Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M.'s Advanced Methods in the Fractional Calculus of Variations PDF
By Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M. Torres
This short offers a normal unifying viewpoint at the fractional calculus. It brings jointly result of numerous fresh 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 basic difficulties during which integer-order integrals are changed through fractional integrals. common theorems are acquired for different types of variational difficulties for which fresh effects built within the literature might be acquired as certain situations. specifically, the authors supply worthy optimality stipulations of Euler–Lagrange kind for the basic and isoperimetric difficulties, transversality stipulations, and Noether symmetry theorems. The lifestyles of recommendations is proven less than Tonelli variety 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 to be able to be priceless for graduate scholars wishing to profit approximately fractional-order structures. The specific motives will curiosity researchers with backgrounds in utilized arithmetic, keep watch over and optimization in addition to in yes parts of physics and engineering.
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Additional resources for Advanced Methods in the Fractional Calculus of Variations
Mediterr J Math 6:215–232 Bourdin L, Odzijewicz T, Torres DFM (2014) Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition—application to fractional variational problems. Differ Integral Equ 27(7–8):743–766 Coimbra CFM (2003) Mechanics with variable-order differential operators. Ann Phys 12(11–12): 692–703 Cresson J (2007) Fractional embedding of differential operators and Lagrangian systems. J Math Phys 48(3):033504, 34 pp Diaz G, Coimbra CFM (2009) Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation.
28 (Generalized partial fractional derivative of Riemann–Liouville type) The generalized partial fractional derivative of Riemann–Liouville type with respect to the ith variable ti is given by A Pi := ∂t∂i ◦ K Pi . 29 (Generalized partial fractional derivative of Caputo type) The generalized partial fractional derivative of Caputo type with respect to the ith variable ti is given by B Pi := K Pi ◦ ∂t∂i . 30 Similarly, as in the one-dimensional case, partial operators K , A and B reduce to the standard partial fractional integrals and derivatives.
To be more precise, let us define the following operator first introduced in Cresson (2007). 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 .
Advanced Methods in the Fractional Calculus of Variations by Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M. Torres