Weighted Generalized Fractional Integration by Parts and the Euler–Lagrange Equation

Abstract

Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann–Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag–Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler–Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.

The Power Fractional Calculus: First Definitions and Properties with Applications to Power Fractional Differential Equations

“…Advanced mathematical results have recently been proved in the framework of fractional calculus: see, e.g., [7][8][9][10][11] and the references therein. However, to effectively describe realistic phenomena, all available definitions suffer from some limitations, depending on the application at hand, which has motivated us to propose here new, more general, notions, containing the key power parameter p. The currently introduced power fractional calculus enables the generalization and unification of many of the cited results, allowing engineers, researchers, and scientists to select the appropriate fractional derivative with respect to the phenomenon under study in a natural way via the presence of the parameter p in our new definitions.…”

Efficient and convenient integration by parts method ‘fixed UV, top derivative and bottom product’

“…With the new general formula, we obtain an appropriate weighted Euler-Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework[7] . For approximations in the spaceL 2 (ℝ + d ) by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm-Liouville operator, we prove the Jackson inequality with sharp constant and optimal argument in the modulus of continuity.…”