zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On exact solutions of a class of fractional Euler-Lagrange equations. (English) Zbl 1170.70328
Summary: In this paper, first a class of fractional differential equations is obtained by using the fractional variational principles. We find a fractional Lagrangian $L(x(t), \text{ where } {}_{a}^{c}D_{t}^{\alpha} x(t))$ and $0<\alpha <1$, such that the following is the corresponding Euler-Lagrange $${}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}\bigr)x(t)+b\bigl(t,x(t)\bigr)\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+f\bigl(t,x(t)\bigr)=0.\tag 1$$ At last, exact solutions for some Euler-Lagrange equations are presented. In particular, we consider the following equations $${}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)=\lambda x(t)\quad (\lambda\in R), \tag 2$$ $${}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+g(t)_{a}^{c}D_{t}^{\alpha}x(t)=f(t), \tag 3$$ where $g(t)$ and $f(t)$ are suitable functions.

70H30Other variational principles (mechanics of particles and systems)
26A33Fractional derivatives and integrals (real functions)
Full Text: DOI arXiv
[1] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[2] Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005) · Zbl 1083.37002
[3] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[4] Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Connecticut (2006)
[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) · Zbl 1054.35156
[6] Tenreiro Machado, J.A.: A probabilistic interpretation of the fractional-order differentiation. Frac. Calc. Appl. Anal. 8, 73--80 (2003) · Zbl 1035.26010
[7] Tenreiro Machado, J.A.: Discrete-time fractional-order controllers. Frac. Calc. Appl. Anal. 4, 47--66 (2001) · Zbl 1111.93307
[8] Tofighi, A.: The intrinsic damping of the fractional oscillator. Phys. A 329, 29--34 (2006)
[9] Trujillo, J.J.: On a Riemann--Liouville generalized Taylor’s formula. J. Math. Anal. Appl. 231, 255--265 (1999) · Zbl 0931.26004 · doi:10.1006/jmaa.1998.6224
[10] Lim, S.C., Muniady, S.V.: Stochastic quantization of nonlocal fields. Phys. Lett. A 324, 396--405 (2004) · Zbl 1123.81376 · doi:10.1016/j.physleta.2004.02.073
[11] Stanislavsky, A.A.: Fractional oscillator. Phys. Rev. E 70, 051103 (2004) · Zbl 1178.26008 · doi:10.1103/PhysRevE.70.051103
[12] Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890--1899 (1996) · doi:10.1103/PhysRevE.53.1890
[13] Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581--3592 (1997) · doi:10.1103/PhysRevE.55.3581
[14] Klimek, M.: Fractional sequential mechanics-models with symmetric fractional derivatives. Czech. J. Phys. 51, 1348--1354 (2001) · Zbl 1064.70507 · doi:10.1023/A:1013378221617
[15] Klimek, M.: Lagrangian and Hamiltonian fractional sequential mechanics. Czech. J. Phys. 52, 1247--1253 (2002) · Zbl 1064.70013 · doi:10.1023/A:1021389004982
[16] El-Nabulusi, R.A.: A fractional approach to nonconservative Lagrangian dynamics. Fiz. A 14(4), 289--298 (2005)
[17] Agrawal, O.P.: Formulation of Euler--Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368--379 (2002) · Zbl 1070.49013 · doi:10.1016/S0022-247X(02)00180-4
[18] Agrawal, O.P.: Fractional variational calculus and the transversality conditions. J. Phys. A: Math. Gen. 39, 10375--10384 (2006) · Zbl 1097.49021 · doi:10.1088/0305-4470/39/33/008
[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)
[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) · Zbl 1104.70012 · doi:10.1016/j.jmaa.2006.04.076
[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) · Zbl 1149.70320 · doi:10.1016/j.jmaa.2004.09.043
[22] Baleanu, D.: Fractional Hamiltonian analysis of irregular systems. Signal Process. 86(10), 2632--2636 (2006) · Zbl 1172.94362 · doi:10.1016/j.sigpro.2006.02.008
[23] Baleanu, D., Muslih, S.I.: Formulation of Hamiltonian equations for fractional variational problems. Czech. J. Phys. 55(6), 633--642 (2005) · Zbl 1181.70017 · doi:10.1007/s10582-005-0067-1
[24] Baleanu, D., Muslih, S.: Lagrangian formulation of classical fields within Riemann--Liouville fractional derivatives. Phys. Scr. 72(2--3), 119--121 (2005) · Zbl 1122.70360 · doi:10.1238/Physica.Regular.072a00119
[25] Baleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann--Liouville fractional derivatives. Nuovo Cimento B 119, 73--79 (2004) · Zbl 1120.26001
[26] Tenreiro-Machado, J.A.: Discrete-time fractional-order controllers. Frac. Calc. Appl. Anal. 4(1), 47--68 (2001) · Zbl 1111.93307
[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)
[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) · Zbl 1154.70011 · doi:10.1016/j.chaos.2006.07.053