Reproducing kernel Hilbert space method for high-order linear Fredholm integro-differential equations with variable coefficients

In this study, a novel reproducing kernel Hilbert space (RKHS) method is introduced to show that high-order linear Fredholm integro-differential equations (IDEs) with variable coefficients can be transformed into ordinary differential equation (ODEs). The RKHS method constructs multiple types of RKHSs related to the given terms based on the H-$H_K$ formulation, which are utilized to determine solutions of the Fredholm IDEs. Then analytical and numerical solutions of the Fredholm IDEs with variable coefficients are obtained by an algorithm. Finally, the effectiveness and feasibility of RKHS method have been provided to confirm our theoretical findings by some numerical results and comparisons.