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.