An approximate approach for the generalized variable-order fractional pantograph equation

This study advances variable-order (VO) fractional delay differential models in the pantograph type introduced in the Caputo sense. A method utilizing the Chebyshev cardinal functions (CCFs) is formulated to find an accurate result. In the proposed scheme, we first expand the solution in terms of the CCFs including unknown coefficients. Then, by inserting this expansion into the problem, employing the VO fractional differentiation operational matrix (OM) and the delay OM, we construct a system with algebraic equations. The generated algebraic systems are mainly sparse due to the cardinality of these functions and it significantly reduces the cost of computation.