Muntz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations

This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Muntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Muntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Muntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.