Rigorous Integration of Burgers Equation

• Autorzy: Cyranka J.
• Języki publikacji: EN

Abstrakty

EN

This paper presents techniques that allows to rigorously integrate dissipative partial differential equations. A full case study of an application to the Burgers equation on the line with periodic boundary conditions is presented.

Słowa kluczowe

EN

rigorous numerics; Lohner algorithm; Burgers equation; rigorous integration; dissipative partial differential equations; interval arithmetics

• Wydawca: Foundation for Young Scientists
• Czasopismo: Challenges of Modern Technology
• Rocznik: 2011
• Tom: Vol. 2, no. 3
• Strony: 3--7
• Opis fizyczny: Bibliogr. 12 poz., tab.

Twórcy

autor

Cyranka J.

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

Bibliografia

• [1] Lohner, R.J. “Computation of Guaranteed Enclosures for the Solutions of Ordinary Initial and Boundary Value Problems”. Computational Ordinary Differential Equations, J.R. Cash, I. Gladwell Eds., Oxford: Clarendon Press, 1992.
• [2] Zgliczyński, P. “C1-Lohner algorithm”. Foundations of Computational Mathematics 2 (2002): 429–465.
• [3] Mrozek, M., and K. Mischaikow. “Chaos in Lorenz equations: a computer assisted proof ”. Bull. Amer. Math. Soc. (N.S.), 33 (1995): 66–72.
• [4] Kapela, T., and P. Zgliczyński. „The existence of simple choreographies for the N-body problem — a computer assisted proof ”. Nonlinearity 16 (2003): 1899–1918.
• [5] Tucker, W. “A Rigorous ODE Solver and Smale’s 14th Problem”. Found. Comput. Math. 2 (2002): 53–117.
• [6] Zgliczyński, P. and K. Mischaikow. „Rigorous Numerics for Partial Differential Equations: the Kuramoto--Sivashinsky equation”. Foundations of Computational Mathematics 1 (2001): 255–288.
• [7] Zgliczyński, P. “Attracting fixed points for the Kuramoto--Sivashinsky equation — a computer assisted proof ”. SIAM Journal on Applied Dynamical Systems 1(2), 2000: 215–288.
• [8] Zgliczyński, P. “Rigorous numerics for dissipative Partial Diff erential Equations II. Periodic orbit for the Kuramoto-Sivashinsky PDE -a computer assisted proof”. Foundations of Computational Mathematics 4 (2004): 157–185.
• [9] Zgliczyński, P. “Rigorous Numerics for Dissipative PDEs III. An effective algorithm for rigorous integration of dissipative PDEs”. Topological Methods in Nonlinear Analysis. Preprint available online http://www.ii.uj.edu.pl/zgliczyn/papers/publ.htm
• [10] Cyranka, J. “Rigorous integration of Burgers equation” [in preparation].
• [11] CAPD — Computer Assisted Proofs in Dynamics, a package for rigorous numeric, http://capd.ii.uj.edu.pl.
• [12] Fefferman, Ch. “Existence and Smoothness of the Navier-Stokes Equation”. The Millenium problem description. Available online http://www.claymath.org/millennium/NavierStokes_Equations/.