Authors: Atanacković, Teodor
Janev, Marko 
Konjik, Sanja
Pilipović, Stevan
Zorica, Dušan 
Title: Expansion formula for fractional derivatives in variational problems
Journal: Journal of Mathematical Analysis and Applications
Volume: 409
Issue: 2
First page: 911
Last page: 924
Issue Date: 15-Jan-2014
Rank: M21
ISSN: 0022-247X
DOI: 10.1016/j.jmaa.2013.07.071
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.
Keywords: Approximation | Expansion formula | Fractional derivatives | Fractional variational principles
Publisher: Elsevier
Project: Viscoelasticity of fractional type and shape optimization in a theory of rods 
Methods of Functional and Harmonic Analysis and PDE with Singularities 
Development of Dialogue Systems for Serbian and Other South Slavic Languages 
Integrated system for detection and estimation of fire development by real-time monitoring of critical parameters 
Provincial Secretariat for Science, Grant 114-451-2648

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