A formulation of Noether's theorem for fractional problems of the calculus of variations

By Gastão S. F. Frederico, Delfim F. M. Torres. Journal of Mathematical Analysis and Applications, 2007

281 citations recorded by the source corpus.

Abstract Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler–Lagrange obtained in 2002. Here we use the notion of Euler–Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7fee2a81f336e824520221616d8bef7686045afd

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Citations in the corpus

Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem (2007, methodology): “While here Dω t depends both on Caputo and Riemann-Liouville derivatives, the ones introduced in [11, 13] involve Riemann-Liouville derivatives only.”
Fractional calculus of variations for a combined Caputo derivative (2011, background): “, [1, 5, 8, 12, 17, 19]), the Caputo fractional derivative (see, e.”
A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials (2016, methodology): “The fractional Euler–Lagrange equation has recently been presented in the literature (Almeida and Torres, 2009; Atanacković et al., 2007; Bastos et al., 2011; Frederico and Torres, 2007; Malinowska and Torres, 2010; Muslih and Baleanu, 2005; Stanislavsky, 2006; Tarasov and Zaslavsky, 2005, 2006).”
A fractional calculus of variations for multiple integrals with application to vibrating string (2010, background): “An interesting open question consists to generalize the fractional Noether-type theorems obtained in [7, 21, 22] to the case of several independent variables.”
Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem (2015, background): “In the last decades, Noether’s theorems have been formulated in various contexts: see [2, 3, 5, 8, 9, 10, 11, 19, 24, 25, 27, 33, 35, 37] and references therein.”
FRACTIONAL h-DIFFERENCE EQUATIONS ARISING FROM THE CALCULUS OF VARIATIONS (2011, methodology): “The presence of the h parameter is particularly interesting from the numerical point of view, because when h tends to zero the solutions of the fractional difference equations can be seen as approximations to the solutions of corresponding Riemann–Liouville fractional differential equations [9, 15] ”
Necessary optimality conditions for fractional action‐like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β) (2007, unknown): “In the fractional case it is necessary to change the definition of constant of motion in a proper way [9, 10, 11].”
NECESSARY OPTIMALITY CONDITIONS FOR FRACTIONAL DIFFERENCE PROBLEMS OF THE CALCULUS OF VARIATIONS (2010, background): “The study of fractional problems of the Calculus of Variations and respective Euler-Lagrange equations is a fairly recent issue – see [1, 2, 5, 7, 8, 11, 13, 15, 16, 21] and references therein – and include only the continuous case.”
Fractional actionlike variational problems (2008, background): “Constants of motion for fractional action-like variational problems were discussed in [15, 16] (see also [17, 18]); FALVA problems with higher-order derivatives are studied in [16, 19]; and some encouraging results obtained and discussed in [10, 11, 12, 13, 14].”
Conservation laws for time-fractional subdiffusion and diffusion-wave equations (2014, background): “Using these results, several fractional generalizations of Noether’s theorem were proved [25–30], and some number of fractional conservation laws were calculated for equations and systems having different fractional Lagrangians [26,31,32].”
Existence of a weak solution for fractional Euler-Lagrange equations (2012, background): “Furthermore, one can find a comprehensive literature regarding necessary optimality conditions and Noether’s theorem, see [1, 3, 4, 6, 13, 25].”