Spectral Collocation Method for Handling Integral and Integrodifferential Equations of n-th Order via Certain Combinations of Shifted Legendre Polynomials

In this study, an accurate and efficient numerical method based on spectral collocation is presented to solve integral equations and integrodifferential equations of n-th order. The method is developed using compact combinations of shifted Legendre polynomials as a spectral basis and shifted Legendre–Gauss–Lobatto nodes as collocation points to construct the appropriate algorithm that makes simple systems easy to solve. The technique treats both types of equations: linear and nonlinear equations. The study aims to provide the relevant spectral basis by the use of compact combinations, which allows us to take advantage of shifted Legendre polynomials and to reduce the dimension of the space of approximation. The reliability of the proposed algorithms is proven via different examples of several cases and the results are discussed to confirm the effectiveness of the spectral approach.