An approximation method to solve fractional dynamical system by Bernoulli polynomials

In this paper, we introduce the Bernoulli operational matrices of Caputo fractional derivative and Reimann-Liouville fractional integration and operational matrix of multiplication. Then these matrices are used to approximate numerical solution of fractional-order dynamical systems such as fractional variational problems and nonlinear fractional optimal control problems. These problems are solved by direct method without solving fractional Euler-Lagrange equations and reduced to a system of algebraic equations. The numerical results of four examples are presented to show validity of our method.