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فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری
پایگاه استنادی علوم جهان اسلام (ISC)

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Dehghan Nezhad, Akbar, Moghaddam Zeabadi, Mina. (1402). Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions. سامانه مدیریت نشریات علمی, 12(2), 266-286. doi: 10.22034/cmde.2023.54283.2268
Akbar Dehghan Nezhad; Mina Moghaddam Zeabadi. "Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions". سامانه مدیریت نشریات علمی, 12, 2, 1402, 266-286. doi: 10.22034/cmde.2023.54283.2268
Dehghan Nezhad, Akbar, Moghaddam Zeabadi, Mina. (1402). 'Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions', سامانه مدیریت نشریات علمی, 12(2), pp. 266-286. doi: 10.22034/cmde.2023.54283.2268
Dehghan Nezhad, Akbar, Moghaddam Zeabadi, Mina. Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions. سامانه مدیریت نشریات علمی, 1402; 12(2): 266-286. doi: 10.22034/cmde.2023.54283.2268

Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions

Computational Methods for Differential Equations
مقاله 5، دوره 12، شماره 2، خرداد 2024، صفحه 266-286    اصل مقاله (941.79 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2023.54283.2268
نویسندگان
Akbar Dehghan Nezhad* ؛ Mina Moghaddam Zeabadi
School of Mathematics and Computer Science, Iran University of Science and Technology, Narmak, Tehran, Iran.
چکیده
We present a novel computational approach for computing invariant manifolds that correspond to equilibrium solutions of nonlinear parabolic partial differential equations (or PDEs). Our computational method combines Lie symmetry analysis with the parameterization method. The equilibrium solutions of PDEs and the solutions of eigenvalue problems are exactly obtained. As the linearization of the studied nonlinear PDEs at equilibrium solutions yields zero eigenvalues, these solutions are non-hyperbolic, and some invariant manifolds are center manifolds. We use the parameterization method to model the infinitesimal invariance equations that parameterize the invariant manifolds. We utilize Lie symmetry analysis to solve the invariance equations. We apply our framework to investigate the Fisher equation and the Brain Tumor growth differential equation. 
کلیدواژه‌ها
Lie symmetry analysis؛ Parameterization method؛ Equilibrium solution؛ Eigenvalue problem؛ Invariant manifolds؛ Invariance equation؛ Tanh method
مراجع
  • [1] X. Cabr´e, E. Fontich, and R. De la Llave, The parameterization method for invariant manifolds I: manifolds associated to non-resonant subspaces, Indiana University mathematics journal, (2003), 283–328.
  • [2] X. Cabr´e, E. Fontich, and R. De la Llave, The parameterization method for invariant manifolds II: regularity with respect to parameters, Indiana University mathematics journal, (2003), 329–360.
  • [3] X. Cabr´e, E. Fontich, and R. De la Llave, The parameterization method for invariant manifolds III: overview and applications, Journal of Differential Equations, 218(2) (2005), 444-515.
  • [4] M. Canadell and A. Haro,´ Parameterization method for computing quasi-periodic reducible normally hyperbolic invariant tori, In Advances in differential equations and applications, (2014), 85–94.
  • [5] R. Castelli, JP. Lessard, and JD. Mireles James, Parameterization of invariant manifolds for periodic orbits I: Efficient numerics via the Floquet normal form, SIAM Journal on Applied Dynamical Systems, 14(1) (2015), 132–167.
  • [6] JL. Figueras and A. Haro,´ Reliable computation of robust response tori on the verge of breakdown, SIAM Journal on Applied Dynamical Systems, 11(2) (2012), 597–628.
  • [7] JL. Figueras and A. Haro,´ Triple collisions of invariant bundles, Discrete, Continuous Dynamical Systems-B, 18 (2013), 2069–2082.
  • [8] JL. Figueras, M. Gameiro , JP. Lessard, and R. de la Llave, A framework for the numerical computation and a posteriori verification of invariant objects of evolution equations, SIAM Journal on Applied Dynamical Systems, 16(1) (2017), 687–728.
  • [9] RA. Fisher, The wave of advance of advantageous genes, Annals of eugenics, 7(4) (1937), 355–369.
  • [10] A. Haro and R. De la Llave,´ A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: rigorous results, Journal of Differential Equations, 228(2) (2006), 530–579.
  • [11] A. Haro and R. De la Llave,´ A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: numerical algorithms, Discrete, Continuous Dynamical Systems-B, 6(6) (2006), 62–121.
  • [12] A. Haro and R. De la Llave,´ A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: explorations and mechanisms for the breakdown of hyperbolicity, SIAM Journal on Applied Dynamical Systems, 6(1)(2007), 142–207.
  • [13] A. Haro, M. Canadell, JL. Figueras , A. Luque , and JM. Mondelo,´ The parameterization method for invariant manifolds, Applied mathematical sciences, 195 (2016),1–95.
  • [14] PE. Hydon, Symmetry methods for differential equations: a beginner’s guide, Cambridge University Press, (22) (2000).
  • [15] B. Krauskopf, HM. Osinga, EJ. Doedel, ME. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge, A survey of methods for computing (un) stable manifolds of vector fields, International Journal of Bifurcation and Chaos, 15(3) (2005), 763–791.
  • [16] JD. Mireles James and M. Murray, Chebyshev–Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications, International Journal of Bifurcation and Chaos, 27(14) (2017), 1730050.
  • [17] PJ. Olver, Applications of Lie groups to differential equations, Springer Science, Business Media, 107 (1993).
  • [18] C. Reinhardt and JM. James, Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation, Indagationes Mathematicae, 30(1) (2019), 39-80.
  • [19] AN. Timsina and JM. James, Parameterized stable/unstable manifolds for periodic solutions of implicitly defined dynamical systems, Chaos, Solitons & Fractals, 161 (2022), 112345.
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