Numerical solution of quasilinear singularly perturbed ordinary differential equation without turning points (Q1178002)

For the quasilinear problem \[ \varepsilon{d^ 2u \over dx^ 2}+{d \over dx}f(u)-g(x,u)=0,\qquad 0<x<1, \] \(u(0)=A\), \(u(1)=B\), the authors give (via integration) an equivalent formulation which allows the iteration method \[ \varepsilon{dy_{n+1} \over dx}+My_{n+1}=\bar G(x)-\bar K- f(y_ n)+My_ n,\quad y_{n+1}(0)=A,\qquad n=0,1,\dots. \] These problems are solved by exponentially fitted difference schemes. For the error one has \(| u(x_ i)-y^{(i)}_ n|\leq c(\varepsilon+h+(1- \alpha M^{-1})^ n)\), that means the method works well for sufficiently small \(\varepsilon\).