A uniformly convergent difference method for the periodical boundary value problem

The periodical boundary value problem for linear second-order ordinary differential equation with small parameter by the first and second derivatives is considered. By the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form, an exponentially fitted difference scheme is constructed in a uniform mesh which gives first-order uniform convergence in the discrete maximum norm. Numerical experiments support these theoretical results