An efficient numerical approximation for the linear class of Fredholm integro-differential equations based on Cattani’s method

This paper provides with a generalization of the work by Cattani (Math. Probl. Eng. (2008) 1–24), who has introduced the connection coefficients of the Shannon wavelets. We apply the Shannon wavelets approximation based on Cattani’s connection coefficients together the collocation points for solving the linear Fredholm integro-differential equations. Finally, numerical results are included to demonstrate the validity and applicability of the method and some comparisons are made with existing results.

Research highlights
► By using the collocation method and the connection coefficients, Fredholm integro-differential Equation (1) transform to an algebraic system.
► Basis in expansion method are usually chosen so that f˜(t) automatically satisfies the boundary conditions. It means, we can compute an approximate solution by using Eq. (1). In this case, Eq. (2) converts to an additional equation and it seems to be ignored in the algebraic system.
► We observe significant improvements have been obtained compared with the numerical results reported by others.