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 ${}_a^c{D}_t^{\alpha }x(t))$ and $0<\alpha <1$, such that the following is the corresponding Euler-Lagrange % \begin{equation}_tD_b^α(_a^cD_t^α) x(t)+ b(t,x(t))(_a^cD_t^αx(t))+f(t,x(t))=0. \end{equation} % At last, exact solutions for some Euler-Lagrange equations are presented. In particular, we consider the following equations % \begin{equation}_tD_b^α(_a^cD_t^αx(t))=λx(t), (λ\in R) \end{equation} % \begin{equation}_tD_b^α(_a^cD_t^αx(t))+g(t)_a^cD_t^αx(t)=f(t), \end{equation} where g(t) and f(t) are suitable functions.

10 pages, LATEX. in press, Nonlinear Dynamics