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

D-t(b)alpha(D-c(a)t(alpha))x(t) + b(t, x(t)) ((c)(a)D(t)(alpha)x(t)) + f(t, x(t)) = 0. (1)

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

D-t(b)alpha(D-c(a)t(alpha))x(t) = lambda x(t) (lambda is an element of R), (2)

D-t(b)alpha(D-c(a)t(alpha))x(t) + g(t) D-c(a)t(alpha) x(t) = f(t), (3)