Continuation of global solution curves using global parameters

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 their 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 (2015), and use this transformation for more accurate numerical computations. While for p>2 the solutions are not of class C2, we show that they are of the form w(rp2(p−1)), where w(z) is of class C2. 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.