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 acDtαx(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange 1$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}\bigr)x(t)+b\bigl(t,x(t)\bigr)\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+f\bigl(t,x(t)\bigr)=0.}$$ At last, exact solutions for some Euler–Lagrange equations are presented. In particular, we consider the following equations 2$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)=\lambda x(t)\quad (\lambda\in R),}$$3$$\lefteqn{{}_{t}D_{b}^{\alpha}\bigl({}_{a}^{c}D_{t}^{\alpha}x(t)\bigr)+g(t)_{a}^{c}D_{t}^{\alpha}x(t)=f(t),}$$ where g(t) and f(t) are suitable functions.

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