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.