Reconstruction of exponentially rate of convergence to Legendre collocation solution of a class of fractional integro-differential equations

In this paper, Legendre Collocation method, an easy-to-use variant of the spectral methods for the numerical solution of a class of fractional integro-differential equations (FIDE’s), is researched. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by using Legendre Gauss quadrature formula. An efficient convergence analysis of the proposed method is given and rate of convergence is established in the L2L2-norm. Due to the fact that the solutions of FIDE’s usually have a weak singularity at origin, we use a variable transformation to change the original equation into a new equation with a smooth solution. We prove that after this regularization technique, numerical solution of the new equation by adopting the Legendre collocation method has exponentially rate of convergence. Numerical results are presented which clarify the high accuracy of the proposed method.