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

## Abstract

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.

## Keywords

Fredholm fractional integro-differential equation Right-sided Caputo’s fractional derivative Bernoulli polynomials## Mathematics Subject Classification

Primary 45B05 Secondary 65R20## Notes

### Acknowledgements

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

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