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Fractional (non-integer) derivatives and integrals play an important role in theory and applications. The fractional calculus of variations is a rather recent subject with the first results from 1996. This paper presents necessary and sufficient optimality conditions for fractional problems of the calculus of variations with a Lagrangian density depending on the free end-points. The fractional ...

Abstract This paper presents a class of nonmonotone Direct Search Methods that converge to stationary points unconstrained and boxed constrained mixed-integer optimization problems. A new concept is introduced: the quasi-descent direction. point x on set search directions if there exists no feasible qdd set. The method does not require computation derivatives nor explicit manipulation asymptoti...

Nowadays, a number of classical order results are being analyzed in the sense fractional derivatives. In this research work, we discuss two different boundary value problems. first half paper, generalize an integer-order problem into fractional-order and then demonstrate existence uniqueness solution subject to Caputo derivative. First, recall some justify our main with proofs given theorems. W...

The widely used backpropagation algorithm based on stochastic gradient descent suffers from typically slow convergence to either local or global minimum error. This backpropagation algorithm bears great resemblance to a classic proportional integral derivative (PID) control system. Fractional calculus shows promise for improving stability and response in feedback control through the use of non-...

We provide a general theorem for evaluating trigonometric Dirichlet series of the form ∑ n>1 f(πnτ) ns , where f is an arbitrary product of the elementary trigonometric functions, τ a real quadratic irrationality and s an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating th...

Abstract Starting from kicked equations of motion with derivatives of non-integer orders, we obtain ‘fractional’ discrete maps. These maps are generalizations of wellknown universal, standard, dissipative, kicked damped rotator maps. The main property of the suggested fractional maps is a long-term memory. The memory effects in the fractional discrete maps mean that their present state evolutio...

We obtain a new decomposition of the Riemann–Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class Cn, n ∈ N, and allow us to develop suitable numerical approximations with known estimations for the error. The usefulness of the obtained results, in solving fractional integral equations and fractional...

Abstract. We consider fractional order optimal control problems in which the dynamic control system involves integer and fractional order derivatives and the terminal time is free. Necessary conditions for a state/control/terminaltime triplet to be optimal are obtained. Situations with constraints present at the end time are also considered. Under appropriate assumptions, it is shown that the o...

Abstract. Differential equations involving derivatives of non-integer order have shown to be adequate models for various physical phenomena in areas like damping laws, diffusion processes, etc. A small number of algorithms for the numerical solution of these equations has been suggested, but mainly without any error estimates. In this paper, we propose an implicit algorithm for the approximate ...