Abstarct: In this thesis, the theoretical foundations and numerical applications of Bernoulli polynomials in solving multi-term fractional differential equations are investigated. First, a new determinantal definition of these polynomials baxsed on the Hessenberg matrix is introduced, and its equivalence with classical approaches is demonstrated. Subsequently, by employing fractional operational matrices and the collocation method, a numerical method is developed for solving multi-term fractional differential equations, which exhibits high accuracy and efficiency. Furthermore, by combining Bernoulli polynomials with block pulse functions, a novel hybrid method is presented that achieves faster convergence and lower error in solving linear and nonlinear fractional equations. The results of this research indicate the high capability of Bernoulli polynomials in providing stable and accurate solutions for fractional problems in both pure and applied mathematics.