Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations

In this study, the numerical solution of Fredholm-Volterra integro-differential equations for two-point, second-order periodic boundary value problems is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the periodic boundary conditions of the problem are satisfied. The exact solution u(x) is represented in the form of series in the space W23. In the mean time, the n-term approximate solution ^un(x) is obtained and is proved to converge to the exact solution u(x). Furthermore, we present an iterative method for obtaining the solution in the space W23. Some examples are displayed to demonstrate the validity and applicability of the proposed method. The numerical result indicates that the proposed method is straightforward to implement, efficient, and accurate for solving linear and nonlinear equations.