Singular perturbation of nonlinear boundary value problems (Q1061288)

The authors obtain the formal asymptotic solution of the following nonlinear boundary value problem: \(\epsilon y''=f(x,y,y',\epsilon,\mu)\) \((\mu <x<1-\mu)\), \(y(x,\epsilon,\mu)|_{x=\mu}=\phi_ 0(\epsilon,\mu)\), \(y(x,\epsilon,\mu)|_{x=1-\mu}=\phi (\epsilon,\mu)\) where \(\epsilon >0\) and \(\mu >0\) are two small parameters, and \(f_ y\leq -k<0\) (k is a positive constant). For \(\epsilon =0\), \(\mu =0\) the perturbed problem degenerates into the unperturbed problem: \(f(x,y,y',0,0)=0\) \((0<x<1)\), \(y(1,0,0)=\phi_ 1(0,0)\). Further, it is assumed that y(x,\(\epsilon\),\(\mu)\), f(x,y,y',\(\epsilon\),\(\mu)\), \(\phi_ 1(\epsilon,\mu)\) have asymptotic double series expansions in \(\epsilon\) and \(\mu\), valid as these parameters both tend to zero.