Abstract: Based on a class of orthogonal polynomials, alternative Legendre polynomials (ALPs), we proposed a numerical calculation method for fractional pantograph delay differential equations. Firstly, we obtained the numerical approximation results of fractional calculus by using the properties of ALPs. Secondly, we  transformed the fractional pantograph delay differential equations into the  algebraic system for obtaining solution. Thirdly, we analyzed  the error  of the method  and obtained the convergence results of the method. Finally, some numerical examples were given to verify the effectiveness and accuracy of the proposed method.

Key words: alternative Legendre polynomials (ALPs), fractional order, pantograph, delay differential equation, numerical solution