University of TabrizComputational Methods for Differential Equations2345-398212220240301Lie symmetry analysis for computing invariant manifolds associated with equilibrium solutions2662861692110.22034/cmde.2023.54283.2268ENAkbarDehghan NezhadSchool of Mathematics and Computer Science, Iran University of Science and Technology,
Narmak, Tehran, Iran.0000-0003-4200-6892MinaMoghaddam ZeabadiSchool of Mathematics and Computer Science, Iran University of Science and Technology,
Narmak, Tehran, Iran.Journal Article20221129We 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. https://cmde.tabrizu.ac.ir/article_16921_27fa5a1ef04691fef4b60494658a6f54.pdf