Abstract
In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.
Highlights
Fractional calculus is a mathematical area that deals with the generalization of the classical notions of derivative and integral to a noninteger order
Since the beginning of the fractional calculus in 1695, numerous definitions of fractional integrals and derivatives were introduced by important mathematicians such as Leibniz, Euler, Fourier, Liouville, Riemann, Letnikov, etc
There are still few works in the literature dedicated to the fractional calculus of variations with time delay
Summary
Fractional calculus is a mathematical area that deals with the generalization of the classical notions of derivative and integral to a noninteger order. In 1969, Caputo introduced the distributed-order fractional integrals and derivatives [11,12] These operators can be seen as a new kind of generalization of the classical fractional operators, since these operators involve a weighted integral of different orders of differentiation. There are still few works in the literature dedicated to the fractional calculus of variations with time delay To fill this gap, we will study in this paper time-delayed variational problems involving distributedorder fractional derivatives with arbitrary smooth kernels. We will study variational problems involving higher-order distributed-order fractional derivatives with arbitrary smooth kernels. We finalize the paper with concluding remarks and mentioning some possibilities for future research
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