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

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 c a D α t 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.

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Correspondence to Dumitru Baleanu.

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D. Baleanu is on leave of absence from Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele-Bucharest, Romania. e-mail: baleanu@venus.nipne.ro.

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Baleanu, D., Trujillo, J.J. On exact solutions of a class of fractional Euler–Lagrange equations. Nonlinear Dyn 52, 331–335 (2008). https://doi.org/10.1007/s11071-007-9281-7

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Keywords

  • Fractional calculus
  • Differential equations of fractional order
  • Fractional variational calculus