This article, presented a shifted Legendre Gauss‐Lobatto collocation (SL‐GL‐C) method which is introduced for solving variable‐order fractional Volterra integro‐differential equation (VO‐FVIDEs) subject to initial or nonlocal conditions. Based on shifted Legendre Gauss‐Lobatto (SL‐GL) quadrature, we treat with integral term in the aforementioned problems. Via the current approach, we convert such problem into a system of algebraic equations. After that we obtain the spectral solution directly for the proposed problem. The high accuracy of the method was proved by several illustrative examples.

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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.