2,188 research outputs found
On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions
We study the non-autonomously forced Burgers equation
on the space interval with two sets of the boundary conditions:
the Dirichlet and periodic ones. For both situations we prove that there exists
the unique bounded trajectory of this equation defined for all . Moreover we demonstrate that this trajectory attracts all
trajectories both in pullback and forward sense. We also prove that for the
Dirichlet case this attraction is exponential
Rigorous numerics for analytic solutions of differential equations : the radii polynomial approach
Judicious use of interval arithmetic, combined with careful pen
and paper estimates, leads to effective strategies for computer assisted analysis
of nonlinear operator equations. The method of radii polynomials is an
efficient tool for bounding the smallest and largest neighborhoods on which
a Newton-like operator associated with a nonlinear equation is a contraction
mapping. The method has been used to study solutions of ordinary, partial,
and delay differential equations such as equilibria, periodic orbits, solutions
of initial value problems, heteroclinic and homoclinic connecting orbits in the
Ck category of functions. In the present work we adapt the method of radii
polynomials to the analytic category. For ease of exposition we focus on studying
periodic solutions in Cartesian products of infinite sequence spaces. We
derive the radii polynomials for some specific application problems and give a
number of computer assisted proofs in the analytic framework
Rigorous numerical computations for 1D advection equations with variable coefficients
This paper provides a methodology of verified computing for solutions to 1D advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few results of verified numerical computations to initial-boundary value problems of hyperbolic PDEs. Our methodology is based on the spectral method and semigroup theory. The provided method in this paper is regarded as an efficient application of semigroup theory in a sequence space associated with the Fourier series of unknown functions. This is a foundational approach of verified numerical computations for hyperbolic PDEs. Numerical examples show that the rigorous error estimate showing the well-posedness of the exact solution is given with high accuracy and high speed
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