Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method

In this study, the numerical solution of Fredholm integro-differential equation is discussed in a reproducing kernel Hilbert space. A reproducing kernel Hilbert space is constructed, in which the initial condition of the problem is satisfied. The exact solution ux is represented in the form of series in the space W22[a,b]. 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 W22[a,b]. 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 Fredholm integro-differential equations.