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

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