Computer Assisted Proofs in Dissipative Partial Differential Equations

• Autorzy: Cyranka J.
• Języki publikacji: EN

Abstrakty

EN

The goal of this paper is to present a brief survey of our research that has focused on studying the dynamics of dissipative partial differential equations by performing computer as sisted proofs. We provide a description of the main ideas behind the computer assisted proofs that we have performed, along with related topics. The emphasis is given to the case of the vis cous Burgers equation with constant forcing, for which the existence of globally attracting fixed points has been established. To achieve this goal, we used a combination of analytical results with computer assistance.

Słowa kluczowe

EN

rigorous numerics; Lohner algorithm; viscous Burgers equation; interval arithmetic; dissipative partial differential equations

• Wydawca: Foundation for Young Scientists
• Czasopismo: Challenges of Modern Technology
• Rocznik: 2012
• Tom: Vol. 3, no. 3
• Strony: 3--6
• Opis fizyczny: Bibliogr. 8 poz., wykr.

Twórcy

autor

Cyranka J.

• Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

Bibliografia

• [1] Watanabe, Y. “A computer-assisted proof for the Kolmogorov flows of incompressible vis cous fluid”. Journal of Computational and Applied Mathematics 223(2009): 953–966.
• [2] Watanabe, Y. et al. “A numerical verification of nontrivial solutions for the heat convection problem”. J. Math. Fluid Mech. 6 (2004): 1–20.
• [3] Arioli, G., H. Koch. “Integration of dissipative PDEs: a case study”. SIAM J. of Appl. Dyn. Syst. 9 (2010): 1119–1133.
• [4] Zgliczyński, P. “Rigorous Numerics for Dissipative PDEs III. An eff ective algorithm for rig orous integration of dissipative PDEs”. Topological Methods in Nonlinear Analysis 36 (2010): 197–262.
• [5] Zgliczyński, P. “Attracting fixed points for the Kuramoto--Sivashinsky equation -a computer assisted proof”. SIAM Journal on Applied Dynamical Systems 1(2), 2002: 215–288.
• [6] Cyranka, J. “Existence of Globally Attracting Fixed Points of Viscous Burg ers Equation with Constant Forcing. A Computer Assisted Proof ”. Submitted, http://www.ii.uj.edu.pl/~cyranka/articles.
• [7] Program package, http://www.ii.uj.edu.pl/~cyranka/Burgers.
• [8] CAPD — Computer Assisted Proofs in Dynamics, a package for rigorous numeric, http://capd.ii.uj.edu.pl.