Legendre spectral collocation method for Fredholm integro-differential-difference equation with variable coefficients and mixed conditions

In this article, the Legendre spectral collocation method has been applied to solve Fredholm integro-differential-difference equations with variable coefficients. The proposed method is based on the Gauss–Legendre points with the basis functions of Lagrange polynomials. Usually, this type of integral equations are very difficult to solve analytically as well as numerically. The presented method applied to the integral equation reduces to solve the system of algebraic equations. Also the numerical results obtained by Legendre spectral collocation method have been compared with the results obtained by existing methods. Illustrative examples have been discussed to demonstrate the validity and applicability of the presented method.