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

ISBN-10: 3319147552

ISBN-13: 9783319147550

ISBN-10: 3319147560

ISBN-13: 9783319147567

This short provides a basic unifying standpoint at the fractional calculus. It brings jointly result of numerous contemporary methods 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 common difficulties during which integer-order integrals are changed by means of fractional integrals. common theorems are acquired for different types of variational difficulties for which fresh effects constructed within the literature will be bought as unique circumstances. specifically, the authors provide worthwhile optimality stipulations of Euler–Lagrange style for the basic and isoperimetric difficulties, transversality stipulations, and Noether symmetry theorems. The lifestyles of options is confirmed less than Tonelli kind stipulations. the consequences are used to turn out the lifestyles of eigenvalues and corresponding orthogonal eigenfunctions of fractional Sturm–Liouville problems.

Advanced tools within the Fractional Calculus of diversifications is a self-contained textual content with a view to be precious for graduate scholars wishing to benefit approximately fractional-order platforms. The specific causes will curiosity researchers with backgrounds in utilized arithmetic, keep an eye on and optimization in addition to in definite components of physics and engineering.

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

**Sample text**

5 (Dual parameter set) Let P = a, t, b, λ, μ be a given parameter set. We denote by P ∗ the parameter set P ∗ = a, t, b, μ, λ . We say that P ∗ is the dual of P. Our first formula of fractional integration by parts involves the operator K P . 6 Let us assume that k ∈ L q (Δ; R). 5) t L p (a, b; R) to is a linear bounded operator from ing integration by parts formula holds: b L q (a, b; R). 6) a for any f, g ∈ L p (a, b; R). 2, we obtain that K P ∗ is a linear bounded operator from L p (a, b; R) to L q (a, b; R).

36) G k (y(t), K P [y](t), y(t), ˙ B P [y](t), t) dt, where y˙ denotes the classical derivative of y, K P is the generalized fractional integral operator with kernel belonging to L q (Δ; R) and B P = K P ◦ dtd .

Indeed, while a linear ordinary differential equation with constant coefficients presents an exponential function in its solution, in the fractional case the Mittag– Leffler functions emerge (Kilbas et al. 2006). 16 Let 0 < α < q1 . 14) 0 subject to the following boundary conditions: 1 E 1−α −(1 − τ )1−α dτ . 15 is given by F(x1 , x2 , x3 , x4 , t) = 1 + (x3 + x4 − 1)2 . 15) 42 4 Standard Methods in Fractional Variational Calculus One can easily check that (see Kilbas et al. (2006) p. 16) 0 α satisfies y(t) ˙ +C a Dt [y](t) ≡ 1.

### Advanced Methods in the Fractional Calculus of Variations by Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M. Torres

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