Approximate solution for SPP with discontinuous source term

Abstract

In this paper we shall consider self-adjoint singularly perturbed problem described by the ordinary differential equation of second order with small parameter multiplying the highest derivative and discontinuous source term, and the appropriate boundary conditions, which describes steady state of certain flow problems. The solution displays both boundary layers and an interior layer. The domain decomposition will be performed determining layer subintervals which are adapted to polynomial approximation. The division points for the interior layer are determined by the procedure similar to the one for boundary layers using appropriate resemblance function. 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 pseudospectral technique. The rate of convergence is examined and the order-of-magnitude of the error is given, using the principle of inverse monotonicity and the behavior of the pseudospectral approximations. Numerical example is included and it shows the high accuracy of the presented method. .