Orthogonal series approximation for boundary layers

In this paper we shall consider a class of singularly perturbed problems described by the ordinary differential equation of second order with small parameter multiplying the highest derivative and the appropriate boundary conditions, which describes certain flow problems in fluid mechanics. The solution of such problems displays boundary layers where the solution changes its values very rapidly. The domain decomposition will be performed determining layer subintervals which are adapted to the possibility of spectral approximation. The division point for the boundary layer is determined using appropriate resemblance function, so that the length of the layer subinterval varies with the degree of the truncated orthogonal series. The solution out of boundary layer is approximated by the solution of the reduced problem, and the layer solutions is approximated by truncated orthogonal series giving a smooth approximate solution upon the entire interval. The coefficients of the truncated series are evaluated using collocation technique. The order-of-magnitude of the error is estimated using the principle of inverse monotonicity and the rate of convergence for spectral approximations.