Numerical Algorithms

, Volume 77, Issue 4, pp 1283–1305 | Cite as

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

  • P. Rahimkhani
  • Y. Ordokhani
  • E. Babolian
Original Paper


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 Müntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Müntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-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.


Müntz-Legendre wavelet Fractional pantograph differential equations Caputo derivative Operational matrix 


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Authors are very grateful to one of the reviewers for carefully reading the paper and for his(her) comments and suggestions which have improved the paper.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesAlzahra UniversityTehranIran
  2. 2.Department of Computer Science, Faculty of Mathematical Sciences and ComputerKharazmi UniversityTehranIran

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