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Fractional variational calculus in terms of Riesz fractional derivatives. (English) Zbl 1125.26007

Despite several results available (in the literature) for fractional calculus (i.e. derivatives and integrals of arbitrary order) and its applications in various disciplines of physics, mathematics and engineering, the present attempt is appreciable. The direction of approach and analysis of problems are interesting and appear to be maiden. Theme of the paper happens to be investigations of fractional derivatives in general and study to fractional calculus of variation in particular. Generalized Euler-Lagrange equations and the transversality conditions for fractional variational problems, defined in terms of Riesz fractional derivatives, are developed, which extend the concepts of fractional calculus variation. Two definitions are possible for a Riesz fraction derivative, one is analogous to Riemann-Liouville fractional derivative and the second is analogous to Caputo fractional derivative.

Section 2 of the paper is much subjective dealing with complete concepts of fractional calculus and results which are used by the author in later investigations. In Section 6 the author considers the problem of finding the extremum of a functional defined in terms of several functions, not all of which are independent and moreover under reasons mentioned, they are called fractional Lagrange problem. In Section 7, the author discusses the canonical form, namely the Hamiltonian formulation of the Euler-Lagrange equations. One may refer to S. I. Muslih and D. Baleanu [J. Math. Anal. Appl. 304, No. 2, 599–606 (2005; Zbl 1149.70320)] for fractional Lagrangians and Hamiltonian. The author claims that the theorems developed through Sections 3–8 for the one-dimensional case can be extended for the multidimensional one. It appears that due to want of space the author could not accommodate some more work due to others.

MSC:
26A33Fractional derivatives and integrals (real functions)