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

Axioms

Houssine Zine¹  Delfim F. M. Torres¹  El Mehdi Lotfi²  Noura Yousfi² 

[1] Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal;Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, P.O. Box 7955, Sidi Othman, Casablanca 20000, Morocco;

关键词: weighted generalized fractional calculus;    integration by parts formula;    Euler–Lagrange equation;    quantum mechanics;    calculus of variations;   

DOI  :  10.3390/axioms11040178

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown