×

Generalized multiparameters fractional variational calculus. (English) Zbl 1268.26008

Summary: This paper builds upon our recent paper on generalized fractional variational calculus (FVC). Here, we briefly review some of the fractional derivatives (FDs) that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs) which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended.

MSC:

26A33 Fractional derivatives and integrals
49K05 Optimality conditions for free problems in one independent variable
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] C. L. Dym and I. H. Shames, Solid Mechanics: A Variational Approach, McGraw-Hill, 1973. · Zbl 1279.74001
[2] D. G. Bodnar and D .T. Paris, “New variational principle in electromagnetics,” IEEE Transactions on Antennas and Propagation, vol. 18, no. 2, pp. 216-223, 1970.
[3] H. Goldstein, Classical Mechanics, Addison-Wesley, 2nd edition, 1980. · Zbl 0491.70001
[4] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, New York, NY, USA, 1994.
[5] M. J. Esteban, M. Lewin, and E. Séré, “Variational methods in relativistic quantum mechanics,” ABulletin of the American Mathematical Society, vol. 45, no. 4, pp. 535-593, 2008. · Zbl 1288.49016
[6] G. A. Bliss, Lectures on the Calculus of Variations, University of Chicago Press, Chicago, Ill, USA, 1946. · Zbl 0063.00459
[7] I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963. · Zbl 0127.05402
[8] J. N. Reddy, An Introduction to the Finite Element Method, McGraw-Hill, 3rd edition, 2005.
[9] J. Jin, The Finite Element Method in Electromagnetics, Wiley-Interscience, New York, NY, USA, 2nd edition, 2002. · Zbl 1001.78001
[10] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. · Zbl 0292.26011
[11] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[12] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Longhorne, Pa, USA, 1993. · Zbl 0818.26003
[13] V. Kiryakova, Generalized Fractional Calculus and Applications, vol. 301, Longman Scientific & Technical, Harlow, NY, USA, 1994. · Zbl 0882.26003
[14] A. Carpinteri and F. Mainardi, Eds., Fractals and Fractional Calculus in Continuum Mechanics, vol. 378, Springer, Vienna, Austria, 1997. · Zbl 0917.73004
[15] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999. · Zbl 0924.34008
[16] R. Hilfer, Ed., Applications of Fraction Calculus in Physics, World Scientific, Singapore, 2000. · Zbl 0998.26002
[17] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003.
[18] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, Conn, USA, 2006.
[19] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[20] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, UK, 2010. · Zbl 1210.26004
[21] Z. Jiao, Y. Chen, and I. Podlubny, Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Springer Briefs in Electrical and Computer Engineering Control, Automation and Robotics, Springer, New York, NY, USA, 2012. · Zbl 1401.93005
[22] F. Riewe, “Nonconservative Lagrangian and Hamiltonian mechanics,” Physical Review E, vol. 53, no. 2, pp. 1890-1899, 1996.
[23] F. Riewe, “Mechanics with fractional derivatives,” Physical Review E, vol. 55, no. 3, part B, pp. 3581-3592, 1997.
[24] M. Klimek, “Fractional sequential mechanics-models with symmetric fractional derivative,” Czechoslovak Journal of Physics, vol. 51, no. 12, pp. 1348-1354, 2001. · Zbl 1064.70507
[25] M. Klimek, “Lagrangean and Hamiltonian fractional sequential mechanics,” Czechoslovak Journal of Physics, vol. 52, no. 11, pp. 1247-1253, 2002. · Zbl 1064.70013
[26] O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368-379, 2002. · Zbl 1070.49013
[27] O. P. Agrawal, “A general formulation and solution scheme for fractional optimal control problems,” Nonlinear Dynamics, vol. 38, no. 1-4, pp. 323-337, 2004. · Zbl 1121.70019
[28] D. Baleanu and S. I. Muslih, “Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives,” Physica Scripta, vol. 72, no. 2-3, pp. 119-121, 2005. · Zbl 1122.70360
[29] O. P. Agrawal, “Fractional variational calculus and the transversality conditions,” Journal of Physics A, vol. 39, no. 33, pp. 10375-10384, 2006. · Zbl 1097.49021
[30] O. P. Agrawal, “Fractional variational calculus in terms of Riesz fractional derivatives,” Journal of Physics A, vol. 40, no. 24, pp. 6287-6303, 2007. · Zbl 1125.26007
[31] O. P. Agrawal and D. Baleanu, “A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1269-1281, 2007. · Zbl 1182.70047
[32] E. M. Rabei, K. I. Nawafleh, R. S. Hijjawi, S. I. Muslih, and D. Baleanu, “The Hamilton formalism with fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 891-897, 2007. · Zbl 1104.70012
[33] T. M. Atanacković, S. Konjik, and S. Pilipović, “Variational problems with fractional derivatives: euler-Lagrange equations,” Journal of Physics A, vol. 41, no. 9, Article ID 095201, 2008. · Zbl 1175.49020
[34] R. Almeida and D. F. M. Torres, “Calculus of variations with fractional derivatives and fractional integrals,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1816-1820, 2009. · Zbl 1183.26005
[35] J. Cresson, “Inverse problem of fractional calculus of variations for partial differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 4, pp. 987-996, 2010. · Zbl 1221.35447
[36] G. S. F. Frederico and D. F. M. Torres, “Fractional Noether’s theorem in the Riesz-Caputo sense,” Applied Mathematics and Computation, vol. 217, no. 3, pp. 1023-1033, 2010. · Zbl 1200.49019
[37] M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of Czestochowa, University of Technology, Czestochowa, Poland, 2009.
[38] O. P. Agrawal, S. I. Muslih, and D. Baleanu, “Generalized variational calculus in terms of multi-parameters fractional derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4756-4767, 2011. · Zbl 1236.49030
[39] V. Kiryakova, “A brief story about the operators of the generalized fractional calculus,” Fractional Calculus & Applied Analysis, vol. 11, no. 2, pp. 203-220, 2008. · Zbl 1153.26003
[40] O. P. Agrawal, “Generalized variational problems and Euler-Lagrange equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1852-1864, 2010. · Zbl 1189.49029
[41] A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd edition, 2008. · Zbl 1154.45001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.