Ritz-generalized Pell wavelet method: Application for two classes of fractional pantograph problems

Abstract

In this work, a computational method for solving fractional pantograph differential equations and fractional pantograph optimal control problems is proposed. The present technique is based on a new set of wavelet functions and a combination of Ritz and collocation methods. To this aim, we construct generalized Pell wavelets (GPws). An extra Caputo pseudo-operational matrix and pantograph operational matrix of GPws as new achievements are presented. To more easily calculate the fractional derivative of GPws, we define generalized piecewise Taylor functions (GPTfs). Then, utilizing these matrices, the Ritz method, and the collocation method, we find an approximate solution for each of the considered problems. An error analysis is proposed. Finally, some illustrative numerical tests are given to display the accuracy and effectiveness of the developed scheme.