On exact solutions of a class of fractional Euler-Lagrange equations

TL;DR

This paper derives exact solutions for a class of fractional Euler-Lagrange equations using fractional variational principles, expanding understanding of fractional differential equations in variational calculus.

Contribution

It introduces a fractional Lagrangian framework for Euler-Lagrange equations and provides explicit solutions for specific fractional differential equations.

Findings

Exact solutions for fractional Euler-Lagrange equations are obtained.

A fractional variational principle leads to new classes of fractional differential equations.

The paper demonstrates solution methods for equations involving fractional derivatives.

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 $_a^cD_t^\alpha x(t))$ and $0<\alpha< 1$, such that the following is the corresponding Euler-Lagrange % \begin{equation}_tD_b^\alpha(_a^cD_t^\alpha) x(t)+ b(t,x(t))(_a^cD_t^\alpha x(t))+f(t,x(t))=0. \end{equation} % At last, exact solutions for some Euler-Lagrange equations are presented. In particular, we consider the following equations % \begin{equation}_tD_b^\alpha(_a^cD_t^\alpha x(t))=\lambda x(t), (\lambda\in R) \end{equation} % \begin{equation}_tD_b^\alpha(_a^cD_t^\alpha x(t))+g(t)_a^cD_t^\alpha x(t)=f(t), \end{equation} where g(t) and f(t) are suitable functions.