Finite difference schemes for second order periodical boundary value problems

In this study, second- order periodical boundary value problems are considered. Difference schemes are constructed by using finite difference methods in a uniform mesh. The algorithms are those consist of classical difference schemes and for singularly perturbed problem, 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. It is proved that the schemes converge uniformly to the solition of differential equations. Numerical experiments support these teorical results.