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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.

MSC:
70H30 Other variational principles in mechanics
26A33 Fractional derivatives and integrals
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