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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),where a c D t α x(t)) and 0<α<1, such that the following is the corresponding Euler-Lagrange

t D b α a c D t α x(t)+bt , x ( t ) a c D t α x (t)+ft , x ( t )=0·(1)

At last, exact solutions for some Euler-Lagrange equations are presented. In particular, we consider the following equations

t D b α a c D t α x (t)=λx(t)(λR),(2)
t D b α a c D t α x (t)+g(t) a c D t α x(t)=f(t),(3)

where g(t) and f(t) are suitable functions.

MSC:
70H30Other variational principles (mechanics of particles and systems)
26A33Fractional derivatives and integrals (real functions)
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