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 fresh 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 by means of fractional integrals. normal theorems are got for various kinds of variational difficulties for which fresh effects built within the literature may be acquired as exact instances. particularly, the authors provide valuable optimality stipulations of Euler–Lagrange sort for the elemental and isoperimetric difficulties, transversality stipulations, and Noether symmetry theorems. The lifestyles of suggestions is verified less than Tonelli variety stipulations. the implications are used to turn out 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 as a way to be priceless for graduate scholars wishing to profit approximately fractional-order structures. The specific motives will curiosity researchers with backgrounds in utilized arithmetic, regulate 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
Tn ) dτ ai =: ai Dtαi i [ f ](t), 1 B Pi [ f ](t) = Γ (1 − αi ) ti (ti − τ )−αi ai ∂ f (t1 , . . , ti−1 , τ , ti+1 , . . , tn ) dτ ∂τ αi =: C ai Dti [ f ](t). If Pi = ai , ti , bi , 0, 1 , then ∂ −1 −A Pi [ f ](t) = Γ (1 − αi ) ∂ti =: (τ − ti )−αi f (t1 , . . , ti−1 , τ , ti+1 , . . , tn ) dτ ti αi ti Dbi [ f ](t), −1 −B Pi [ f ](t) = Γ (1 − αi ) =: bi bi ti C αi ti Dbi [ f ](t). (τ − ti )−αi ∂ f (t1 , . . , ti−1 , τ , ti+1 , . . , tn ) dτ ∂τ Moreover, one can easily check that also variable order partial fractional integrals and derivatievs are particular cases of operators K Pi , A Pi and B Pi .
Direct calculations show that ⎞ ⎛ t 1 1 1 2 − τ2 2 − t2 t τ K P  dt = ⎝ dτ − dτ ⎠ dt (t 2 + τ 2 )2 (t 2 + τ 2 )2 2 0 0 1 ⎛ 0 1 ⎝ = 0 0 ⎛ τ ⎞ 2 t t2 − τ2 dτ ⎠ dt = (t 2 + τ 2 )2 1 1 π dt = t2 + 1 4 0 and 1 1 ⎝− K P ∗  dτ = 0 0 1 ⎛ =− 0 1 0 − t2 (t 2 + τ 2 )2 0 ⎝ τ2 1 dt + ⎞ τ ⎞ t2 − τ2 (t 2 + τ 2 )2 τ2 − t2 dt ⎠ dτ = − (t 2 + τ 2 )2 1 0 τ2 dt ⎠ dτ π 1 dτ = − . 6) does not hold. Observe that in this 1 1 case 0 0 |k(t, τ )|2 dτ dt = ∞. 6 gives the following result. 8 Let 1 p < α < 1. If f, g ∈ L p (a, b; R), then b b g(t) · a Itα [ f ](t) dt a f (t) · t Ibα [g](t) dt.
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.
Advanced Methods in the Fractional Calculus of Variations by Agnieszka B. Malinowska, Tatiana Odzijewicz, Delfim F.M. Torres