Variational principle of stationary action for fractional nonlocal media and fields
Tóm tắt
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.
Tài liệu tham khảo
Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems, pp. 368–379 (2002).
Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A. 24, 6287–6303 (2007).
Agrawal, O.P.: Generalized multiparameters fractional variational calculus. Int. J. Differential Equations. 2012, 521750 (2012).
Almeida, R., Torres, D.F.M.: Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22(12), 1816–1820 (2009). (arXiv:0907.1024).
Almeida, R., Malinowska, A.B., Torres, D.F.M.: A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51(3), 033503 (2010). (arXiv:1001.2722).
Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles. Wiley-ISTE, London, Hoboken (2014).
Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions Volume 1. McGraw-Hill, New York, (1953), and Krieeger, Melbourne, Florida, (1981).
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002).
Jonscher, A.K.: The universal dielectric response. Nature. 267, 673–679 (1977).
Jonscher, A.K.: Universal Relaxation Law. Chelsea Dielectrics, London (1996).
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations (2006).
Korabel, N., Zaslavsky, G.M., Tarasov, V.E.: Coupled oscillators with power-law interaction and their fractional dynamics analogues. Commun. Nonlin. Sci. Numeric. Simul. 12(8), 1405–1417 (2007). (arXiv:math-ph/0603074).
Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi F (eds.)Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien and New York (1997). (arXiv:1201.0863).
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific, Singapore (2010).
Malinowska, A.B., Torres, D.F.M.: Fractional calculus of variations for a combined Caputo derivative. Fractional Calculus Appl. Anal. 14(4), 523–537 (2011).
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Rational Mech. Anal. 16(1), 51–78 (1964).
Mindlin, R.D.: Theories of elastic continua and crystal lattice theories. In: Kroner, E. (ed.)Mechanics of Generalized Continua, pp. 312–320. Springer-Verlag, Berlin (1968).
Nasrolahpour, H.: Fractional Lagrangian and Hamiltonian formulations in field theory Generalized multiparameters fractional variational calculus. Prespacetime J. 4(3), 604–608 (2013).
Odzijewicz, T., Malinowska, AB., Torres, D. F. M.: Fractional variational valculus with vlassical and vombined Caputo derivatives. Nonlinear Anal. 75(3), 1507–1515 (2012). (arXiv:1101.2932).
Riesz, M.: L’intégrale de Riemann-Liouville et le probléme de Cauchy. Acta Math. 81(1), 1–222 (1949). in French.
Rogula, D.: Nonlocal Theory of Material Media. Springer-Verlag, New York (1983).
Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A., (Eds): Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007).
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Integrals and Derivatives of Fractional Order and Applications (Nauka i Tehnika, Minsk, 1987); and Fractional Integrals and Derivatives Theory and Applications Gordon and Breach, New York (1993).
Sedov, L.I.: Mathematical methods for constructing new models of continuous media. Russ. Math. Surv. 20(5), 123–182 (1965).
Sedov, L.I.: Models of continuous media with internal degrees of freedom. J. Appl. Math. Mech. 32(5), 803–819 (1968).
Sedov, L.I., Tsypkin, A.G.: Principles of the Microscopic Theory of Gravitation and Electromagnetism, Nauka, Moscow (1989). in Russian.
Tarasov, V.E.: Universal electromagnetic waves in dielectrics. J. Phys.: Condensed Matter. 20(17), 175223 (2008). (arXiv:0907.2163).
Tarasov, V.E.: Fractional integro-differential equations for electromagnetic waves in dielectric media. Theor. Math. Phys. 158(3), 355–359 (2009). (arXiv:1107.5892).
Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York (2011).
Tarasov, V.E.: Review of some promising fractional physical models. Int. J. Modern Phys. B. 27(9), 1330005 (2013). (arXiv:1502.07681).
Tarasov, V.E.: Lattice model with power-law spatial dispersion for fractional elasticity. Central Eur. J. Phys. 11(11), 1580–1588 (2013). (arXiv:1501.01201).
Tarasov, V.E.: Lattice model of fractional gradient and integral elasticity: Long-range interaction of Grünwald-Letnikov-Riesz type. Mech. Mater. 70(1), 106–114 (2014). (arXiv:1502.06268).
Tarasov, V.E.: Lattice with long-range interaction of power-law type for fractional non-local elasticity. Int. J. Solids Struct. 51, 2900–2907 (2014). (arXiv:1502.05492).
Tarasov, V.E.: Fractional gradient elasticity from spatial dispersion law. ISRN Condensed Matter Phys. 2014. Article ID 794097, 13 pages (2014). (arXiv:1306.2572).
Tarasov, V.E.: Fractional quantum field theory: From lattice to continuum. Adv. High Energy Phys. 2014, 957863 (2014). 14 pages.
Tarasov, V.E.: General lattice model of gradient elasticity. Modern Phys. Lett. B. 28(7), 1450054 (2014). (arXiv:1501.01435).
Tarasov, V.E.: Toward lattice fractional vector calculus. J. Phys. A. 47(35), 355204 (2014). (51 pages).
Tarasov, V.E.: Non-linear fractional field equations: weak non-linearity at power-law non-locality. Nonlinear Dynam. 80(4), 1665–1672 (2015).
Tarasov, V.E.: Lattice fractional calculus. Appl. Math. Comput. 257, 12–33 (2015).
Tarasov, V.E.: Three-dimensional lattice models with long-range interactions of Grünwald-Letnikov type for fractional generalization of gradient elasticity. Meccanica. 50 (2015). doi:10.1007/s11012-015-0190-4.
Tarasov, V.E.: Lattice model with nearest-neighbor and next-nearest-neighbor interactions for gradient elasticity. Discontinuity, Nonlinearity, Complexity. 4(1), 11–23 (2015). (arXiv:1503.03633).
Valerio, D., Trujillo, J.J., Rivero, M., Tenreiro Machado, J.A., Baleanu, D.: Fractional calculus: A survey of useful formulas. Eur. Phys. J. Spec. Topics. 222(8), 1827–1846 (2013).
Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014).