Exact and Numerical Solution of Abel Integral Equations by Orthonormal Bernoulli Polynomials

  • Mithilesh SinghEmail author
  • Shivani Singhal
  • Nidhi Handa
Original Paper


In this study, we use the Bernoulli polynomials, Bernoulli number to construct the orthonormal polynomials by using Gram–Schmidt orthogonalization. We use the function approximation on orthonormal polynomials, apply the integration operator on these orthonormal polynomials and obtain tri-diagonal operational matrix of integration. Apply the tri-diagonal operational matrix on many Abel-type integral equations and change all integral equations to the algebraic equation solutions. The exact and approximate solution of Abel-type integral equations by this new method will be found out. To illustrate the applicability, efficiency and accuracy of suggested scheme, some numerical examples of Abel-type integral equations are implemented and the comparisons are given by exact solutions.


Bernoulli polynomials Orthonormal Bernoulli Polynomials Tri-diagonal operational matrix Abel integral equations 



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© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of Applied ScienceRajkiya Engineering CollegeSonbhadraIndia
  2. 2.Department of Mathematics, (KGC)Gurukula Kangari VishwavidyalayaHaridwarIndia

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