2013/10/13 by Jacek Cyranka, Cyranka, Jacek · 1 citation
Engineering · Mathematics · #35B40. Secondary: 35B41 #Advanced Differential Equations and Dynamical Systems #Computational Fluid Dynamics and Aerodynamics #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods for differential equations #Primary: 65M99 #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1310.3514
openalex publication_date 2013/10/13 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We present a computer assisted method for proving the existence of globally\nattracting fixed points of dissipative PDEs. An application to the viscous\nBurgers equation with periodic boundary conditions and a forcing function\nconstant in time is presented as a case study. We establish the existence of a\nlocally attracting fixed point by using rigorous numerics techniques. To prove\nthat the fixed point is, in fact, globally attracting we introduce a technique\nrelying on a construction of an absorbing set, capturing any sufficiently\nregular initial condition after a finite time. Then the absorbing set is\nrigorously integrated forward in time to verify that any sufficiently regular\ninitial condition is in the basin of attraction of the fixed point.\n