An operational approach for solving fractional pantograph differential equation

The aim of the current paper is to construct the shifted fractional-order Jacobi functions (SFJFs) based on the Jacobi polynomials to numerically solve the fractional-order pantograph differential equations. To achieve this purpose, first the operational matrices of integration, product, and pantograph, related to the fractional-order basis, are derived (operational matrix of integration is derived in Riemann–Liouville fractional sense). Then, these matrices are utilized to reduce the main problem to a set of algebraic equations. Finally, the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, some theorems are presented on existence of solution of the problem under study and convergence of our method.