Numerical Solution of Fredholm Fractional Integro-differential Equation with Right-Sided Caputo’s Derivative Using Bernoulli Polynomials Operational Matrix of Fractional Derivative

  • Jian Rong Loh
  • Chang PhangEmail author


In this article, fractional integro-differential equation (FIDE) of Fredholm type involving right-sided Caputo’s fractional derivative with multi-fractional orders is considered. Analytical expressions of the expansion coefficient \(c_{k}\) by Bernoulli polynomials approximation have been derived for both approximation of single- and double-variable function. The Bernoulli polynomials operational matrix of right-sided Caputo’s fractional derivative \(\mathbf {P}^{\alpha }_{-;B}\) is derived. By approximating each term in the Fredholm FIDE with right-sided Caputo’s fractional derivative in terms of Bernoulli polynomials basis, the equation is reduced to a system of linear algebraic equation of the unknown coefficients \(c_{k}\). Solving for the coefficients produces the approximate solution for this special type of FIDE.


Fredholm fractional integro-differential equation Right-sided Caputo’s fractional derivative Bernoulli polynomials 

Mathematics Subject Classification

Primary 45B05 Secondary 65R20 



This research is supported by the Ministry of Education Malaysia under the Fundamental Research Grant Scheme (FRGS) Vot K072.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversiti Tun Hussein Onn MalaysiaJohorMalaysia
  2. 2.Foundation in Engineering Faculty of Science and EngineeringThe University of Nottingham MalaysiaSelangorMalaysia

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