The dependence of critical parameter bounds on the monotonicity of a Newton sequence

Abstract

Summary. A new method is proposed for the inclusion of the critical parameter 2* of some convex operator equation u=2 Tu (appearing e.g. in thermal explosion theory). It is based on the fact that for a fixed 2 Newton's method starting with a suitable subsolution is not monotonically if and only if 2 > 2*. Several numerical examples arising from nonlinear boundary value problems illustrate the efficiency of the method.

Exact bounds for the solution branches of nonlinear eigenvalue problems

“…By an analytic formula, 2*= 3.5138307 .... With m=2560 equidistant intervals and n = 10 iteration steps for r= 1.186 we obtain from (2.11) the lower bound 2* > 3.513827. Mooney-Voss-Werner [7] obtained for the critical value 22/20 associated to a with h = 1/20 discretisized analogue of (3.6):0.6863 < 2*/2 o ~ 0.6864. Our algorithm yields with m = 1280 equidistant intervals, n = t0 steps and r = 0.7000 for the exact 2" the bound 2* >0.686319.…”