Analysis and numerical simulation of positive and dead-core solutions of singular two-point boundary value problems

We investigate the solvability of the Dirichlet boundary value problem u″(t)=λg(u(t)),λ≥0,u(0)=1,u(1)=1, where λλ is a nonnegative parameter. We discuss the existence of multiple positive solutions and show that for certain values of λλ, there also exist solutions that vanish on a subinterval [ρ,1−ρ]⊂(0,1)[ρ,1−ρ]⊂(0,1), the so-called dead-core solutions. In order to illustrate the theoretical findings, we present computational results for g(u)=1/u, computed using the collocation method implemented in bvpsuite, a new version of the standard MATLAB code sbvp1.0.