Saber Tavan - Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.
Mohammad Jahangiri Rad - Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.
Ali Salimi Shamloo - Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran.
Yaghoub Mahmoudi - Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

The object of this paper devotes on offering an indirect scheme based on time-fractional Bernoulli functions in the sense of Rieman-Liouville fractional derivative which ends up to the high credit of the obtained approximate fractional Bessel solutions. In this paper, the operational matrices of fractional Rieman-Liouville integration for Bernoulli polynomials are introduced. Utilizing these operational matrices along with the properties of Bernoulli polynomials and the least squares method, the fractional Bessel differential equation converts into a nonlinear system of algebraic. To solve these nonlinear algebraic equations which are a prominent the problem, there is a need to employ Newton’s iterative method. In order to elaborate the study, the synergy of the proposed method is investigated and then the accuracy and the efficiency of the method are clearly evaluated by presenting numerical results.

Fractional-order differential equation, Caputo and Rieman-Liouville fractional derivative and integral, Convergence analysis, Bernoulli functions, Least square method