%0 Journal Article
%T Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions
%J Computational Methods for Differential Equations
%I University of Tabriz
%Z 2345-3982
%A Dehghan Nezhad, Akbar
%A Moghaddam Zeabadi, Mina
%D 2024
%\ 03/01/2024
%V 12
%N 2
%P 266-286
%! Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions
%K Lie symmetry analysis
%K Parameterization method
%K Equilibrium solution
%K Eigenvalue problem
%K Invariant manifolds
%K Invariance equation
%K tanh method
%R 10.22034/cmde.2023.54283.2268
%X 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.
%U https://cmde.tabrizu.ac.ir/article_16921_27fa5a1ef04691fef4b60494658a6f54.pdf