain, Cubic B-splines collocation method for solving nonlinear parabolic partial differential equations with Neumann boundary conditions. Commun. Nonlinear Sci. Numer. Simulat. 17, 4616–4625 (2012) K.B. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974) I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999) H. Ramos, A. Kaur, V. Kanwar, Using a cubic B-spline method in conjunction with a one-step optimized hybrid block approach to solve nonlinear partial differential equations. Comput. Appl. Math. 41–34 (2022) W. Rui, X. Zhang, Lie symmetries and conservation laws for the time fractional Derrida-Lebowitz-Speer-Spohn equation. Commun. Nonlinear Sci. Numer. Simul. 34, 38–44 (2016) M.G. Sakar, O. Saldir, F. Erdogan, An iterative approximation for time-fractional Cahn-Allen equation with reproducing kernel method. Comput. Appl. Math. 37, 5951–5964 (2018) S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivative: Theory and Applications (Gordon and Breach Science Publishers, Yverdon, 1993) K. Shah, H. Khalil, R.A. Khan, Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos Solitons Fractals 77, 240–246 (2015) H.S. Shukla, M. Tamsir, Extended modified cubic B-spline algorithm for nonlinear Fisher’s reaction-diffusion equation. Alex. Eng. J. 55, 2871–2879 (2016) H. Tariq, G. Akram, New traveling wave exact and approximate solutions for the nonlinear Cahn-Allen equation: evolution of a nonconserved quantity. Nonlinear Dyn. 88, 581–594 (2017) H. Tariq, G. Akram, New approach for exact solutions of time fractional Cahn-Allen equation and time fractional Phi-4 equation. Physica A (2017). https://doi.org/10.1016/j.physa.2016.12.081 F. Tascan, A. Bekir, Travelling wave solutions of the Cahn-Allen equation by using first integral method. Appl. Math. Comput. 207, 279–282 (2009) A.M. Wazwaz, The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188, 1467–1475 (2007) M.Y. Xu, W.C. Tan, Intermediate processes and critical phenomena: theory method and progress of fractional operators and their applications to modern mechanics. Sci. China Ser. G: Phys. Mech. Astron. 49, 257–272 (2006) P. Yue, C. Zhou, J.J. Feng, C.F. Ollivier-Gooch, H.H. Hu, Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. J. Comput. Phys. 219, 47–67 (2006) S. Zhai, Z. Weng, X. Feng, Fast explicit operator splitting method and time-step adaptivity for fractional nonlocal Allen-Cahn model. Appl. Math. Model. 40, 1315–1324 (2016) S. Zhang, H.Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A 375, 1069–1073 (2011) B. Zheng, \(G^{\prime }/G\)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 58, 623–630 (2012)