Nonlinear Dynamics

, Volume 52, Issue 4, pp 331–335

# On exact solutions of a class of fractional Euler–Lagrange equations

Original Paper

## Abstract

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where a c D t α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange
$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}\bigr)x(t)+b\bigl(t,x(t)\bigr)\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+f\bigl(t,x(t)\bigr)=0.}$$
(1)
At last, exact solutions for some Euler–Lagrange equations are presented. In particular, we consider the following equations
$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)=\lambda x(t)\quad (\lambda\in R),}$$
(2)
$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+g(t)_{a}^{c}D_{t}^{\alpha}x(t)=f(t),}$$
(3)
where g(t) and f(t) are suitable functions.

## Keywords

Fractional calculus Differential equations of fractional order Fractional variational calculus

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
2. 2.
Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)
3. 3.
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
4. 4.
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Connecticut (2006) Google Scholar
5. 5.
Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space–time fractional diffusion equation. Frac. Calc. Appl. Anal. 4(2), 153 (2001)
6. 6.
Tenreiro Machado, J.A.: A probabilistic interpretation of the fractional-order differentiation. Frac. Calc. Appl. Anal. 8, 73–80 (2003)
7. 7.
Tenreiro Machado, J.A.: Discrete-time fractional-order controllers. Frac. Calc. Appl. Anal. 4, 47–66 (2001)
8. 8.
Tofighi, A.: The intrinsic damping of the fractional oscillator. Phys. A 329, 29–34 (2006) Google Scholar
9. 9.
Trujillo, J.J.: On a Riemann–Liouville generalized Taylor’s formula. J. Math. Anal. Appl. 231, 255–265 (1999)
10. 10.
Lim, S.C., Muniady, S.V.: Stochastic quantization of nonlocal fields. Phys. Lett. A 324, 396–405 (2004)
11. 11.
Stanislavsky, A.A.: Fractional oscillator. Phys. Rev. E 70, 051103 (2004)
12. 12.
Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)
13. 13.
Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)
14. 14.
Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivatives. Czech. J. Phys. 51, 1348–1354 (2001)
15. 15.
Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czech. J. Phys. 52, 1247–1253 (2002)
16. 16.
El-Nabulusi, R.A.: A fractional approach to nonconservative Lagrangian dynamics. Fiz. A 14(4), 289–298 (2005) Google Scholar
17. 17.
Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)
18. 18.
Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A: Math. Gen. 39, 10375–10384 (2006)
19. 19.
Agrawal, O.P.: Generalized Euler–Lagrange equations and transversality conditions for FVPs in terms of Caputo Derivative. In: Tas, K., Tenreiro Machado, J.A., Baleanu, D. (eds.) Proc. MME06, Ankara, Turkey, 27–29 April 2006, to appear in J. Vib. Control (2007) Google Scholar
20. 20.
Rabei, E.M., Nawafleh, K.I., Hijjawi, R.S., Muslih, S.I. Baleanu, D.: The Hamiltonian formalism with fractional derivatives. J. Math. Anal. Appl. 327, 891–897 (2007)
21. 21.
Muslih, S., Baleanu, D.: Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives. J. Math. Anal. Appl. 304(3), 599–603 (2005)
22. 22.
Baleanu, D.: Fractional Hamiltonian analysis of irregular systems. Signal Process. 86(10), 2632–2636 (2006)
23. 23.
Baleanu, D., Muslih, S.I.: Formulation of Hamiltonian equations for fractional variational problems. Czech. J. Phys. 55(6), 633–642 (2005)
24. 24.
Baleanu, D., Muslih, S.: Lagrangian formulation of classical fields within Riemann–Liouville fractional derivatives. Phys. Scr. 72(2–3), 119–121 (2005)
25. 25.
Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004) Google Scholar
26. 26.
Tenreiro-Machado, J.A.: Discrete-time fractional-order controllers. Frac. Calc. Appl. Anal. 4(1), 47–68 (2001)
27. 27.
Agrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. In: Tas, K., Tenreiro Machado, J.A., Baleanu, D. (eds.) Proc. MME06, Ankara, Turkey, 27–29 April 2006, to appear in J. Vib. Control (2007) Google Scholar
28. 28.
Jumarie, G.: Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor’s series of nondifferntiable functions. Chaos Solitons Fractals 32, 969–987 (2007)

## Authors and Affiliations

1. 1.Department of Mathematics and Computer Sciences, Faculty of Arts and SciencesÇankaya UniversityAnkaraTurkey
2. 2.Departamento de Análisis MatemáticoUniversity of La LagunaLa LagunaSpain