Solving fractional time-delay diffusion equation with variable-order derivative based on shifted Legendre–Laguerre operational matrices

Abstract This article adopts a novel technique to numerical solution for fractional time-delay diffusion equation with variable-order derivative (VFDDEs). As matter of fact, the (VFD) that has been used is in Caputo sense. The first step this constructive on construction using shifted Legendre–Laguerre polynomials unknown coefficients. second involves combination collocation method and operational matrices (OMs) polynomials, as well Newton–Cotes nodal points, find final focuses solving resulting algebraic equations by employing Newton’s iterative method. To illustrate demonstrate technique’s efficacy applicability, two examples have provided.