Continuation of global solution curves using global parameters

TL;DR

This paper develops a method for computing global solution curves of differential equations by continuation in global parameters, including numerical techniques and applications to various equations.

Contribution

It introduces a direct continuation method for global solution curves, incorporating a simplified regularizing transformation for better numerical accuracy.

Findings

Global solution curves can be computed automatically as parameters vary.

The regularizing transformation improves numerical accuracy for p-Laplace equations.

Bifurcation diagrams are obtained for non-autonomous and fourth order elastic beam problems.

Abstract

This paper provides both the theoretical results and numerical calculations of global solution curves, by continuation in global parameters. Each point on the solution curves is computed directly as the global parameter is varied, so that all of the turns that the solution curves make, as well as its different branches, appear automatically on the computer screen. For radial $p$-Laplace equations we present a simplified derivation of the regularizing transformation from P. Korman [15], and use this transformation for more accurate numerical computations. While for $p>2$ the solutions are not of class $C^2$, we show that they are of the form $w(r^{\frac{p}{2(p-1)}})$, where $w(z)$ is of class $C^2$. Bifurcation diagrams are also calculated for non-autonomous problems, and for the fourth order equations modeling elastic beams. We show that the first harmonic of the solution can also serve as a global parameter.