An accurate computation of the global solution curve for the Gelfand problem through a two point approximation

摘要

When one solves numerically the Gelfand boundary value problem in two dimensions (with u=u(x,y), r=x2+y2 and λ a positive parameter)Δu+λeu=0forr<1,u=0whenr=R.with 10 mesh points, two spurious turns appear on the solution curve. When one increases the number of mesh points, the spurious turns persist, occurring at smaller values of λ. It turns out that spurious turns are avoided in the other direction, by using two to six mesh points. We prove this in case of two mesh points. This gives us the correct form of the solution curve, and the accuracy can then be improved by using a linear search on u(0), combined with shooting.