A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm–Volterra type

In this paper, the numerical solution of periodic Fredholm–Volterra integro–differential equations of first-order is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the periodic condition of the problem is satisfied. The exact solution u(x)ux is represented in the form of series in the space W22. In the mean time, the n  -term approximate solution un(x)unx is obtained and is proved to converge to the exact solution u(x)ux. Furthermore, we present an iterative method for obtaining the solution in the space W22. 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.