By Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M. Torres
This short offers a basic unifying viewpoint at the fractional calculus. It brings jointly result of a number of 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 besides nonetheless extra normal difficulties within which integer-order integrals are changed by way of fractional integrals. normal theorems are acquired for various kinds of variational difficulties for which fresh effects built within the literature might be got as specific situations. specifically, the authors provide precious optimality stipulations of Euler–Lagrange kind for the elemental and isoperimetric difficulties, transversality stipulations, and Noether symmetry theorems. The lifestyles of recommendations is proven lower than Tonelli style stipulations. the implications are used to end up the lifestyles 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 worthwhile for graduate scholars wishing to benefit approximately fractional-order structures. The distinct motives will curiosity researchers with backgrounds in utilized arithmetic, keep an eye on and optimization in addition to in convinced parts of physics and engineering.
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Additional resources for Advanced Methods in the Fractional Calculus of Variations
In the sense of the calculus of variations, we can formulate the soap bubble problem in the following way: we want to minimize the variational functional b y(t) 1 + y(t) ˙ 2 dt J (y) = subject to y(a) = A, y(b) = B. 2) with F(u, v, t) = u 1 + v 2 . Let y(t) > 0, ∀t. It is not difficult to verify that the Euler–Lagrange equation is given by y(t) ¨ = 1 + y(t) ˙ 2 y(t) and its solution is the catenary curve given by y(t) = k cosh t +c , k where c, k are certain constants. This book is devoted to the fractional calculus of variations and its generalizations.
A β β Dtα N [y](t),t Db 1 [y](t), . . ,t Db N [y](t), y(t), t dt, a with r , N , and N being natural numbers. Using the fractional variational principle he obtained the following Euler–Lagrange equation: N αi t Db N βi a Dt [∂i F] + i=1 ∂i+N F + ∂ N +N +1 F = 0. 4) i=1 Riewe illustrated his results through the classical problem of linear friction. 5) where the first term in the sum represents kinetic energy, the second one represents potential energy, the last one is linear friction energy, and i 2 = −1.
Note that for P = a, t, b, 1, 0 and the kernel k α (t, τ ) = Γ (1−α) 0 < α < q1 , we have k α ∈ L q (Δ; R). 16 we shall make use of the Mittag–Leffler function of one parameter. Let α > 0. We recall that the Mittag–Leffler function is defined by ∞ E α (z) = k=0 zk , Γ (αk + 1) where z ∈ C. This function appears naturally in the solutions of fractional differential equations, as a generalization of the exponential function (Camargo et al. 2009). 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.
Advanced Methods in the Fractional Calculus of Variations by Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M. Torres