scholarly journals Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

Optimization ◽  
2015 ◽  
Vol 64 (6) ◽  
pp. 1381-1391 ◽  
Author(s):  
Dina Tavares ◽  
Ricardo Almeida ◽  
Delfim F.M. Torres
Keyword(s):  
Optimality Conditions ◽  
Caputo Derivative ◽  
Variational Problems ◽  
Variable Order ◽  
Fractional Variational Problems

Noether’s theorem for fractional variational problems of variable order

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Tatiana Odzijewicz ◽  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Optimality Condition ◽  
Variational Problems ◽  
Time Variable ◽  
Variable Order ◽  
Necessary Optimality Condition ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Fractional Variational Problems ◽  
Derivatives Of

AbstractWe prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether’s theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

scholarly journals Fractional calculus of variations for a combined Caputo derivative

2011 ◽  
Vol 14 (4) ◽  
Author(s):  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Caputo Derivative ◽  
Convex Combination ◽  
Variational Problems ◽  
Lagrange Equations ◽  
Transversality Conditions ◽  
Fractional Variational Problems ◽  
Fractional Calculus Of Variations ◽  
The Right

AbstractWe generalize the fractional Caputo derivative to the fractional derivative C D γα,β, which is a convex combination of the left Caputo fractional derivative of order α and the right Caputo fractional derivative of order β. The fractional variational problems under our consideration are formulated in terms of C D γα,β. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

scholarly journals Constrained fractional variational problems of variable order

2017 ◽  
Vol 4 (1) ◽  
pp. 80-88 ◽  
Author(s):  
Dina Tavares ◽  
Ricardo Almeida ◽  
Delfim F. M. Torres
Keyword(s):  
Variational Problems ◽  
Variable Order ◽  
Fractional Variational Problems

Numerical Methods for Fractional Variational Problems Depending on Indefinite Integrals

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Dongling Wang ◽  
Aiguo Xiao
Keyword(s):  
Numerical Methods ◽  
Caputo Derivative ◽  
Variational Problems ◽  
Variational Integrators ◽  
Lagrange Equations ◽  
Numerical Examples ◽  
Fractional Variational Problems

In this paper, the fractional variational integrators for fractional variational problems depending on indefinite integrals in terms of the Caputo derivative are developed. The corresponding fractional discrete Euler–Lagrange equations are derived. Some fractional variational integrators are presented based on the Grünwald–Letnikov formula. The fractional variational errors are discussed. Some numerical examples are given to illustrate these results.

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