Abstract
Derivatives and integrals of non-integer orders have a wide application to describe complex properties of physical systems and media including nonlocality of power-law type and long-term memory. We suggest an extension of the standard variational principle for fractional nonlocal media with power-law type nonlocality that is described by the Riesz-type derivatives of non-integer orders. As examples of application of the suggested variational principle, we consider an N-dimensional model of waves in anisotropic fractional nonlocal media, and a one-dimensional continuum (string) with power-law spatial dispersion. The main advantage of the suggested fractional variational principle is that it is connected with microstructural lattice approach and the lattice fractional calculus, which is recently proposed.
Highlights
Derivatives and integrals of non-integer orders [11, 13, 23, 42, 43] have wide application in physics and mechanics [6, 14, 22, 29, 30]
This fractional variational principle is based on the Riesz-type fractional derivatives
(4) The Riesz-type fractional derivative RT Dαj j is a derivative with respect to one coordinate xj ∈ R1 contrary to the usual Riesz derivative [11, 23], which is a fractional generalization of N-dimensional Laplacian
Summary
Derivatives and integrals of non-integer orders [11, 13, 23, 42, 43] have wide application in physics and mechanics [6, 14, 22, 29, 30]. In fractional nonlocal theory, we should use the fractional-order derivatives that are directly connected with models of lattices with longrange interactions. One of the methods to describe the fractional nonlocal continua is based on the variational principle with Lagrangian density that contains fractional derivatives with respect to coordinates. Taking into account the direct connection of the Riesz-type derivatives with microstructural lattice approach, we suggest to use these fractional derivatives to formulate new fractional variational principle that is compatible with the microstructural lattice approach. In addition to "lattice" motivation, it is useful to have a variational principle that allows us to derive fractional differential equations of motion that can be solved for a wide class of Lagrangian densities. Using the suggested variational principle, we derive the fractional differential equations and some solutions of these equations are obtained
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