A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition

Fogelklou, Oswald; Tucker, Warwick; Kreiss, Gunilla; Siklosi, Malin

doi:10.1016/j.cnsns.2010.07.008

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…Other techniques reduce the problem to compute the norm of an operator (as in Example 1.2) and apply a fixed point theorem, or use topological methods (Conley index). They have been successfully applied for instance in computing the following: Conley index for Kuramoto-Sivashinsky [142], bifurcation diagram for stationary solutions of Kuramoto-Sivashinsky [3], stationary solutions of viscous 1D Burgers with boundary conditions [69], traveling wave solutions for 1D Burgers equation [68], periodic orbits of Kuramoto-Sivashinsky [66,141,67,73,4], Conley index for the Swift-Hohenberg equation [51], bifurcation diagram of the Ohta-Kawasaki equation [136], stability of periodic viscous roll waves of the KdV-KS equation [6], existence of hexagons and rolls for a pattern formation model [134], self-similar solutions of a 1D model of 3D axisymmetric Euler [97], and many others.…”

Section: Historymentioning

confidence: 99%

Computer-assisted proofs in PDE: a survey

Gómez-Serrano¹

2018

Preprint

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Section: Historymentioning

confidence: 99%

Computer-assisted proofs in PDE: a survey

Gómez-Serrano¹

2018

Preprint

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“…The methods in [22,12,30,33,27,9,8] focused on second order equations; [22] worked with an equivalent integral formulation while [33,27] used a finite element based method. [8] treated an integral boundary condition.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error bounds for two point boundary value problems: A green's function approach

Birrell¹

2015

Journal of Computational Dynamics

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We present a computer assisted method for generating existence proofs and a posteriori error bounds for solutions to two point boundary value problems (BVPs). All truncation errors are accounted for and, if combined with interval arithmetic to bound the rounding errors, the computer generated results are mathematically rigorous. The method is formulated for n-dimensional systems and does not require any special form for the vector field of the differential equation. It utilizes a numerically generated approximation to the BVP fundamental solution and Green's function and thus can be applied to stable BVPs whose initial value problem is unstable. The utility of the method is demonstrated on a pair of singularly perturbed model BVPs and by using it to rigorously show the existence of a periodic orbit in the Lorenz system.

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“…Let us, however, mention a few key papers to briefly sketch what kind of methods have been developed by the rigorous numerics community. In [11,21,9,45,2,1] functional analytic methods, similar in spirit to ours, are used to solve BVPs: the differential equation is reformulated into an equivalent fixed-point problem and is solved by verifying the conditions of the Contraction Mapping Principle with the aid of a computer. Fundamentally different approaches based on topological rather than functional analytic methods, such as the Conleyindex and covering relations, have been proven to be very effective as well (see e.g.…”

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confidence: 99%

Rigorous numerics for ODEs using Chebyshev series and domain decomposition

Berg¹,

Sheombarsing²

2021

JCD

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<p style='text-indent:20px;'>In this paper we present a rigorous numerical method for validating analytic solutions of nonlinear ODEs by using Chebyshev-series and domain decomposition. The idea is to define a Newton-like operator, whose fixed points correspond to solutions of the ODE, on the space of geometrically decaying Chebyshev coefficients, and to use the so-called radii-polynomial approach to prove that the operator has an isolated fixed point in a small neighborhood of a numerical approximation. The novelty of the proposed method is the use of Chebyshev series in combination with domain decomposition. In particular, a heuristic procedure based on the theory of Chebyshev approximations for analytic functions is presented to construct efficient grids for validating solutions of boundary value problems. The effectiveness of the proposed method is demonstrated by validating long periodic and connecting orbits in the Lorenz system for which validation without domain decomposition is not feasible.</p>

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A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition

Cited by 4 publications

References 15 publications

Computer-assisted proofs in PDE: a survey

Computer-assisted proofs in PDE: a survey

A posteriori error bounds for two point boundary value problems: A green's function approach

Rigorous numerics for ODEs using Chebyshev series and domain decomposition

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