SeMA Journal

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Explicit representation of orthonormal Bernoulli polynomials and its application for solving Volterra–Fredholm–Hammerstein integral equations

  • Farshid MirzaeeEmail author
  • Nasrin Samadyar


Bernoulli polynomials have many useful properties, but the main disadvantage of these polynomials is that they are not orthogonal. In this paper, we offer an explicit representation of orthonormal Bernoulli polynomials (OBPs) for the first time. We show that these polynomials can be created from a linear combination of standard basis polynomials. Although these polynomials can be applied to solve various problems, in this article we will use them to solve nonlinear Volterra–Fredholm–Hammerstein integral equations. By using this approach, nonlinear integral equation converts to a systems of nonlinear equations which can be solved via an appropriate numerical method such as Newton’s method. Also, we prove some theorems and use them to get an upper error bound of proposed method. Finally, some numerical examples are given to demonstrate pertinent features of this method. Also, we compare obtained results from this method with the achieved results from relevant studies.


Volterra–Fredholm–Hammerstein integral equations Bernoulli polynomials Gram–Schmidt process Error analysis 

Mathematics Subject Classification

45B05 45D05 45G10 65D30 65R20 



The authors would like to express our very great appreciation to anonymous reviewers for their valuable comments and constructive suggestions which have helped to improve the quality and presentation of this paper.


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Copyright information

© Sociedad Española de Matemática Aplicada 2019

Authors and Affiliations

  1. 1.Faculty of Mathematical Sciences and StatisticsMalayer UniversityMalayerIran

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