Existence - uniqueness result for a certain equation of motion in fractional mechanics

Bulletin of the Polish Academy of Sciences: Technical Sciences

Volumn 58, Issue 4, 2010, Pages 573-581

  Klimek, M ^a  

^a CZESTOCHOWA UNIVERSITY OF TECHNOLOGY   (Poland)

Author keywords

Approximate solution; Euler Lagrange equation; Fixed point theorem; Fractional derivative; Fractional differential equation

Indexed keywords

APPROXIMATE SOLUTION; EULER-LAGRANGE EQUATION; FIXED POINT THEOREM; FRACTIONAL DERIVATIVES; FRACTIONAL DIFFERENTIAL EQUATIONS;

BOUNDARY CONDITIONS; EIGENVALUES AND EIGENFUNCTIONS; LAGRANGE MULTIPLIERS; MATHEMATICAL OPERATORS; PARTIAL DIFFERENTIAL EQUATIONS; SOLAR CELL ARRAYS; THEOREM PROVING;

EQUATIONS OF MOTION;

EID: 79951597861     PISSN: 02397528     EISSN: None     Source Type: Journal    
DOI: 10.2478/v10175-010-0058-7     Document Type: Article

References (34)

1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore 2000.
2. B.J. West, M. Bologna, and P. Grigolini, Physics of Fractional Operators, Springer-Verlag, Berlin, 2003.
3. O.P. Agrawal, J.A. Tenreiro Machado, and J. Sabatier, Fractional Derivatives and Their Application: Nonlinear Dynamics, Springer-Verlag, Berlin, 2004.
4. R. Metzler and J. Klafter, "The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics", J. Phys. A 37, R161-R208 (2004).
5. R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Redding, 2006.
6. R. Herrmann, "The fractional symmetric rigid rotor", J. Phys. G: Nuc. Phys. 34, 607-625 (2007).
7. J. Sabatier, O.P. Agrawal, and J.A. Tenreiro Machado, Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering, Springer-Verlag, Berlin, 2007.
8. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.
9. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
10. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
11. M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of the Czestochowa University of Technology, Czestochowa, 2009.
12. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives, Gordon & Breach, Amsterdam, 1993.
13. F. Riewe, "Nonconservative Lagrangian and Hamiltonian mechanics", Phys. Rev. E 53, 1890-1899 (1996).
14. F. Riewe, "Mechanics with fractional derivatives", Phys. Rev. E 55, 3581-3592 (1997).
15. M. Klimek, "Fractional sequential mechanics - models with symmetric fractional derivative", Czech. J. Phys. 51, 1348-1354 (2001).
16. M. Klimek, "Lagrangean and Hamiltonian fractional sequential mechanics", Czech J. Phys. 52, 1247-1253 (2002).
17. O.P. Agrawal, "Formulation of Euler-Lagrange equations for fractional variational problem", J. Math. Anal. Appl. 272, 368-379 (2002).
18. D. Baleanu and T. Avkar, "Lagrangians with linear velocities within Riemann-Liouville fractional derivatives", Nuovo Cimento 119, 73-79 (2004).
19. D. Baleanu and S.I. Muslih, "Formulation of Hamiltonian equations for fractional variational problems", Czech. J. Phys. 55, 633-642 (2005).
20. D. Baleanu, "Fractional Hamiltonian analysis of irregular systems", Signal Processing 86, 2632-2636 (2006).
21. J. Cresson, "Fractional embedding of differential operators and Lagrangian systems", J. Math. Phys. 48, 033504 (2007).
22. O.P. Agrawal, "Fractional variational calculus and the transversality conditions", J. Phys. A 39, 10375-10384 (2006).
23. T.M. Atanackovic, S. Konjik, and S. Pilipovic, "Variational problems with fractional derivatives: Euler-Lagrange equations", J. Phys. A: Math. & Theor. 41 (095201), CD-ROM (2008).
24. V.E. Tarasov, "Fractional variations for dynamical systems: Hamilton and Lagrange approaches", J. Phys. A: Math. & Gen. 39, 8409 (2006).
25. V.E. Tarasov, "Psi-series solution of fractional Ginzburg-Landau equation", J. Phys. A: Math. & Gen. 39, 8395-8397 (2006).
26. G. Jumarie, "Lagrangian mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor's series of nondifferentiable functions", Chaos, Solitons & Fractals 32, 969-973 (2007).
27. M. Klimek, "Lagrangian fractional mechanics - a noncommutative approach", Czech. J. Phys. 55, 1447-1454 (2005).
28. O.P. Agrawal, "Analytical schemes for a new class of fractional differential equations", J. Phys. A 40, 5469-5476 (2007).
29. D. Baleanu and J.J. Trujillo, "On exact solutions of a class of fractional Euler-Lagrange equations", Nonlinear Dyn. 52, 9281-9287 (2008).
30. M. Klimek, "Solutions of Euler-Lagrange equations in fractional mechanics", AIP Conf. Proc. 956, XXVI Workshop on Geometric Methods in Physics 1, 73-78 (2007).
31. M. Klimek, "G-Meijer functions series as solutions for certain fractional variational problem on a finite time interval", J. Eu. des Systèmes Automatisé s 42, 653-664 (2008).
32. T.M. Atanackovic and B. Stankovic, "On a class of differential equations with left and right fractional derivatives", Zamm 87, 537-539 (2007).
33. T. Maraaba, D. Baleanu, and F. Jarad, "Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives", J. Math. Phys. 49, 083507 (2008).
34. M. Klimek, "On analogues of exponential functions for antisymmetric fractional derivatives", Comp. & Math. Appl. 59, 1709-1717 (2010).