Expansion formula for fractional derivatives in variational problems

Abstract

We modify the expansion formula introduced in [T.M. Atanacković, B. Stanković, An expansion formula for fractional derivatives and its applications, Fract. Calc. Appl. Anal. 7 (3) (2004) 365-378] for the left Riemann-Liouville fractional derivative in order to apply it to various problems involving fractional derivatives. As a result we obtain a new form of the fractional integration by parts formula, with the benefit of a useful approximation for the right Riemann-Liouville fractional derivative, and derive a consequence of the fractional integral inequality ∫0Ty{dot operator}0Dtαydt≥0. Further, we use this expansion formula to transform fractional optimization (minimization of a functional involving fractional derivatives) to the standard constrained optimization problem. It is shown that when the number of terms in the approximation tends to infinity, solutions to the Euler-Lagrange equations of the transformed problem converge, in a weak sense, to solutions of the original fractional Euler-Lagrange equations. An illustrative example is treated numerically.