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

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

This short provides a normal 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 besides nonetheless extra normal difficulties during which integer-order integrals are changed through fractional integrals. basic theorems are got for different types of variational difficulties for which fresh effects built within the literature might be bought as designated instances. specifically, the authors supply worthwhile optimality stipulations of Euler–Lagrange kind for the elemental and isoperimetric difficulties, transversality stipulations, and Noether symmetry theorems. The life of strategies is established less than Tonelli variety stipulations. the implications are used to end up the life 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 that allows you to be priceless for graduate scholars wishing to benefit approximately fractional-order structures. The designated reasons will curiosity researchers with backgrounds in utilized arithmetic, regulate and optimization in addition to in yes components of physics and engineering.

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Extra resources for Advanced Methods in the Fractional Calculus of Variations

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N ) are in Rn . We shall present definitions of generalized partial fractional integrals and derivatives. Let ki : Δi → R, i = 1, . . , n and t ∈ Ωn . 27 (Generalized partial fractional integral) For any function f defined almost everywhere on Ωn with value in R, the generalized partial integral K Pi is defined for almost all ti ∈ (ai , bi ) by: ti K Pi [ f ](t) : = λi ki (ti , τ ) f (t1 , . . , ti−1 , τ , ti+1 , . . , tn ) dτ ai bi + μi ki (τ , ti ) f (t1 , . . , ti−1 , τ , ti+1 , . .

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.

Moreover, we assume that • K P ∗ [∂2 G(y(τ ), K P [y](τ ), y(τ ˙ ), B P [y](τ ), τ )] ∈ C([a, b]; R), ˙ B P [y](t), t) ∈ C 1 ([a, b]; R), • t → ∂3 G(y(t), K P [y](t), y(t), ˙ ), B P [y](τ ), τ )] ∈ C 1 ([a, b]; R). 9) subject to the isoperimetric constraint J (y) = ξ. In the next theorem, we provide a necessary optimality condition for this type of problem. 25 Suppose that y¯ is a minimizer of functional I in the class Aξ (ya , yb ) := {y ∈ A(ya , yb ) : J (y) = ξ} . 26) ¯ K P [y](t), ¯ y˙¯ (t), B P [y](t), ¯ t).

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