Computer Assisted Fourier Analysis in Sequence Spaces of Varying Regularity

Abstract

This work treats a functional analytic framework for computer assisted Fourier analysis which can be used to obtain mathematically rigorous error bounds on numerical approximations of solutions of differential equations. An abstract a-posteriori theorem is employed in order to obtain existence and regularity results for C k problems with 0 < k ≤ ∞ or k = ω. The main tools are certain infinite sequence spaces of rapidly decaying coefficients: we employ sequence spaces of algebraic and exponential decay rates in order to characterize the regularity our results. We illustrate the implementation and effectiveness of the method in a variety of regularity classes. We also examine the effectiveness of spaces of algebraic decays for studying solutions of problems near the breakdown of analyticity.

Computer-assisted proofs for some nonlinear diffusion problems

“…For problems where analyticity is too much too ask for, be it because the solution itself is not analytic, or because we care about local uniqueness but proving the analyticity a priori is hard, it should be noted that one can easily replicated the whole procedure presented here in a sequence space corresponding to functions of lower regularity, see e.g. [17].…”

“…These nonlinear diffusion terms can give rise to a repulsive effect which leads to spatial segregation: the two species co-exist but mostly concentrate in different regions of the domain Ω. Mathematically, this corresponds to the existence of non-homogeneous steady states of (17) exhibiting this type of patterns. The existence and stability analysis of such steady states has been studied extensively, through a wide variety of techniques such as bifurcation theory, singular perturbation theory or fixed point index theory, see e.g.…”

“…The computer-assisted study of the steady states of the SKT system started first via a system with linear diffusion approximating (17) in [9], and then for (17) itself in [4], but only in the so-called triangular case. This case corresponds to taking the self-diffusion coefficients d 11 and d 22 equal to 0, and one of the cross-diffusion coefficient (say d 21 ) equal to 0.…”

A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

“…In the latter case, when the vector field is not analytic, one should change the weights in the norm to be algebraic rather than exponential, cf. [27]. We are confident that either approach can be readily combined with the methods presented in the current paper.…”

“…. , M`N we have (27) with the (unbounded) diagonal operator k defined by (with w P C Z ) pkwq k " kw k for k P Z.…”