An Introduction to Weighted Operators via Composition and Selected Properties, Aimed at Numerical Implementation

This work introduces a novel class of weighted fractional operators constructed through the composition of differential and integral operators. In particular, we propose the operator q^¯D_x^μ, which generalizes classical fractional derivatives while maintaining essential properties such as linearity. Although the semigroup property and the Leibniz rule do not hold in their traditional forms, we derive analogous formulations by combining the proposed operator with the Riemann-Liouville derivative. Furthermore, a numerical representation based on the Grünwald-Letnikov method is developed, enabling efficient discretization and simulation of the weighted operator in cases where analytical solutions are intractable. The approach also considers the interplay between Laplace transforms and convolutions, which is crucial for real-world applications in control and signal processing.