A spectral shifted gegenbauer collocation method for fractional pantograph partial differential equations and its error analysis

This study presents a new numerical approach for solving fractional-order pantograph partial-differential equations, in which the fractional derivatives are expressed in the Caputo sense. New operational matrices are obtained by introducing the two-variable Gegenbauer polynomials. Using these matrices with the collocation method, solving the fractional order pantograph partial differential equation is converted into solving a system of algebraic equations. An error bound is computed for this method. Also, some examples are presented that show our proposed method has a better agreement with the exact solution in comparison with methods such as the homotopy perturbation and natural decomposition methods so that we can say that it is about 103\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{3}$$\end{document} times more precise than the two mentioned methods. In general, our method provides a useful tool for solving these equations.