Abstract
In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.
Highlights
It is worth mentioning that, since these types of fractional derivatives are generalizations of several fractional derivatives and our variational problem is a generalization of different types of fractional variational problems, many results available in the literature are corollaries of the results proven in this paper
We consider a functional depending on time, on the state function x, its α,ψ β,ψ fractional derivatives C Da+ x and C Db− x of orders α, β ∈]0, 1[, the values x ( a) and x (b), and a free parameter ζ
We proved necessary and sufficient conditions of optimality, where the Lagrangian function depends on a general form of fractional derivative, a free parameter, and the state values
Summary
2021, 5, Non-integer calculus, known as fractional calculus, deals with integrals and derivatives with arbitrary real or complex orders [1,2]. It has developed in the past decades, becoming an important tool in applied sciences and engineering. In [2], we find the concept of fractional derivative with respect to another function. We denote the fractional order by α ∈ R+ , and let ψ ∈ C1 ([ a, b], R) be a function with ψ0 (t) > 0, for all t ∈ [ a, b]. Given an integrable function x : [ a, b] → R, the left-sided and the right-sided Riemann–Liouville fractional integrals of x with kernel ψ are defined as
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