By Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M. Torres

This short provides a basic unifying point of view at the fractional calculus. It brings jointly result of numerous contemporary ways in generalizing the least motion precept and the Euler–Lagrange equations to incorporate fractional derivatives.

The dependence of Lagrangians on generalized fractional operators in addition to on classical derivatives is taken into account in addition to nonetheless extra normal difficulties during which integer-order integrals are changed by way of fractional integrals. normal theorems are got for different types of variational difficulties for which fresh effects constructed within the literature may be got as distinct instances. particularly, the authors provide worthwhile optimality stipulations of Euler–Lagrange style for the elemental and isoperimetric difficulties, transversality stipulations, and Noether symmetry theorems. The life of strategies is established lower than Tonelli sort stipulations. the consequences are used to turn out the life of eigenvalues and corresponding orthogonal eigenfunctions of fractional Sturm–Liouville problems.

Advanced equipment within the Fractional Calculus of diversifications is a self-contained textual content on the way to be necessary for graduate scholars wishing to profit approximately fractional-order platforms. The targeted factors will curiosity researchers with backgrounds in utilized arithmetic, regulate and optimization in addition to in sure parts of physics and engineering.

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**Additional info for Advanced Methods in the Fractional Calculus of Variations**

**Example text**

1 Fractional Euler–Lagrange Equations 27 The next examples are borrowed from Klimek (2009). 3 (cf. 1 of (Klimek 2009)) Let 0 < α < 1 and y be a minimizer of the functional b 1 y(t)a Dtα [y](t) dt. 2 J (y) = a Then y is a solution to the following Euler–Lagrange equation: 1 2 α C α a Dt [y] + t Db [y] = 0. 4 (cf. 2 of (Klimek 2009)) Let 0 < α < 1. 10) J (y) = 2 2 a and is determined by the equation y + ω 2 y = 0. 9) the Euler–Lagrange equation has the following form: α −C t Db α a Dt [y] + ω 2 y = 0.

Gordon and Breach, Yverdon Samko SG, Ross B (1993) Integration and differentiation to a variable fractional order. Integral Transform Spec Funct 1(4):277–300 Chapter 3 Fractional Calculus of Variations Abstract We review a few main approaches to the fractional calculus of variations. Keywords Calculus of variations · Fractional calculus of variations · Lagrangian · Fractional Euler–Lagrange equations · Fractional embedding · Coherence problem The calculus of variations is a beautiful and useful field of mathematics that deals with the problems of determining extrema (maxima or minima) of functionals (Dacorogna 2004; Malinowska and Torres 2012, 2014).

5 (Fractional operator of order (α, β)) Let a, b ∈ R, a < b and μ ∈ C. We define the fractional operator of order (α, β), with α > 0 and β > 0, by Dμα,β = 1 2 α a Dt β −t Db + iμ 2 α a Dt β +t Db . 13) In particular, for α = β = 1 one has Dμ1,1 = dtd . Moreover, for μ = −i we recover the left Riemann–Liouville fractional derivative of order α, α,β D−i = a Dtα , and for μ = i the right Riemann–Liouville fractional derivative of order β: α,β Di = −t Dbα . Now, let us consider the following variational functional: b F(Dμα,β [y](t), y(t), t) dt J (y) = a β defined on the space of continuous functions such that a Dtα [y] together with t Db [y] exist and y(a) = ya , y(b) = yb .