Spectral Galerkin schemes for a class of multi-order fractional pantograph equations

In this paper, we study and present a spectral numerical technique for solving a general class of multi-order fractional pantograph equations with varying coefficients and systems of pantograph equations. In this study, the spectral Galerkin approach in combination with the properties of shifted Legendre polynomials is used to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with fractional pantograph equations via spectral Galerkin approach. The errors and convergence of the adopted approach are rigorously analyzed. The efficiency and accuracy of the technique are tested by considering five different examples, to ensure that the suggested approach is more accurate than the existing other techniques. The obtained results in this paper are comparing favorably with those published by other researchers and with the existing exact solutions, whenever possible.