A fractional calculus of variations for multiple integrals with application to vibrating string

By Ricardo Almeida, Agnieszka B. Malinowska, Delfim F. M. Torres. 2010

169 citations recorded by the source corpus.

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/ed0049895010a7ffad23dae8d5092fabc61f73e7

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

Minimal modified energy control for fractional linear control systems with the Caputo derivative (2010, background): “e concept of fractional (e., non-integer) derivative and integr al is increasingly being recognized as a good tool to model the behavior of complex systems in various ﬁelds of science and engineering [2, 3, 4, 6, 9, 20, 23]. This is particularly true in the areas of control theory and control engine”
Fractional calculus of variations for a combined Caputo derivative (2011, background): “ves (see, e.g., [1, 5, 8, 12, 17, 18]), the Caputo fractional derivative (see, e.g., [2, 10, 27]), the symmetric fractional derivative (see, e.g., [25]), the Jumarie fractional derivative (see, e.g., [4, 7, 22, 23, 31]), and others (see, e.g., [3, 15, 16, 20]). Although the literature of FCV is alre”
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].”
Fractional Variational Calculus with Classical and Combined Caputo Derivatives (2011, background): “t is a subject of strong current research due to its many applications in science and engineering, including mechanics, chemistry, biology, economics, and control theory (see, e.g., the recent papers [2, 5, 6, 7, 8, 13, 19, 21, 23, 24]).1 Following [25], we consider here that the highest derivative ”
Exact solutions for nonlinear partial fractional differential equations (2012, methodology): “org variation approach with several variables [33], the fractional variational iteration method [34], the fractional variational approach with natural boundary conditions [35] and the fractional Lie group method [36].”
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.”