Fractional-order shifted Legendre collocation method for solving non-linear variable-order fractional Fredholm integro-differential equations

Fractional differential equations have been adopted for modeling many real-world problems, namely those appearing in biological systems since they can capture memory and hereditary effects. In this paper, an efficient and accurate method for solving both one-dimensional and systems of nonlinear variable-order fractional Fredholm integro-differential equations with initial conditions is proposed. The method is based on the fractional-order shifted Legendre–Gauss–Lobatto collocation technique for fractional-order Riemann–Liouville derivative. The effectiveness and validity of the numerical approach are illustrated by solving four distinct problems.