New operational matrix for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative

Subject Areas : Statistics

M.  alipour ¹ (Department of Mathematics, Babol Noshirvani University of Technology, Shariati Ave., Babol, Iran)
P.  allahgholi ² (Department of Mathematics, Babol Noshirvani University of Technology, Shariati Ave., Babol, Iran)

Received: 2017-03-10 Accepted : 2017-05-23 Published : 2017-05-01

Keywords: مسائل کنترل بهینه کسری, چندجمله‌ای‌های برنشتاین, ماتریس عملیاتی, مشتق کسری ریمان–لیوویل اصلاح ش, حساب دیفرانسیل کسری,

Abstract :

In this paper, we apply spectral method based on the Bernstein polynomials for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative. In the first step, we introduce the dual basis and operational matrix of product based on the Bernstein basis. Then, we get the Bernstein operational matrix for the Jumarie’s modified Riemann-Liouville fractional derivative, which has not been undertaken before. By using the function approximations based on the Bernstein basis and mentioned operational matrices, the optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative is reduced to a system of algebraic equations that easily solvable by Newton’s iteration method. We apply the proposed method for solving two examples. The numerical results show that present method is simple in implementation and the approximate solutions are in high accuracy. Some comparisons with other method guarantee that the results are reasonable. Also, the obtained solutions approach to classical solutions as the order of the fractional derivatives approach to 1, as expected.