A new application of Taylor expansion for approximate solution of systems of Fredholm integral equations

Systems of Fredholm integral equations are essential in engineering fields for modeling complex systems involving dynamic behavior, material properties, and interactions between different forces or fluids. In this research, the Taylor expansion approach is used to solve the systems of linear Fredholm integral equations (SLFIEs) of the second kind in a novel manner. By combining the mth-order Taylor polynomial of unknown functions at an arbitrary point and employing the repeated integration method; the given system of linear Fredholm integral equations is transformed into a new system of linear equations of unknown functions and their derivatives. Eventually, this new system is solved to obtain demanded mth-order approximate solutions. An error analysis is given as well as several numerical examples to demonstrate the efficiency, accuracy, and ability of the proposed method. Furthermore, this method always leads to the exact solution if the exact solution is a polynomial of degree less than or equal to m.