A fast collocation method for solving the weakly singular fractional integro-differential equation

In the present paper, we propose a spectral collocation method based on Pell polynomials to obtain the solution of a variable-order fractional integro-differential equation with a weakly singular kernel. Fractional integro-differential equations are used in many mathematical models, including heat conduction in memory materials, nuclear reactor dynamics, and chemical kinetics. To provide a numerical scheme, we consider Pell polynomials and use operational matrices to approximate variable-order Caputo fractional derivative as well as integral terms in the main equation. A linear system of equations is formed by collocating the resulting approximate equations. The existence and uniqueness of the solution to the main equation have been proved. The convergence of the approximation solution has been discussed. Several test problems are reported to demonstrate the validity of the method.