Asymptotic expansion method for the two point boundary value problem with small periodic structure (Q1763327)

The authors find an asymptotic expansion for the solution of the following boundary value problem \[ {\frac{d}{dx}}\left[a\left( {\frac{x}{\varepsilon}}\right)\cdot{\frac{du}{dx}}\right]=f(x);\quad u(c)=u_0,\;u(d)=u_1, \] where the function \(a\) is assumed to be \(1\)-periodic, \(f\in C^\infty(c,d)\) and \(\varepsilon\) is a small parameter.