Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation

Lessard, Jean‐Philippe

doi:10.1016/j.jde.2009.11.008

Cited by 52 publications

(81 citation statements)

References 19 publications

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“…In the context of infinite dimensional problems like PDEs and DDEs, the continuation algorithms have to be performed on a finite dimensional projection, which raises the natural question of the validity of the outputs. In order to address this fundamental question, rigorous one-parameter continuation methods have been proposed to compute global branches of solutions of PDEs and DDEs [1,2,3,4,5]. While these methods have been applied to compute one-dimensional manifolds, we are not aware of any rigorous method aiming at computing solution manifolds of dimension greater than one.…”

Section: Introductionmentioning

confidence: 99%

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Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

2015

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In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a twodimensional manifold of equilibria of the Cahn-Hilliard equation.

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Section: Introductionmentioning

confidence: 99%

“…The problem (1) is reduced to a finite dimensional one using a spectral Galerkin projection. See (5) for the details of this finite dimensional projection.…”

Section: Introductionmentioning

confidence: 99%

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

2015

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“…Since then, it was adapted to many different situations, e.g. to the study of higher-dimensional PDEs [9], delay equations [17], Euler-Lagrange equations [4], radially symmetric localized solutions of PDEs [22] and many more. In these previous work, the computer-assisted proofs were all obtained in Banach spaces of solutions with low regularity.…”

Section: Introductionmentioning

confidence: 99%

Blow-up profile for solutions of a fourth order nonlinear equation

D’Ambrosio

Lessard

Pugliese

2015

Nonlinear Analysis: Theory, Methods & Applications

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It is well known that the nontrivial solutions of the equationblow up in finite time under suitable hypotheses on the initial data, κ and f . These solutions blow up with large oscillations. Knowledge of the blow-up profile of these solutions is of great importance, for instance, in studying the dynamics of suspension bridges. The equation is also commonly referred to as extended Fisher-Kolmogorov equation or Swift-Hohenberg equation.In this paper we provide details of the blow-up profile. The key idea is to relate this blow-up profile to the existence of periodic solutions for an auxiliary equation.

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“…For instance, time periodic solutions of ODEs [2,3], stationary solutions of PDEs with periodic or Neumann boundary conditions [4,5,6,7], time periodic solutions of delay differential equations [8,9] and invariant sets of infinite dimensional maps [10] have been successfully computed using Fourier series and rigorous numerics. However, to the best of our knowledge, this is the first time that a method based on Chebyshev series is presented to rigorously compute solutions of nonlinear differential equations.…”

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confidence: 99%

“…The goal is to develop a rigorous computational method based on Chebyshev series to compute solutions of (1.7). Given the Chebyshev expansion (1.2) of u with a = (a k ) k≥0 the infinite vector of Chebyshev coefficients, consider the Chebyshev expansion of Ψ(u) given by 8) where…”

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Rigorous Numerics for Nonlinear Differential Equations Using Chebyshev Series

Lessard¹,

Reinhardt²

2014

SIAM J. Numer. Anal.

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Abstract. A computational method based on Chebyshev series to rigorously compute solutions of initial and boundary value problems of analytic nonlinear vector fields is proposed. The idea is to recast solutions as fixed points of an operator defined on a Banach space of rapidly decaying Chebyshev coefficients and to use the so-called radii polynomials to show the existence of a unique fixed point nearby an approximate solution. As applications, solutions of initial value problems in the Lorenz equations and symmetric connecting orbits in the Gray-Scott equation are rigorously computed. The symmetric connecting orbits are obtained by solving a boundary value problem with one of the boundary values in the stable manifold.Key words. Rigorous numerics, chebyshev series, nonlinear ODEs, boundary value problems, initial value problems, contraction mapping theorem AMS subject classifications. 34B99, 34A34, 34A12, 42A10, 65L05, 65L101. Introduction. In this paper, we propose a rigorous numerical method based on the Chebyshev polynomials to compute solutions of nonlinear differential equations. More explicitly, the field of rigorous numerics develops algorithms that provide approximate solutions to a problem together with precise bounds within which exact solutions are guaranteed to exist in the mathematically rigorous sense. In this context, the main idea of our proposed approach is to expand the solution of a given differential equation using its Chebyshev series, plug the expansion in the equation, obtain an equivalent infinite dimensional problem of the form f (x) = 0 to solve in a Banach space of rapidly decaying Chebyshev coefficients and to get the existence, via a fixed point argument, of a genuine solution of f (x) = 0 nearby a numerical approximation of a finite dimensional projection of f . The fixed point argument is solved by using the radii polynomials (e.g. see [1]), which provide an efficient way of constructing a set on which the contraction mapping theorem is applicable.Before proceeding further, it is worth mentioning that a similar approach based on Fourier series is widely used in the field of rigorous numerics to compute solutions of differential equations with periodic profiles. For instance, time periodic solutions of ODEs [2,3], stationary solutions of PDEs with periodic or Neumann boundary conditions [4,5,6,7], time periodic solutions of delay differential equations [8,9] and invariant sets of infinite dimensional maps [10] have been successfully computed using Fourier series and rigorous numerics. However, to the best of our knowledge, this is the first time that a method based on Chebyshev series is presented to rigorously compute solutions of nonlinear differential equations. Since a large class of solutions of differential equations are non periodic (e.g. solutions of initial value problems (IVPs) and boundary value problems (BVPs) with non periodic boundary values), we believe that our proposed approach is a valuable contribution to the field of rigorous numerics. Also, since Chebyshev seri...

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Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation

Cited by 52 publications

References 19 publications

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

Blow-up profile for solutions of a fourth order nonlinear equation

Rigorous Numerics for Nonlinear Differential Equations Using Chebyshev Series

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