10. Inner and outer approximations.

Again consider the problem

It can be showed that the exact solution y=y(t) is

We now try to find inner and outer approximations to the problem:

1) If we let we get the unperturbed problem

By searching for the solution that fulfills the outer boundary condition

we get an outer approximation

that well agrees with the solution for “large” t (t close to 1).

2) For “small” t (t close to 0, belongs to the boundary layer) a good inner approximation is

We will in part 12 examine how to find this.