In this article, we introduce a fractional Hankel transform specifically adapted to a new class of generalized functions within the Colombeau algebra, called the space of tempered \( H^F \) -generalized functions, denoted by \( \mathcal {H}_{\mu ,\alpha ,\tau } \) . We rigorously establish fundamental properties of this transform, including Parseval’s identity and its interaction with the Kepinski-type operator \( \Delta _{\mu ,\alpha ,x} \) . To illustrate the practical relevance of our construction, we apply the transform to the resolution of partial differential equations involving singularities, particularly those associated with \( \Delta _{\mu ,\alpha ,x} \) . This development expands the toolkit available for addressing problems with non-smooth data in applied mathematics and engineering.
