Global smooth solution curves using rigorous branch following

Abstract

Abstract. In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f : R l 1 ×B 1 → R l 2 ×B 2 , where B 1 and B 2 are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.

Stationary Coexistence of Hexagons and Rolls via Rigorous Computations

“…All errors due to truncation are estimated analytically (see section 4), with all bounds expressed explicitly in terms of analytically known constants and in terms of the data of u num . In these estimates we keep the radius of the ball as a parameter (as in [11,14,15,16,17,18]) in order to retain the flexibility to tune the radius. With the assistance of the computer we then check that the operator is indeed a contraction on balls with small (but not too small) radius (see section 2.3) leading to a unique fixed point, hence a unique heteroclinic solution in a small neighborhood around u num .…”

Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

“…2 and we use the explicit radii polynomials constructed in Sect. 3 to the Cahn-Hilliard equation (1), for the case c = 0 in (2). Taking the domain as where L j = π/ j , for j = 1, .…”

“…To define the bounds Y k , one needs to compute ∂ 2 μ k ∂λ 2 (λ 0 ), μ (3) k from (19), β k (x) and β k (ẋ) given by (20), y (1) k , y (2) k and y (3) k given respectively by (21), (22) and (23), and finally μ * k given by (24). First notice that…”

“…That raises the question of computational efficiency and applicability of these techniques to higher-dimensional PDEs. The rigorous continuation method developed in [1,2,[9][10][11]15] was precisely introduced to address these kind of computational efficiency issues.…”

“…However, to the best of our knowledge, these techniques have never been applied to equations defined on domains of dimension greater than two. Our new proposed method combines the theory of rigorous branch following introduced in [2] and the analytic estimates constructed in [10]. Using this method, one can obtain proofs of existence of equilibria for continuous parameter ranges, existence and smoothness of global solution curves and about local uniqueness of the solutions on the curves.…”